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\date{}
\markboth{H. Esnault, E. Viehweg: Lectures on Vanishing
Theorems}{}
\makeindex
\begin{document}
\setcounter{page}{-3}
\thispagestyle{empty}
\vspace*{4cm}
{\Huge\noindent\sf
H\'{e}l\`{e}ne Esnault\\[.2cm]
Eckart Viehweg\\[.2cm]
{\bf Lectures on \\[.2cm]
Vanishing Theorems}}\\[9cm]
{\large{\bf
1992}}
\newpage
\thispagestyle{empty}
\noindent
H\'el\`ene Esnault, Eckart Viehweg\\
Fachbereich 6, Mathematik \\
Universit\"at-Gesamthochschule Essen\\
D-45117 Essen, Germany\\[.3cm]
esnault@uni-essen.de\\[.15cm]
viehweg@uni-essen.de\\[11cm]
ISBN 3-7643-2822-3 (Basel)\\[.15cm]
ISBN 0-8176-2822-3 (Boston)\\[.5cm]
\small{$\bigcirc\hspace{-.24cm} c$ \ \ 1992 Birkh\"auser Verlag Basel,
P.O. Box 133, CH-4010 Basel}\\[.5cm]
We cordially thank Birkh\"auser-Verlag for their permission to
make this book available on the web. The page
layout might be slightly different from the printed version.
\normalsize
\newpage
\thispagestyle{empty}
\noindent
{\large\bf Acknowledgement}\\[1cm]
These notes grew out of the DMV-seminar on algebraic geometry
(Schlo\ss\ Reisensburg, October 13 - 19, 1991).
We thank the DMV (German Mathematical
Society) for giving us the opportunity to organize this seminar and to present
the theory of vanishing theorems to a group of younger mathematicians. We
thank all the participants for their interest, for their useful comments
and for the nice atmosphere during the seminar.\\[3cm]
\tableofcontents
\newpage
\thispagestyle{empty}
\ \
\newpage
\paragraph*{Introduction} \addcontentsline{toc}{paragraph}{Introduction}
\ \\
K. Kodaira's vanishing theorem, saying that the inverse of an
ample invertible sheaf
on a projective complex manifold $X$ has no cohomology below
the dimension of $X$
and its generalization, due to Y. Akizuki and S. Nakano, have
been proven originally
by methods from differential geometry (\cite{Kod53} and \cite{AN54}).

Even if, due to J.P. Serre's
GAGA-theorems \cite{Ser56} and base change for field extensions the
algebraic analogue was obtained for
projective manifolds over a field $k$ of characteristic $p=0$,
for a long time no algebraic
proof was known and no generalization to $p > 0$, except for
certain lower dimensional manifolds. Worse, counterexamples
due to M. Raynaud \cite{Ray78} showed that in characteristic $p > 0$ some
additional assumptions were needed.\\

This was the state of the art until P. Deligne and L. Illusie
\cite{DI87} proved the degeneration of the Hodge to de
Rham spectral sequence for projective manifolds $X$ defined over a field $k$
of characteristic $p > 0$ and liftable to the second Witt vectors $W_{2} (k)$.

Standard degeneration arguments allow to deduce the degeneration of the
Hodge to de Rham spectral sequence in characteristic zero, as well, a
result which again could only be obtained by analytic and differential
geometric methods beforehand.
As a corollary of their methods M. Raynaud (loc. cit.) gave an easy proof
of Kodaira vanishing in all characteristics, provided that $X$ lifts
to $W_{2} (k)$.\\

Short time before \cite{DI87} was written the two authors studied
in \cite{EV86} the relations
between logarithmic de Rham complexes and vanishing
theorems on complex algebraic manifolds
and showed that quite generally vanishing theorems follow from the
degeneration of certain Hodge to de Rham type spectral sequences. The interplay
between topological and algebraic vanishing theorems thereby obtained is
also reflected in J. Koll\'{a}r's work \cite{Kol89}
and in the vanishing theorems
M. Saito obtained as an application of his theory of mixed Hodge
modules (see \cite{Sai89}).

It is obvious that the combination of \cite{DI87} and \cite{EV86} give
another algebraic approach to vanishing theorems and it is one of the aims
of these lecture notes to present it in all details. Of course, after
the Deligne-Illusie-Raynaud proof of the original Kodaira and Akizuki-Nakano
vanishing theorems, the main motivation to present the methods of \cite{EV86}
along with those of \cite{DI87} is that they imply as well some of the known
generalizations.

Generalizations have been found by D. Mumford \cite{Mum67}, H. Grauert and
O. Riemenschneider \cite{GR70}, C.P. Ramanujam \cite{Ram72} (in whose
paper the method of coverings already appears), Y. Miyaoka \cite{Miy79}
(the first who works with integral parts of $\Q$ divisors, in the surface
case), by Y. Kawamata \cite{Kaw82} and the second author \cite{Vie82}. All
results mentioned replace the condition ``ample'' in
Kodaira's result by
weaker conditions. For Akizuki-Nakano type theorems
A. Sommese (see for example \cite{SS85}) got some improvement, as well as F. Bogomolov and A.
Sommese (as explained in \cite{Bog78} and \cite{SS85}) who showed
the vanishing of the global sections in certain cases.\\

Many of the applications of vanishing theorems of Kodaira type rely on
the surjectivity of the adjunction map
\[  H^{b} (X, {\cal L} \otimes \omega_{X} (B)) \>>> H^{b} (B,
{\cal L} \otimes \omega_{B}) \]
where $B$ is a divisor and ${\cal L}$ is ample or is belonging to the class
of invertible sheaves considered in the generalizations.

J. Koll\'{a}r \cite{Kol86}, building up on partial results by Tankeev, studied
the adjunction map directly and gave criteria for ${\cal L}$ and $B$
which imply the surjectivity.

This list of generalizations is probably not complete and its composition
is evidently influenced by the fact that all the results mentioned and
some slight improvements have been obtained in \cite{EV86} and \cite{EV89}
as corollaries of two vanishing theorems for sheaves of differential
forms with values in ``integral parts of $\Q$-divisors'', one for the cohomology
groups and one for restriction maps between cohomology groups.\\

In these notes we present the algebraic proof of Deligne and Illusie
\cite{DI87} for the degeneration of the Hodge to de Rham spectral sequence
(Lecture  10). Beforehand, in Lectures 8 and 9, we worked out the
properties of liftings of schemes and of the Frobenius morphism to the second
Witt vectors \cite{DI87} and the properties
of the Cartier operator \cite{Kat70} needed
in the proof. Even if some of the elegance of the original arguments
is lost thereby, we avoid using the derived category. The necessary facts
about hypercohomology and spectral sequences are shortly recalled in the
appendix, at the end of these notes.

During the first seven lectures we take the degeneration of the Hodge to
de Rham spectral sequence for granted and we develop the interplay between cyclic
coverings, logarithmic de Rham complexes and vanishing theorems (Lectures
2 - 4).\\

We try to stay as much in the algebraic language as possible.
Lectures 5 and 6 contain the geometric interpretation of the vanishing
theorems obtained, i.e. the generalizations mentioned above. Due to
the use of H. Hironaka's embedded resolution of singularities, most
of those require the assumption that the manifolds considered are defined
over a field of characteristic zero.

Raynaud's elegant proof of the Kodaira-Akizuki-Nakano vanishing theorem
is reproduced in Lecture 11, together with some generalization. However,
due to the non-availability of desingularizations in
characterisitic $p$, those generalizations seem to be useless for
applications in geometry over fields of characteristic $p>0$.

In characteristic zero the generalized vanishing theorems for integral parts
of $\Q$-divisors and J. Koll\'{a}r's vanishing for restriction maps
turned out to be powerful tools in higher dimensional algebraic geometry. Some
examples, indicating ``how to use vanishing theorems'' are contained in the
second half of Lecture 6, where we discuss higher direct images and the
interpretation of vanishing theorems on non-compact manifolds, and
in Lecture 7. Of course, this list is determined by our own taste and
restricted by our lazyness. In particular, the applications of vanishing
theorems in the birational classification theory and in the minimal
model program is left out. The reader is invited to consult the
survey's of S. Mori \cite{Mor87} and of Y. Kawamata, K. Matsuda and
M. Matsuki \cite{KMM86}.\\
\ \\
There are, of course, more subjects belonging to the circle of
ideas presented in these notes which we left aside:
\begin{itemize}
\item L. Illusie's generalizations of \cite{DI87} to variations
of Hodge structures \cite{Ill91}.
\item J.-P. Demailly's analytic approach to generalized vanishing theorems
\cite{Dem89}.
\item M. Saito's results on ``mixed Hodge modules and vanishing theorems''
\cite{Sai89}, related to J. Koll\'{a}r's program \cite{Kol89}.
\item The work of I. Reider, who used unstability of rank two
vector bundles (see \cite{Bog78}) to show that certain invertible sheaves
on surfaces are generated by global sections \cite{Rei88} (see however
(7.23)).
\item Vanishing theorems for vector bundles.
\item Generalizations of the vanishing theorems for integral parts of
$\Q$-divisors (\cite{Ara91}, \cite{AJ89}, \cite{BK90}, \cite{Mae90} and
\cite{Mae91}).
\end{itemize}
\ \\
However, we had the feeling that we could not pass by the
generic vanishing theorems of M. Green and R. Lazarsfeld \cite{GL87}.
The general picture of ``vanishing theorems'' would be incomplete without
mentioning this recent development. We include in Lectures 12 and 13
just the very first results in this direction. In particular, the
more explicit description and geometric interpretation of the ``bad locus
in $\Pic^{0} (X)$ '', contained in
A. Beauville's paper \cite{Bea88} and Green and Lazarsfeld's second paper
\cite{GL91} on this subject is missing. During the preparation of these
notes C. Simpson \cite{Sim92} found a quite complete
description of such ``degeneration loci''.\\

The first Lecture takes possible proofs of Kodaira's vanishing theorem
as a pretext to introduce some of the key words and methods, which will
reappear throughout these lecture notes and to give a more
technical introduction to its subject.\\
\ \\
Methods and results due to P. Deligne and Deligne-Illusie have
inspired and influenced our work. We cordially thank L. Illusie
for his interest and several conversations
helping us to understand \cite{DI87}.
\newpage
\Paragraph{Kodaira's vanishing theorem, a general discussion}
\ \\
Let $X$ be a projective manifold defined over an algebraically closed
field $k$ and let ${\cal L}$ be an invertible sheaf on $X$. By explicit
calculations of the \v{C}ech-cohomology of the projective space one obtains:\\
\begin{satz}{Theorem (J.~P.~Serre \cite{Ser55})}
\index{Vanishing theorem!Serre}
If ${\cal L}$ is ample and ${\cal F}$ a coherent sheaf, then there is
some $\nu_{0} \in \N$ such that
\[ H^{b} (X, {\cal F} \otimes {\cal L}^{\nu}) = 0 \ \
\mbox{for} \ \ b > 0 \ \ \mbox{and} \ \ \nu \geq \nu_{0} \]
In particular, for ${\cal F} = {\cal O}_{X}$, one obtains the vanishing
of $H^{b}(X, {\cal L}^{\nu})$ for $b > 0$ and $\nu$ sufficiently
large.\\
\end{satz}
If $\Char (k) = 0$, then ``$\nu$ sufficiently large'' can be made more precise.
For example, it is enough to choose $\nu$ such that ${\cal A} =
{\cal L}^{\nu} \otimes \omega^{-1}_{X}$ is ample, where $\omega_{X} =
\Omega^{n}_{X}$ is the canonical sheaf of $X$, and to use:\\
\begin{satz}{Theorem (K. Kodaira \cite{Kod53})}
\index{Vanishing theorem!Kodaira}
Let $X$ be a complex
projective manifold and $\cal A$ be an ample
invertible sheaf. Then
\[
\begin{array}{ll}
\mbox{a) \ \ $H^{b} (X, \omega_{X} \otimes {\cal A}) = 0$}
& \ \ \mbox{for} \ b > 0\\
\ \\
\mbox{b) \ \ $H^{b '} (X, {\cal A}^{-1}) = 0$}
& \ \ \mbox{for} \  b' < n = \dim \ X.
\end{array}
\]
\end{satz}
Of course it follows from Serre-duality that a) and b) are equivalent.
Moreover, since every algebraic variety in characteristic $0$ is defined
over a subfield of $\C$, one can use flat base change to extend (1.2) to
manifolds $X$ defined over any algebraically closed field of characteristic
zero.\\
\begin{satz}{Theorem (Y. Akizuki, S. Nakano \cite{AN54})}
\index{Vanishing theorem!Akizuki Kodaira Nakano}
Under the assumptions
made in (1.2), let $\Omega_{X}^{a}$ denote the sheaf of
$a$-differential forms. Then
\[
\begin{array}{ll}
\mbox{a) \ \ $H^{b}(X, \Omega^{a}_{X} \otimes {\cal A}) = 0$}
& \ \ \mbox{for} \ a + b > n\\
\ \\
\mbox{b) \ \ $H^{b'}(X, \Omega^{a'}_{X} \otimes {\cal A}^{-1}) = 0$}
& \ \ \mbox{for} \ a' + b' < n.
\end{array}
\]
\end{satz}

For a long time, the only proofs known for (1.2) and (1.3) used methods of
complex analytic differential geometry, until in 1986 P. Deligne and L. Illusie
found an elegant algebraic approach to prove (1.2) as well
as (1.3), using characteristic $p$ methods. About one year earlier, trying
to understand several generalizations of (1.2), the two authors obtained
(1.2) and (1.3) as a direct consequence of the decomposition of the
de Rham-cohomology $H^{k}(Y,\C )$ into a direct sum
$$\bigoplus_{b+a=k} H^{b}(Y, \Omega^{a}_{Y})$$
or, equivalently, of the degeneration of
the ``Hodge to de Rham'' spectral sequence, both applied to cyclic covers
$\pi : Y \>>> X$.

As a guide-line to the first part of our lectures, let us sketch two possible
proofs of (1.2) along this line.
\begin{Beweis}{1. Proof: With Hodge decomposition for non-compact manifolds and
topological vanishing} For sufficiently large $N$ one can find a non-singular
primedivisor $H$ such that ${\cal A}^{N} = {\cal O}_{X}(H)$.
Let $s \in H^{0}(X,  {\cal A}^{N})$ be the corresponding
section. We can regard $s$ as a rational function, if we fix some divisor
$A$ with ${\cal A} = {\cal O}_{X}(A)$ and take
$$s \in \C(X) \mbox{ \ with \ }
(s) + N \cdot  A = H.$$ The field $L = \C (X)(\sqrt[N]{s})$ depends
only on $H$. Let $\pi : Y \>>> X$ be the covering obtained
by taking the normalization of $X$ in $L$ (see (3.5) for
another construction).\\
\ \\
An easy calculation (3.13) shows that $Y$ is non-singular as well as\\
$ D = (\pi^{\ast} H)_{red}$ and that $\pi : Y \>>> X$ is unramified
outside of $D$. One has
\[ \pi^{\ast} \Omega^{a}_{X}(\log \ H) = \Omega^{a}_{Y}(\log \ D) \]
where $\Omega_{X}^{a}(\log \ H)$ denotes the sheaf of $a$-differential
forms with logarithmic poles along $H$ (see (2.1)). Moreover
\[ \pi_{*}{\cal O}_{Y} \  = \bigoplus_{i=0}^{N-1} {\cal A}^{-i} \mbox{
\ \ \ \ \ \ \ and}\]
\[ \pi_{*}\Omega^{n}_{Y}(\log \ D) \ =
\bigoplus_{i=0}^{N-1} \Omega^{n}_{X}(\log \ H)\otimes {\cal A}^{-i}
= \bigoplus_{i=0}^{N-1} \Omega^{n}_{X}\otimes {\cal A}^{N-i} \]
Deligne \cite{Del72} has shown that
\[ H^{k}(Y-D, \C) \cong \bigoplus_{b+a=k}
H^{b}(Y,\Omega^{a}_{Y}(\log \ D)). \]
Since $X - H$ is affine, the same
holds true for $Y - D$ and hence $H^{k}(Y-D, \C) = 0$ for $k > n$.
Altogether one obtains for $b > 0$
\[0 = H^{b}(Y,\Omega^{n}_{Y}(\log \ D)) =
\bigoplus_{i=0}^{N-1}H^{b}(X,\Omega^{n}_{X}\otimes {\cal A}^{N-i}). \]
\end{Beweis}
In fact, a similar argument shows as well that
$$H^{b}(X, \Omega^{a}_{X}(\log \ H) \otimes {\cal A}^{-1}) = 0$$
for $a+b > n$ . We can deduce (1.3)
from this statement by induction on $\dim \ X$ using the residue
sequence (as will be explained in (6.4)).\\
\ \\
The two ingredients of the first proof can be interpretated in a different
way. First of all, since the de Rham complex
on $Y-D$ is a resolution of the constant
sheaf one can use GAGA \cite{Ser56}
and Serre's vanishing to obtain the topological
vanishing used above. Secondly, the decomposition of the de Rham
cohomology of $Y$ into the direct sum of $(a,b)$-forms, implies
that the differential
\[ d: \Omega^{a}_{Y} \>>> \Omega^{a+1}_{Y} \]
induces the zero map
\[ d: H^{b} (Y, \Omega^{a}_{Y}) \>>> H^{b}(Y, \Omega^{a+1}_{Y}). \]
Using this one can give another proof of (1.2):
\begin{Beweis}{2. Proof: Closedness of global $(p,q)$ forms and Serre's
vanishing theorem}
Let us return to the covering $\pi: Y \rightarrow X$ constructed in the first
proof. The Galois-group $G$ of $\C (Y)$ over $\C (X)$ is cyclic of order $N$.
A generator $\sigma$ of $G$ acts on $Y$ and $D$ and hence on the
sheaves $\pi_{\ast} \Omega^{a}_{Y}$ and $\pi_{\ast} \Omega^{a}_{Y}(\log \ D)$.
Both sheaves decompose in a direct sum of sheaves of eigenvectors of $\sigma$ and,
if we choose the $N$-th root of unity carefully, the \ $i$-th summand of
\[ \pi_{\ast} \Omega^{a}_{Y}(\log \ D) = \Omega^{a}_{X}(\log \ H) \otimes
\pi_{\ast} {\cal O}_{Y} = \bigoplus^{N-1}_{i=0} \Omega^{a}_{X} (\log \ H)
\otimes {\cal A}^{-i} \]
consists of eigenvectors with eigenvalue $e^{i}$. For $e^{i} \neq 1$ the
eigenvectors of $\pi_{\ast} \Omega^{a}_{Y}$ and of $\pi_{\ast} \Omega^{a}_{Y}
(\log \ D)$ coincide, the difference of both sheaves is just living in
the invariant parts $\Omega^{a}_{X}$ and $\Omega_{X}^{a} (\log \ H)$.
Moreover, the differential
\[ d: {\cal O}_{Y} \>>> \Omega^{1}_{Y} \]
is compatible with the $G$-action and we obtain a $\C$-linear
map (in fact a connection)
\[ \nabla_{i}: {\cal A}^{-i} \>>> \Omega^{1}_{X} (\log \ H)
\otimes {\cal A}^{-i}. \]
Both properties follow from local calculations. Let us show first,
that
\[ \pi_{\ast} \Omega^{a}_{Y} = \Omega^{a}_{X} \oplus
\bigoplus^{N-1}_{i=1} \Omega^{a}_{X} (\log \ H)
\otimes {\cal A}^{-i} \ . \]
Since $H$ is non-singular one can choose local parameters $x_{1},
\ldots, x_{n}$ such that $H$ is defined by $x_{1} = 0$. Then
$$y_{1} = \sqrt[N]{x_{1}} \mbox{ \ and \ } x_{2}, \ldots , x_{n}$$
are local parameters on $Y$. The local generators
$$N \cdot  \frac{dx_{1}}{x_{1}}, dx_{2}, \ldots, dx_{n}
 \mbox{ \ of \ } \Omega^{1}_{X} (\log \ H)$$ lift to local generators
$$\frac{dy_{1}}{y_{1}}, dx_{2}, \ldots, dx_{n} \mbox{ \  of \ } \Omega^{1}_{Y}(\log
\ D).$$ The $a$-form $$\phi = s \cdot  \frac{dy_{1}}{y_{1}} \wedge dx_{2} \wedge
\ldots \wedge dx_{a}$$ (for example) is an eigenvector with eigenvalue
$e^{i}$ if and only if the same holds true for $s$, i.e. if
$s \in {\cal O}_{X} \cdot  y_{1}^{i}$.
If $\phi $ has no poles, $s$ must be divisible by $y_{1}$. This condition is
automatically satisfied as long as $i > 0$. For $i = 0$ it implies that $s$
must be divisible by $y_{1}^{N} = x_{1}$.\\
\ \\
The map $\nabla_{i}$ can be described locally as well. If
$$s = t \cdot  y^{i}_{1} \in {\cal O}_{X} \cdot  y_{1}^{i}$$
then on $Y$ one has
$$ ds = y_1^i \cdot  dt + t \cdot  dy_1^i$$
and therefore $d$ respects
the eigenspaces and $\nabla_{i}$ is given by
\[ \nabla_{i} (s) = (dt + \frac{i}{N}
\cdot  t \frac{dx_{1}}{x_{1}}) \cdot  y_{1}^{i}. \]
If $Res : \Omega^{1}_{X} (\log \ H) \>>> {\cal O}_{H}$
denotes the residue map, one obtains in addition that
\[ (Res \ \otimes id_{{\cal A}^{-1}}) \circ \nabla_{1} : {\cal A}^{-1}
\>>> {\cal O}_{H} \otimes {\cal A}^{-1} \]
is the ${\cal O}_{X}$-linear map $$s \longmapsto \frac{1}{N} s \mid_{H}.$$
Since $d: H^{b} (Y, {\cal O}_{Y}) \>>> H^{b} (Y, \Omega^{1}_{Y})$
is the zero map, the direct summand
\[ \nabla_{1}: H^{b}(X, {\cal A}^{-1}) \>>> H^{b} (X, \Omega_{X}^{1}
(\log \ D) \otimes {\cal A}^{-1})  \]
is the zero map as well as the restriction
map
\[ N \cdot (Res \ \otimes id_{{\cal A}^{-1}}) \circ \nabla_{1} :
H^{b} (X, {\cal A}^{-1}) \>>> H^{b} (H,
{\cal O}_{H}\otimes {\cal A}^{-1}). \]
Hence, for all $b$ we have a surjection
\[ H^{b}(X,{\cal A}^{-N-1}) = H^{b}(X,{\cal O}_{X}(-H)
\otimes {\cal A}^{-1}) \>>> H^{b} (X, {\cal A}^{-1}). \]
Using Serre duality and (1.1) however, $H^{b} (X, {\cal A}^{-N-1}) = 0$
for $b < n$ and $N$ sufficiently large.\\
\end{Beweis}
%
Again, the proof of (1.2) gives a little bit more:\\
\ \\
{\it If} ${\cal A}$ {\it is an invertible sheaf such that}
${\cal A}^{N} = {\cal O}_{X}(H)$ {\it for a non-singular divisor}
$H$, {\it then the restriction map}
\[ H^{b}(X, {\cal A}^{-1}) \>>> H^{b} (H,
{\cal O}_{H}\otimes {\cal A}^{-1}) \]
{\it is zero}.\\
\ \\
This statement is a special case of J. Koll\'{a}r's vanishing
theorem\linebreak[4](\cite{Kol86}, see (5.6,a)).



The main theme of the first part of these notes will be to extend the
methods sketched above to a more general situation:\\
If one allows $Y$ to be any cyclic cover of $X$ whose ramification divisor
is a normal crossing divisor, one obtains vanishing theorems for the
cohomology (or for the restriction maps in cohomology) of a larger class of
locally free sheaves. \\
Or, taking a more axiomatic point of view,
one can consider locally
free sheaves ${\cal E}$ with logarithmic connections
\[ \nabla: {\cal E} \>>> \Omega^{1}_{X} (\log \ H)
\otimes {\cal E} \]
and ask which proporties of $\nabla$ and $H$ force cohomology groups of
${\cal E}$ to vanish.\\
The resulting ``vanishing theorems for integral
parts of $\Q$-divisors'' (5.1)
and (6.2) will imply several generalizations of the Kodaira-Nakano
vanishing theorem (see Lectures 5 and 6), especially those obtained
by Mumford, Grauert and Riemenschneider, Sommese, Bogomolov, Kawamata,
Koll\'{a}r ......


However, the approach presented above is using (beside of algebraic methods)
the Hodge theory of projective manifolds, more precisely the degeneration
of the Hodge to de Rham spectral sequence
\[ E^{ab}_{1} = H^{b} (Y, \Omega^{a}_{Y}(\log \ D)) \Longrightarrow \h^{a+b}
(Y, \Omega^{\bullet}_{Y} (\log \ D)) \]
again a result which for a long time could only be deduced from
complex analytic differential geometry.\\

Both, the vanishing theorems and the degeneration of the Hodge to de Rham
spectral sequence do not hold true  for manifolds defined over a field
of characteristic $p > 0$. However, if $Y$ and $D$ both lift to the
ring of the second
Witt-vectors (especially if they can be lifted to characteristic $0$) and
if $ p \geq \dim \ X $,
P. Deligne and L. Illusie were able to prove the degeneration (see
\cite{DI87}). In fact, contrary
to characteristic zero, they show that the degeneration is induced by
some local splitting:\\
\ \\
If $F_{k}$ and $F_{Y}$ are the absolute Frobenius morphisms one obtains the
geometric Frobenius by
\[
\begin{CDS}
Y  \> {F} >> Y' = Y {\times}_{\Spec\ k} \Spec \ k  \> {\sigma} >> Y \\
\novarr \SE EE \V VV \novarr \V VV \\
\noharr  \Spec \ k \> {F_{k}} >> \Spec \ k
\end{CDS}
\]
with $F_{Y}= \sigma \circ F$. If we write $D' = (\sigma^{\ast} D)_{red}$
then, roughly speaking, they show that $F_{\ast} (\Omega^{\bullet}_{Y}
(\log \ D))$
is quasi-isomorphic to the complex
$$\bigoplus_{a} \Omega^{a}_{Y'} (\log \ D')[-a]$$
with $\Omega^{a}_{Y'}(\log \ D')$ in degree $a$ and with trivial
differentials.\\
\ \\
By base change for $\sigma$ one obtains
\[ \dim \ \h^{k}(Y, \Omega^{\bullet}_{Y}(\log \ D)) = \sum_{a+b=k} \dim \ H^{b}
(Y', \Omega^{a}_{Y'} (\log \ D')) \]
\[ = \sum_{a+b=k} \dim \ H^{b} (Y,
\Omega^{a}_{Y} (\log \ D)). \]
Base change again allows to lift this result to characteristic $0$.\\

Adding this algebraic proof, which can be found in Lectures 8 - 10,
to the proof of (1.2) and its generalizations (Lectures 2 - 6)
one obtains algebraic proofs of most of the vanishing theorems mentioned.\\
\ \\
However, based on ideas of M. Raynaud, Deligne and Illusie give in \cite{DI87}
a short and elegant argument for (1.3) in characteristic $p$ (and, by base
change, in general): \\
By Serre's vanishing theorem one has for some $m\gg 0$
\[ H^{b} (Y, \Omega^{a}_{Y} \otimes {\cal A}^{-p^{\nu}}) = 0 \ \ \
\mbox{ for \ \ \ $ \nu \geq (m+1)$} \]
and $a+b < n$, where ${\cal A}$ is ample on $Y$.
One argues by descending induction on $m$:\\
As
$${\cal A}^{p^{(m+1)}} = F^*({{\cal A}^{\prime}}^{p^m})
 \mbox{ \ for \ } {\cal A}^{\prime} = \sigma^{\ast}{\cal A}$$
and as $\Omega_{Y}^{\bullet}$ is a ${\cal O}_{Y'}$ complex,
$\Omega_{Y}^{\bullet} \otimes {\cal A}^{-p^{(m+1)}}$ is a complex
 of ${\cal O}_{Y'}$ sheaves with
\[ \h^{k} (Y, \Omega^{\bullet}_{Y} \otimes {\cal A}^{-p^{m+1}}) = 0 \ \ \
\mbox{ for \ \ \ $k < n$}.\]
However one has
\[ F_{\ast} (\Omega^{\bullet}_{Y} \otimes {\cal A}^{-p^{m+1}}) = \bigoplus_{a}
\Omega^{a}_{Y'} \otimes {{\cal A}^{\prime}}^{-p^m}[-a] \]
and
\[ 0 = H^{b} (Y', \Omega^{a}_{Y'} \otimes {{\cal A}^{\prime}}^{-p^m}) =
H^{b} (Y, \Omega^{a}_{Y} \otimes {\cal A}^{-p^m}) \]
for $a+b < n$.

Unfortunately this type of argument does not allow to weaken the
assumptions made in (1.2) or (1.3). In order to deduce the generalized
vanishing theorems from the degeneration of the Hodge to de Rham spectral
sequence in characteristic $0$ we have to use H.~Hironaka's theory
of embedded resolution of singularities, at present a serious obstruction
for carrying over arguments from characteristic $0$ to characteristic $p$.
Even the Grauert-Riemenschneider vanishing theorem (replace ``ample'' in
(1.2) by ``semi-ample of maximal Iitaka dimension'') has no known analogue
in characteristic $p$ (see \S 11).\\

M. Green and R. Lazarsfeld observed, that ``ample'' in (1.2) can sometimes be
replaced by ``numerically trivial and sufficiently general''.
To be more precise, they showed that
$H^{b}(X, {\cal N}^{-1}) = 0$ for a general element ${\cal N}
\in Pic^{0} (X)$ if $b$ is smaller than the dimension of the image
of $X$ under its Albanese map $$\alpha : X \>>> \ \mbox{Alb} (X).$$
By Hodge-duality (for Hodge theory with
values in unitary rank one bundles) $H^{b}(X, {\cal N}^{-1})$ can be
identified with $H^{0} (X, \Omega^{b}_{X} \otimes {\cal N})$. If
$H^{b}(X, {\cal N}^{-1}) \neq 0$ for all
${\cal N} \in Pic^{0}(X)$ the deformation theory for cohomology groups,
developed by Green and Lazarsfeld, implies that for
all $\omega \in H^{0} (X, \Omega^{1}_{X})$
the wedge product
\[ H^{0}(X, \Omega^{b}_{X} \otimes {\cal N}) \>>>
H^{0} (X, \Omega^{b+1}_{X} \otimes {\cal N}) \]
is non-trivial. This however implies that the image of $X$ under the Albanese
map, or equivalently the subsheaf of $\Omega^{1}_{X}$ generated by
global sections is small.
\\
For example, if
\[ S^{b} (X) = \{ {\cal N} \in Pic^{0} (X); \ H^{b} (X, {\cal N}^{-1})
\neq 0 \} , \]
then the first result of Green and Lazarsfeld says that
\[ \codim_{Pic^{0}(X)} (S^{b} (X)) \geq \dim (\alpha (X)) - b. \]
It is only this part of their results we include in these notes, together
with some straightforward generalizations due to H. Dunio \cite{Dun91}
(see Lectures 12 and 13). The more detailed description of $S^{b}(X)$,
due to Beauville \cite{Bea88}, Green-Lazarsfeld \cite{GL91} and C. Simpson
\cite{Sim92} is just mentioned, without proof, at the end of Lecture 13.
\ \\
\newpage
\Paragraph{Logarithmic de Rham complexes}
\ \\
In this lecture we want to start with the definition and simple
properties of the sheaf of (algebraic) logarithmic differential
forms and of sheaves with logarithmic integrable connections, developed
in \cite{Del70}.
The main examples of those will arise from cyclic covers (see Lecture 3).
Even if we stay in the algebraic language, the reader is invited (see
2.11) to compare the statements and constructions with the analytic case.\\

Throughout this lecture $X$ will be an algebraic manifold, defined over an
algebraically closed field $k$, and $D = \sum_{j=1}^{r} D_{j}$ a reduced normal
crossing divisor, i.e. a divisor with non-singular components $D_j$
intersecting each other transversally.\\
\ \\
We write $\tau : U = X - D \>>> X$ and
\[ \Omega^{a}_{X}(\ast D) = \lim_{\scriptstyle \>>> \atop \scriptstyle \nu}
\Omega^{a}_{X} (\nu \cdot  D) = \tau_{\ast} \Omega^{a}_{U}. \]
Of course \index{$\Omega^{a}_{X}(\ast D)$}
$(\Omega^{\bullet}_{X} ( \ast D), d)$ is a complex.\\
\begin{defn}{Definition} $\Omega^{a}_{X} (\log \ D)$ denotes the subsheaf
of $\Omega^{a}_{X}(\ast D)$ of {\it differential forms with
logarithmic poles} along $D$, i.e.: if $V \subseteq X$ is open, then
\[\index{$\Omega^{a}_{X} (\log \ D)$}\index{Differential forms!logarithmic}
\begin{array}{ll}
\Gamma (V, \Omega^{a}_{X} (\log \ D)) = \\
\ \\
\{ \ \alpha \in \Gamma (V, \Omega^{a}_{X} (\ast D)); \ \alpha \mbox{ and }
d\alpha \mbox{ have simple poles along  } D \}.
\end{array}
\]
\end{defn}
\begin{defn}{Properties} \\
a) $$(\Omega^{\bullet}_{X}(\log \ D), d) \hookrightarrow
(\Omega^{\bullet}_{X}(*D), d).$$ is a subcomplex. \\
b) $$\Omega^{a}_{X} (\log \ D) = \bigwedge^{a} \Omega^{1}_{X}
(\log \ D)$$
c) $\Omega^{a}_{X} (\log \ D)$ is locally free. More
precisely:\\
For $p \in X$, let us say with $p \in D_{j}$ for $j =1, \ldots, s$
and $p \not\in D_{j}$ for $j = s+1, \ldots, r$, choose local
parameters $f_{1},\ldots, f_{n}$ in $p$ such that $D_{j}$ is defined
by $f_{j} = 0$ for $j= 1, \ldots, s$.
Let us write \[
\delta_{j} =
\left\{
\begin{array}{ll}
\frac{df_{j}}{f_{j}} & \mbox{if \ \ \ $j \leq s$}\\
df_{j} & \mbox{if \ \ \ $j > s$}\\
\end{array}
\right. \]
and for $I = \ \{ j_{1}, \ldots, j_{a} \} \ \subset \ \{ 1,\ldots, n \}$
with $j_{1} < j_{2} \ldots < j_{a}$
\[ \delta_{I} = \delta_{j_{1}} \wedge \ldots \wedge \delta_{j_{a}}.\]
Then $ \{ \delta_{I}; \ \sharp I = a \}$ is a free system of generators
for $\Omega^{a}_{X} (\log \ D)$.
\end{defn}
\begin{Beweis}{Proof} (see \cite{Del70}, II, 3.1 - 3.7).
a) is obvious and b) follows from the explicite form of the generators
given in c).\\
Since $\delta_{j}$ is closed, $\delta_{I}$ is a local section
of $\Omega_{X}^{a} (\log \ D)$. By the Leibniz rule
the ${\cal O}_{X}$-module $\Omega$ spanned by the $\delta_{I}$ is
contained in $\Omega_{X}^{a}(\log \ D)$. $\Omega$ is locally free and,
in order to show that $\Omega = \Omega^{a}_{X}(\log \ D)$ it is enough
to consider the case $s = 1$.
Each local section $\alpha \in \Omega^{a}_{X} (\ast D)$ can be
written as
$$\alpha = \alpha_{1} +
\alpha_{2} \wedge \frac{df_{1}}{f_{1}},$$ where
$\alpha_{1}$ and $\alpha_{2}$ lie in $\Omega_{X}^{a}(\ast D)$ and
$\Omega^{a-1}_{X} (\ast D)$ and where both are in the subsheaves generated
over ${\cal O}(\ast D)$ by wedge products of $df_{2}, \ldots, df_{n}$.\\
$\alpha \in \Omega_{X}^{a} (\log \ D)$ implies that
\[
f_{1} \cdot  \alpha = f_{1} \cdot  \alpha_{1} + \alpha_{2} \wedge df_{1} \ \in
\Omega^{a}_{X} \mbox{ \ and \ }
f_{1}d\alpha = f_{1}d\alpha_{1} + d\alpha_{2} \wedge df_{1} \in
\Omega_{X}^{a+1}.
\]
Hence $\alpha_{2}$ as well as $f_{1}\alpha_{1}$ are without poles. Since
\[ d(f_{1}\alpha_{1}) = df_{1} \wedge \alpha_{1} + f_{1} d\alpha_{1} =
df_{1} \wedge \alpha_{1} + f_{1} d\alpha - d\alpha_{2} \wedge df_{1} \]
the form $df_{1} \wedge \alpha_{1}$ has no poles which implies $\alpha_{1}
\in \Omega^{a}_{X}$.\\
\end{Beweis}
%
Using the notation from (2.2,c) we define
\[ \alpha : \Omega^{1}_{X} (\log \ D) \>>> \bigoplus^{s}_{j=1}
{\cal O}_{D_{j}} \]
by \[ \alpha (\sum_{j=1}^{n} a_{j} \delta_{j}) = \bigoplus_{j=1}^{s}
a_{j}\vert_{D_{j}}. \]
For $a \geq 1$ we have correspondingly a map
\[ \beta_{1} : \Omega^{a}_{X} (\log \ D) \>>> \Omega_{D_{1}}^{a-1}
(\log \ (D-D_{1})\vert_{D_{1}}) \]
given by:\\
If $\varphi$ is a local section of $\Omega^{a}_{X} (\log \ D)$,
we can write $$\varphi = \varphi_{1} + \varphi_{2} \wedge \frac{df_{1}}{f_{1}}$$
where $\varphi_{1}$ lies in the span of the $\delta_{I}$ with $1 \not\in I$
and
\[ \varphi_{2} = \sum_{1 \in I}a_{I} \delta_{I-\{ 1\} }. \]
Then
\[ \beta_{1}(\varphi) = \beta_{1}(\varphi_{2} \wedge \frac{df_{1}}{f_{1}}) =
\sum a_{I} \delta_{I- \{ 1 \} }\vert_{D_{1}}. \]
Of course, $\beta_i$ will denote the corresponding map for the $i$-th
component. Finally, the natural restriction of differential forms gives
\[ \gamma_{1} : \Omega^{a}_{X} (\log \ (D - D_{1})) \>>> \Omega^{a}_{D_{1}}
(\log \ (D-D_{1})\vert_{D_{1}}). \]
Since the sheaf on the left hand side is generated by
$$\{  f_{1} \cdot
\delta_{I}; \ 1 \in I \} \ \cup \ \{\delta_{I}; \ 1 \not\in I \}$$
we can describe $\gamma_{1}$ by
\[ \gamma_{1} ( \sum_{1 \in I} f_{1} a_{I} \delta_{I} + \sum_{1 \not\in I}
a_{I} \delta_{I}) = \sum_{1 \not\in I} a_{I} \delta_{I}\vert_{D_{1}}.\]
Obviously one has
\begin{defn}{Properties}\index{Differential forms!logarithmic!exact
sequences}
One has three exact sequences:\\
a) $$0 \rightarrow \Omega^{1}_{X} \>>> \Omega^{1}_{X} (\log \ D)
\> \alpha >> \bigoplus_{j=1}^{r} {\cal O}_{D_{j}} \rightarrow 0.$$
b) $$ 0 \rightarrow \Omega^{a}_{X} (\log \ (D-D_{1})) \>>>
\Omega^{a}_{X}(\log \ D) \> \beta_{1} >> \Omega^{a-1}_{D_{1}}(\log \ (D-D_{1})
\vert_{D_{1}}) \rightarrow 0.$$
c) $$ 0 \rightarrow \Omega^{a}_{X} (\log \ D)(-D_{1}) \>>>
\Omega^{a}_{X}(\log \ (D-D_{1})) \> \gamma_{1} >> \Omega^{a}_{D_{1}}
(\log \ (D-D_{1})\vert_{D_{1}}) \rightarrow 0.$$
\end{defn}
By (2.2,b) $(\Omega^{\bullet}_{X}(\log \ D), d)$ is a complex.
It is the most simple example
of a logarithmic de Rham complex.
%
\begin{defn}{Definition} Let ${\cal E}$ be a locally free coherent sheaf
on $X$ and let
\[ \nabla: {\cal E} \>>> \Omega^{1}_{X} (\log \ D) \otimes {\cal E} \]
be a $k$-linear map satisfying
\[ \nabla (f \cdot  e) = f \cdot  \nabla (e) + df
\otimes e. \]
One defines
\[ \nabla_{a}: \Omega^{a}_{X}(\log \ D) \otimes
{\cal E} \>>> \Omega^{a+1}_{X} (\log \ D) \otimes {\cal E} \]
by the rule
\[ \nabla_{a} (\omega \otimes  e) = d\omega \otimes e + (-1)^{a}
\cdot  \omega \wedge \nabla (e). \]
We assume that $\nabla_{a+1} \circ \nabla_{a}
=0$ for all $a$.
Such $\nabla$ will be called an {\it integrable logarithmic connection
along} $D$, or just a connection. The complex\index{Connection!logarithmic}
$$(\Omega^{\bullet}_{X} (\log \ D)
\otimes {\cal E}, \nabla_{\bullet})$$ is called the {\it logarithmic
de Rham complex} of $({\cal E}, \nabla)$.\index{de Rham complex!logarithmic}
\end{defn}
\begin{defn}{Definition} For an integrable logarithmic connection
\[ \nabla : {\cal E} \>>> \Omega^{1}_{X} (\log \ D) \otimes
{\cal E} \]
we define the residue map along $D_{1}$ to be the composed map
\[ Res_{D_{1}} (\nabla): {\cal E} \> \nabla >> \Omega^{1}_{X} (\log \ D)
\otimes {\cal E} \> \beta'_{1} = \beta_1 \otimes id_{\cal E}
>> {\cal O}_{D_{1}} \otimes {\cal E}. \]
\index{Residue map}\index{$Res_{D} (\nabla)$}
\end{defn}
%
\begin{satz}{Lemma}\\
a) $Res_{D_{1}} (\nabla)$ is $ {\cal O}_{X}$-linear
and it factors through
\[ {\cal E} \> restr. >> {\cal O}_{D_{1}} \otimes {\cal E} \>>>
{\cal O}_{D_{1}} \otimes {\cal E} \]
where restr. the restriction of ${\cal E}$
to $D_{1}$. By abuse of notations we will call
the second map $Res_{D_{1}} (\nabla)$ again.\\
b) One has a commutative diagram
\[
\begin{CD}
\Omega^{a}_{X} (\log \ (D-D_{1})) \otimes {\cal E} \> (\nabla_{a}) \circ (incl.)>>
\Omega_{X}^{a+1} (\log \ D) \otimes {\cal E}\\
\V V \gamma_{1} \otimes id_{\cal E} V
\V V \beta_{1} \otimes id_{\cal E} = \beta^{\prime}_{1} V \\
\Omega^{a}_{D_{1}} (\log \ (D-D_{1})\mid_{D_{1}}) \otimes {\cal E}
\> ((-1)^{a} \cdot  id) \otimes Res_{D_{1}}(\nabla) >>
\Omega^{a}_{D_{1}} (\log \ (D-D_{1})\mid_{D_{1}}) \otimes {\cal E}
\end{CD}
\]
\end{satz}
\begin{Beweis}{Proof} a) We have $$\nabla (g \cdot  e) = g \cdot  \nabla (e)
+ dg \otimes e \mbox{ \ \ \  and \ \ \ }
\beta^{\prime}_{1} (\nabla (g \cdot  e)) = g \cdot  \beta^{\prime}_{1} (\nabla (e)) .$$
If $f_{1}$ divides $g$ then $g \cdot  \beta'_{1}(\nabla (e)) = 0$.\\
b) For $\omega \in \Omega_{X}^{a} (\log \ (D-D_{1}))$ and
$e \in \cal E$ we have
\[
\begin{array}{lcl}
\beta^{\prime}_{1} (\nabla_{a} (\omega \otimes e)) & = &
\beta^{\prime}_{1} (d \omega \otimes
e + (-1)^{a} \cdot  \omega \wedge \nabla (e))\\
\ & \ & \\
& = & \beta'_{1} ((-1)^{a} \cdot  \omega \wedge \nabla (e)).
\end{array}
\]
If $\omega = f_{1} \cdot  a_{I} \cdot  \delta_{I}$ \ for $1 \in I$, then
\[ (-1)^{a} \omega \wedge \nabla (e)
\in \Omega^{a+1}_{X} (\log \ D) (-D_{1}) \]
and $\beta'_{1} (\nabla_{a} ( \omega \otimes \ e)) = 0$. On the other hand,
$\gamma_{1} (\omega) \otimes e = 0$ by definition.\\
If $\omega = a_{I} \delta_{I}$ for $ 1 \not\in I$, then
\[ \gamma_{1} (\omega) \otimes e = a_{I} \cdot
\delta_{I}\mid_{D_{1}} \otimes e \]
and
\[ \beta^{\prime}_{1}((-1)^{a} \omega \wedge \nabla (e))
= (-1)^{a} \omega|_{D_1} \otimes \ Res_{D_{1}}(\nabla)(e). \]
\end{Beweis}
\begin{satz}{Lemma}
Let $$B = \sum_{j=1}^{r} \mu_{j} D_{i}$$ be
any divisor and $(\nabla, {\cal E})$ as in (2.4).
Then $\nabla$ induces a connection $\nabla^{B}$
with logarithmic poles on $${\cal E} \otimes {\cal O}_{X}
(B) = {\cal E} (B)$$ and the residues satisfy
$$Res_{D_{j}} (\nabla^{B}) = Res_{D_{j}}(\nabla) - \mu_{j} \cdot  id_{D_{j}}.$$
\end{satz}
\begin{Beweis}{Proof} A local section of ${\cal E}(B)$ is of the form
$$ \sigma = \prod_{j=1}^{s} f^{-\mu_{j}}_{j} \cdot  e$$
and
\[ \nabla^{B}(\sigma) = \prod_{j=1}^{s} f^{-\mu_{j}}_{j} \nabla(e) +
d(\prod_{j=1}^{s} f^{-\mu_{j}}_{j}) \otimes e =\]
\[ = \prod^{s}_{j=1} f^{-\mu_{j}}_{j} \nabla (e) + \sum_{k=1}^{s}
(\prod^{s}_{j=1} f_{j}^{-\mu_{j}}) \cdot  (-\mu_{k}) \frac{df_{k}}{f_{k}}
\otimes e. \]
Hence $\nabla^{B} : {\cal E}(B) \>>>
\Omega^{1}_{X}(\log \ D) \otimes {\cal E} (B)$ is well defined.
One obtains
\[ Res_{D_{1}} (\nabla^{B}(\sigma)) = \prod^{s}_{j=1} f^{-\mu_{j}}_{j} \
Res_{D_{1}} (\nabla (e)) + \prod^{s}_{j=1} f_{j}^{-\mu_{j}} (- \mu_{1})
\otimes e\mid_{D_{1}}.\]
\end{Beweis}
%
\begin{defn}{Definition} a) We say that $(\nabla, {\cal E})$ satisfies
the condition $(\ast)$ if for all divisors\index{Condition $(\ast)$}
$$B = \sum_{j=1}^{r} \mu_{j}D_{j} \geq D $$
and all $ j=1 \ldots r$ one has an isomorphism of sheaves
\[ Res_{D_{j}} (\nabla^{B})= Res_{D_{j}} (\nabla) - \mu_{j} \cdot  id_{D_{j}} :
{\cal E}\mid_{D_{j}} \>>> {\cal E}\mid_{D_{j}}. \] b) We say that
$(\nabla, {\cal E})$ satisfies the condition (!) if for all
divisors\index{Condition {\test}}
\[ B = \sum_{j=1}^{r} - \nu_{j}D_{j} \leq 0 \]
and all $j=1, \ldots , r$
\[ Res_{D_{j}}(\nabla^{B}) = Res_{D_{j}} (\nabla) + \nu_{j} \cdot  \
id_{D_{j}} : {\cal E}\mid_{D_{j}} \>>> {\cal E}\mid_{D_{j}} \]
is an isomorphism of sheaves.\\
\ \\
In other words, $(\ast)$ means that no $\mu_{j} \in \Z$, $ \mu_{j} \geq 1$,
is an eigenvalue of $Res_{D_{j}}(\nabla)$ and (!) means the same for
$\mu_{j} \in \Z$, $ \mu_{j} \leq 0$. We will see later, that $(\ast)$
and (!) are only of interest if $\Char \ (k) = 0$.
\end{defn}
%
\begin{defn}{Properties} \\
a)  {Assume that} $({\cal E}, \nabla)$
 {satisfies} $(\ast)$  {and that} $B = \sum  \mu_{j}
D_{j} \geq 0$.  {Then the
natural map }
\[ (\Omega^{\bullet}_{X} (\log \ D) \otimes {\cal E}, \nabla_{\bullet})
\>>>
(\Omega^{\bullet}_{X}(\log \ D) \otimes {\cal E} (B),
\nabla_{\bullet}^{B}) \]
between the logarithmic de Rham complexes is a
 quasi-isomorphism.\\
b) Assume that $({\cal E}, \nabla)$ satisfies
(!) and that $B = \sum - \mu_{j} D_{j} \leq 0$.
Then the natural map
\[ (\Omega^{\bullet}_{X} (\log \ D) \otimes {\cal E}(B), \nabla_{\bullet}^{B})
\>>>
(\Omega^{\bullet}_{X} (\log \ D) \otimes {\cal E} ,
\nabla_{\bullet}) \]
between the logarithmic de Rham complexes
is a  quasi-isomorphism.\\
\end{defn}
%
(2.9) follows from the definition of $(\ast)$ and (!) and from:\\
%
\begin{satz}{Lemma}
 {For} $({\cal E}, \nabla)$  {as in
(2.4) assume that}
\[ Res_{D_{1}} (\nabla): {\cal E}\mid_{D_{1}} \>>>
{\cal E} \mid_{D_{1}} \]
is an isomorphism. Then the inclusion of complexes
\[ (\Omega^{\bullet}_{X} (\log \ D) \otimes {\cal E} (-D_{1}),
\nabla_{\bullet}^{-D_{1}}) \>>>
(\Omega^{\bullet}_{X} (\log \ D) \otimes {\cal E}, \nabla_\bullet) \]
is an quasi-isomorphism.
\end{satz}
%
\begin{Beweis}{Proof} Consider the complexes ${\cal E}^{(\nu)}$:
\[
{\cal E}(-D_{1}) \>>> \Omega^{1}_{X} (\log \ D) \otimes
{\cal E} (-D_{1}) \>>> \ldots \>>> \Omega^{\nu - 1}_{X}
(\log \ D) \otimes {\cal E} (-D_{1}) \>>> \]
\[ \>>> \Omega^{\nu}_{X} (\log \ (D-D_{1})) \otimes {\cal E}
\>>> \Omega^{\nu + 1}_{X} (\log \ D) \otimes {\cal E} \>>>
\ldots \>>> \Omega^{n}_{X}(\log \ D) \otimes
{\cal E} \]
We have an inclusion
\[ {\cal E}^{(\nu + 1)} \>>> {\cal E}^{(\nu)} \]
and, by (2.6,b) the quotient is the complex
\[ 0 \>>> \Omega^{\nu}_{D_{1}}(\log \ (D - D_{1})|_{D_{1}}) \otimes
{\cal E} \> (-1)^{\nu} \otimes Res_{D_{1}} (\nabla) >>
\Omega^{\nu}_{D_{1}} (\log \ (D-D_{1})|_{D_{1}}) \otimes {\cal E}
\>>> 0 \]
Since the quotient has no cohomology all the ${\cal E}^{(\nu)}$ are
quasi-isomorphic, especially ${\cal E}^{(0)}$ and ${\cal E}^{(n)}$,
as claimed.\\
\end{Beweis}
\small
\Abschnitt{The analytic case}
At this point it might be helpful to consider the analytic case for a moment:
 ${\cal E}$ is a locally free sheaf over the sheaf of analytic functions
${\cal O}_{X}$,
\[ \nabla : {\cal E} \>>> \Omega^{1}_{X}
(\log \ D) \otimes {\cal E} \]
is a holomorphic and integrable connection. Then
$\ker (\nabla \mid_{U}) = V$ is a
local constant system. If $(\ast)$ holds
true, i.e. if the residues of $\nabla$ along the $D_{j}$ do not have strictly
positive integers as eigenvalues, then (see \cite{Del70}, II, 3.13 and 3.14)
$$(\Omega^{\bullet}_{X} (\log \ D) \otimes {\cal E}, \nabla_{\bullet})$$
is quasi-isomorphic to $R{\tau_{\ast}} V$. By Poincar\'{e}-Verdier
duality (see \cite{EV86}, Appendix A) the natural map
\[ \tau_{!} V^{\vee} \>>> (\Omega^{\bullet}_{X}(\log \ D) \otimes
{\cal E}^{\vee} (-D), \nabla^{\vee}_{\bullet}) \]
is a quasi-isomorphism.
Hence (!) implies that
the natural map
\[
\tau_{!}V
\>>>
(\Omega^{\bullet}_{X} (\log \ D) \otimes {\cal E}, \ \nabla_{\bullet})
 \]
is a quasi-isomorphism as well.
In particular, topological properties of $U$ give vanishing theorems for
\[ \h^{l} (X, \Omega^{\bullet}_{X} (\log \ D) \otimes {\cal E})  \]
and for some $l$.\index{Cohomological dimension!$r(U)$}  \index{$r(U)$}
More precisely, if we choose $r(U)$ to be the smallest number that satisfies:\\
\begin{quote}
For all local constant systems $V$ on $U$
one has $H^{l}(U,V) = 0$ for $l > n + r(U)$,
\end{quote}
then one gets:
\begin{satz}{Corollary}\\ \index{Vanishing theorem!for local systems}
a) If $({\cal E}, \nabla)$ satisfies $(\ast)$,
then for $l > n + r(U)$
\[ \h^{l} (X, \Omega^{\bullet}_{X}(\log \ D) \otimes {\cal E})
= H^{l} (U,V) = 0 .\]
b)  {If} $({\cal E}, \nabla)$ satisfies $(!)$,
then for $l < n - r (U)$
\[ \h^{l}(X, \Omega^{\bullet}_{X} (\log \ D) \otimes {\cal E})
= H^{l}_{c}(U,V) = 0. \]
\end{satz}
By GAGA (see \cite{Ser56}), (2.12) remains true if we consider
the complex of algebraic differential forms over the complex
projective manifold $X$, even if the number $r(U)$ is
defined in the analytic topology.\\
\ \\
(2.12) is of special interest if
both, $(\ast)$ and $(!)$, are satisfied,  {i.e.} if none of the
eigenvalues of $Res_{D_{j}} (\nabla)$ is an integer. Examples
 of such connections
can be obtained, analytically or algebraically, by cyclic covers.\\
\ \\
If $U$ is affine (or a Stein manifold) one has $r(U) = 0$. For $U$ affine
there is no need to use GAGA and analytic arguments. Considering blowing ups
and the Leray spectral sequence one can obtain (2.12) for algebraic
sheaves from:\\
\normalsize
\begin{satz}{Corollary} Let $X$ be a projective manifold defined over
the algebraically closed field $k$. Let $B$ be an effective ample divisor,
$D = B_{red}$ a normal crossing divisor and $({\cal E}, \nabla)$ a logarithmic
connection with poles along $D$ (as in (2.4)).\\
{\rm a)} If $({\cal E}, \nabla)$ satisfies $(*)$, then for $l > n$
\[ \h^{l} (X, \Omega^{\bullet}_{X} (\log \ D) \otimes {\cal E}) = 0. \]
{\rm b)} If $({\cal E}, \nabla)$ satisfies (!), then for $l < n$
\[ \h^{l} (X, \Omega^{\bullet}_{X} (\log \ D) \otimes {\cal E}) = 0. \]
\end{satz}
\begin{Beweis}{Proof} (2.9) allows to replace ${\cal E}$ by ${\cal E}
(N \cdot  B)$ in case a) or by ${\cal E} (-N \cdot  B)$ in case b) for
$N > 0$. By Serre's vanishing theorem (1.1) we can assume that
\[ H^{b} (X, \Omega^{a}_{X} (\log \ D) \otimes {\cal E}) = 0 \]
for $a + b = l$. The Hodge to de Rham spectral sequence (see (A.25))
implies (2.13).\\
\end{Beweis}
\Paragraph{Integral parts of $\Q$-divisors and coverings}
\ \\
Over complex manifolds the Riemann Hilbert correspondence obtained by
Deligne \cite{Del70} is an equivalence between logarithmic connections
$({\cal E}, \nabla)$ and representations of the fundamental group $\pi_{1}
(X - D)$. For applications in algebraic geometry the most simple
representations, i.e. those who factor through cyclic quotient
groups of $\pi_{1}(X - D)$, turn out to be useful. The induced invertible
sheaves and connections can be constructed directly as summands of the
structure sheaves of cyclic coverings. Those constructions remain valid
for all algebraically closed fields.

Let $X$ be an algebraic manifold defined over the algebraically
closed field $k$.

\begin{defn}{Notation} a) Let us write $\Div(X)$ for the group of divisors on $X$ and
\[ \Div_{\Q} (X) = \Div (X) \otimes_{\Z} \Q .\]
Hence a {\it $\Q$-divisor} $\Delta \in \Div_{\Q}(X)$ is a sum $$\Delta
= \sum_{j=1}^{r} \alpha_{j} D_{j}$$ of irreducible
prime divisors $D_{j}$ with\index{$\Q$-divisors}
coefficients $\alpha_{j} \in \Q$.\\
b) For $\Delta \in \Div_{\Q} (X)$ we write\index{Integral part of
a $\Q$-divisor}\index{$[\Delta]$}
$$ [\Delta ] = \sum_{j=1}^{r} [ \alpha_{j} ] \cdot  D_{j}$$ where for
$\alpha \in \Q$, $[\alpha ]$ denotes the integral part
of $\alpha$, defined as the only integer such that
\[ [ \alpha ] \leq \alpha < [ \alpha ] + 1. \]
$[\Delta ]$ will be called the  \ integral part of $\Delta$.\\
c) For an invertible sheaf ${\cal L}$, an effective divisor
$$D = \sum_{j=1}^{r} \alpha_{j}D{j}$$
and a positive natural number $N$, assume
\index{${\cal L}^{(i)}$}\index{${\cal L}^{(i,D)}$}
that ${\cal L}^{N} = {\cal O}_{X} (D)$. Then we will write for $i \in \N$
\[ {\cal L}^{(i, D)} = {\cal L}^{i} (- [\frac{i}{N} D]) = {\cal L}^{i} \otimes
{\cal O}_{X} (- [ \frac{i}{N} \cdot  D ]). \]
Usually $N$ and $D$ will be fixed and we just write ${\cal L}^{(i)}$ instead
of ${\cal L}^{(i,D)}$.\\
d) If $$D = \sum_{j=1}^{r} \alpha_{j} D_{j}$$ is a normal crossing
divisor, we will write,
for simplictiy, $$\Omega_{X}^{a} (\log \ D) \mbox{ \ instead of \ }
\Omega^{a}_{X}(\log \ D_{red}).$$
In spite of their strange definition the sheaves
${\cal L}^{(i)}$ will turn out to be
related to cyclic covers in a quite natural way. We will need this to
prove:
\end{defn}
\begin{satz}{Theorem} \index{de Rham complex!logarithmic!$E_1$ degeneration}
Let $X$ be a projective manifold,
$$ D = \sum^{r}_{j=1} \alpha_{j} D_{j}$$
be an effective normal crossing divisor,
${\cal L}$ an invertible sheaf and $N \in \N - \{ 0 \}$
prime to $ \Char  (k)$,  such that
$ {\cal L}^{N} = {\cal O}_{X} (D)$. Then
for $i = 0, \ldots, N-1$  \ the sheaf ${\cal L}^{(i)^{-1}}$
has an integrable logarithmic connection
\[ \nabla^{(i)} : {\cal L}^{(i)^{-1}} \>>>
\Omega^{1}_{X} (\log \ D^{(i)}) \otimes {\cal L}^{(i)^{-1}}\]
\[
\mbox{with poles along \ \ \ }
D^{(i)} = \sum^{r}_{\scriptstyle j=1 \atop \scriptstyle \frac{i \cdot
\alpha_{j}}{N} \not\in \Z} D_{j}, \]
satisfying:\\
{\rm a)} The residue of $\nabla^{(i)}$ along $D_{j}$
is given by  multiplication with
\[ (i \cdot  \alpha_{j} - N \cdot  [\frac{i \cdot  \alpha_{j}}{N} ]) \cdot
N^{-1} \ \in k. \]
{\rm b)} Assume that either $\Char  (k) = 0$,  \ or, if
$\Char  (k) = p \neq 0$, that $X$ and $D$ admit a lifting to
$W_{2} (k)$ (see (8.11)) and that $p \geq \dim \ X $. Then
the spectral sequence
$$E_{1}^{ab} = H^{b} (X, \Omega^{a}_{X}(\log \
D^{(i)}) \otimes {\cal L}^{(i)^{-1}}) \Longrightarrow
\h^{a+b}(X,
\Omega^{\bullet}_{X} (\log \ D^{(i)})\otimes {\cal L}^{(i)^{-1}})$$
associated to the logarithmic de Rham
complex
\[ (\Omega^{\bullet}_{X} (\log \ D^{(i)})\otimes {\cal L}^{(i)^{-1}},
\nabla_{\bullet}^{(i)}) \]
degenerates in $E_{1}$.\\
c) Let $A$  and $B$  be reduced divisors
(both having the lifting property (8.11) if $\Char  (k) = p \neq 0$)
such that $B,A$ and $D^{(i)}$ have
pairwise no commom components and such that $A + B + D^{(i)}$
is a normal crossing divisor. Then $\nabla^{(i)}$
induces a logarithmic connection
\[ {\cal O}_{X} (-B)\otimes {\cal L}^{(i)^{-1}} \>>>
\Omega^{1}_{X} (\log \ (A+B+D^{(i)})) (-B)\otimes {\cal L}^{(i)^{-1}} \]
and under the assumptions of b) the spectral sequence
\[ E_{1}^{ab} = H^{b} (X, \Omega^{a}_{X}
(\log \ (A+B+D^{(i)}))(-B)\otimes {\cal L}^{(i)^{-1}}) \Longrightarrow
\ \ \ \ \ \ \ \ \ \ \]
\[ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \h^{a+b} (X,
\Omega^{\bullet}_{X} (\log \ (A+B+D^{(i)}))
(-B)\otimes {\cal L}^{(i)^{-1}}) \]
degenerates in $E_{1}$  as well.
\end{satz}
\begin{defn}{Remarks}
a) In (3.2), whenever one likes, one can assume that $i = 1$. In fact,
one just has to replace ${\cal L}$ by ${\cal L}^{\prime} = {\cal L}^{i}$ and
$D$ by $D^{\prime} = i \cdot  D$ .Then
$${\cal L}^{\prime N} = {\cal O}_{X}(i \cdot  D) = {\cal O}_{X} (D^{\prime})$$
and
\[ {\cal L}^{\prime^{(1,D^{\prime})}} = {\cal L}^{\prime} ( - [ \frac{D^{\prime}}{N} ])
= {\cal L}^{i} ( - [ \frac{i}{N} D] ). \]
b) Next, one can always assume that $ 0 < \alpha_{j} < N$.
In fact, if $\alpha_{1} \geq N$, then
\[ {\cal L}^{\prime} =
{\cal L} (-D_{1}) \ \ \mbox{and \ \ $D^{\prime} = D - N \cdot  D_{1}$} \]
give the same sheaves as ${\cal L}$ and $D$:
\[ {\cal L}^{\prime^{(i, D^{\prime})}} = {\cal L}^{i} (-i \cdot
D_{1} - [ \frac{i}{N} \cdot  D^{\prime} ]) = {\cal L}^{i} (-
[ \frac{i}{N} \cdot  D ]). \]
c) In particular, for $i = 1$ and $ 0 < a_{j} < N$ we have
\[ {\cal L}^{(1)} = {\cal L} \ \  \mbox{and\ \  $D^{(1)} = D$}. \]
Nevertheless, in the proof of (3.2) we stay with the notation, as started. \\
d) Finally, for $i \geq N$ one has
\[ {\cal L}^{(i, D)} = {\cal L}^{i} ( - [ \frac{i}{N} \cdot  D ]) =
{\cal L}^{i-N} (- [ \frac{i-N}{N} \cdot  D ]) = {\cal L}^{(i-N, D)}. \]
The ``${\cal L}^{(i)}$'' are the most natural notation for ``integral parts of $\Q$-
divisors'' if one wants to underline their relations with coverings.
In the literature one finds other equivalent notations, more adapted
to the applications one has in mind:
\end{defn}
\begin{defn}{Remarks}\\
a) Sometimes the integral part $[ \Delta ]$ is denoted by $\lfloor
\Delta \rfloor$.\\
b) One can also consider the round up\index{$\lceil \Delta \rceil$}
$\{ \Delta \} = \lceil \Delta \rceil$ given by $$\{ \Delta \} = - [ - \Delta ]$$
or the fractional part of $ \Delta $ given by
\[ < \Delta > = \Delta - [ \Delta ]. \]\index{$< \Delta >$}
c) For ${\cal L}$, $N$ and $D$ as in (3.1,c) one can write
\[ {\cal L} = {\cal O}_{X} (C) \]
for some divisor $C$. Then
\[ \Delta = C - \frac{1}{N} \cdot  D \ \in \Div_{\Q} (X) \]
has the property that $N \cdot  \Delta $ is a divisor
linear equivalent to zero. One has
\[ {\cal L}^{(i,D)} = {\cal O}_{X} (i \cdot  C - [ \frac{i}{N} \cdot  D ] ) =
{\cal O}_{X} (- [ - i \cdot  \Delta ] ) = {\cal O}_{X} ( \{ i \cdot
\Delta \} ).\]
d) On the other hand, for $\Delta \in \Div_{Q} (X)$ and $N > 0$ assume that
$N \cdot  \Delta$ is a divisor linear equivalent to zero.
Then one can choose a divisor $C$ such that $C - \Delta$ is effective.
For ${\cal L} = {\cal O}_{X}(C)$ and $D = N \cdot  C - N \cdot  \Delta
\ \in \ \Div(X)$ one has
\[ {\cal L}^{N} = {\cal O}_{X} (N \cdot  C) = {\cal O}_{X} (D) \]
and
\[ {\cal L}^{(i,D)} = {\cal O}_{X}(i \cdot  C - [ \frac{i}{N} D ]) = \]
\[ {\cal O}_{X} (-[ -i \cdot  C + \frac{i}{N} \cdot  D ] ) = {\cal O}_{X}
( \{ i \cdot  \Delta \} ). \]
e) Altogether, (3.2) is equivalent to:\\
\ \\
{\it For $\Delta  \in  \Div_{\Q} (X)$ such that $N \cdot  \Delta$ is a divisor linear
equivalent to zero, assume that
$ < \Delta > $ is supported in $D$ and that $D$ is a normal
crossing divisor. Then ${\cal O}_{X} ( \{ \Delta \} )$ has a logarithmic
integrable connection with poles along $D$ which satisfies a residue
condition similar to (3.2,a) and the $E_{1}$-degeneration.}\\
\ \\
We leave the exact formulation and the translation as an
exercise.
\end{defn}
\begin{defn}{Cyclic covers} \index{Cyclic cover}
Let ${\cal L}, N$ and
$$D = \sum^{r}_{j=1} \alpha_{j} D_{j}$$ be as in (3.1,c) and let
$ s \ \in \ H^{0} (X, {\cal L}^{N}) $
be a section whose zero divisor is $D$.
The dual of
$$s: {\cal O}_{X} \>>> {\cal L}^{N} \mbox{\ , i.e. \ }
s^{\vee}: {\cal L}^{-N} \>>> {\cal O}_{X},$$
defines a ${\cal O}_{X}$-algebra structure on
\[ {\cal A}^{\prime} = \bigoplus_{i=0}^{N-1} \ {\cal L}^{-i}. \]
In fact, $${\cal A}^{\prime} = \bigoplus_{i=0}^{\infty} \ {\cal L}^{-i} / I$$
where $I$ is the ideal-sheaf generated locally by
$$\{ s^{\vee} (l) - l, \ l \mbox{  local section of } {\cal L}^{-N} \}.$$
Let $$Y^{\prime} = {\bf Spec}_{X} ({\cal A}^{\prime})
\stackrel{\pi^{\prime}}{\>>>} X$$
be the spectrum of the ${\cal O}_{X}$-algebra
${\cal A}^{\prime}$, as defined in \cite{Har77}, page 128, for example.\\
Let $\pi : Y \rightarrow X$ be the finite morphism obtained by normalizing
$Y^{\prime} \rightarrow X$. To be more precise, if $Y^{\prime}$ is reducible, $Y$
will be the disjoint union of the normalizations of the components of
$Y^{\prime}$ in their function fields. We will call $Y$ {\it the cyclic
cover obtained by taking the} $n$-{\it th root out of} $s$ (or {\it out of}
$D$, if ${\cal L}$ is fixed). \index{Cyclic cover!$n$-th root
out of $D$}\\
\ \\
Obviously one has:
\end{defn}
\begin{defn}{Claim} $Y$ is uniquely determined by:\\
a) $\pi: Y \rightarrow X$ is finite.\\
b) $Y$ is normal.\\
c) There is a morphism $\phi: {\cal A}^{\prime} \rightarrow \pi_{*}
{\cal O}_{Y}$ of ${\cal O}_{X}$-algebras, isomorphic over some dense open
subscheme of $X$.
\end{defn}
\begin{defn}{Notations} For $D$, $N$ and ${\cal L}$ as in (3.1,c) let us write
$${\cal A} = \bigoplus_{i=0}^{N-1} {\cal L}^{(i)^{-1}}.$$
The inclusion
$${\cal L}^{-i} \>>> {\cal L}^{(i)^{-1}} = {\cal L}^{-i} ( [
\frac{i}{N} \cdot  D ] )$$
gives a morphism of ${\cal O}_{X}$-modules
\[ \phi: {\cal A}^{\prime} \>>> {\cal A}. \]
\end{defn}
\begin{defn}{Claim} ${\cal A}$ has a structure of an ${\cal O}_{X}$-algebra,
such that $\phi$ is a homomorphism of algebras.
\end{defn}
\begin{Beweis}{Proof} The multiplication in ${\cal A}^{\prime}$ is nothing
but the multiplication $${\cal L}^{-i} \times {\cal L}^{-j} \>>>
{\cal L}^{-i-j}$$
$$\mbox{ \  composed with \ } s^{\vee}: {\cal L}^{-i-j} \>>>
{\cal L}^{-i-j+N} \ ,$$ in case that $i+j \geq N$. For $i,j \geq 0$ one has
\[ [ \frac{i}{N} \cdot  D ] + [ \frac{j}{N} \cdot  D ] \leq
[ \frac{i+j}{N} \cdot  D ] \]
and, for $i+j \geq N$, one has
\[ {\cal L}^{(i+j)} = {\cal L}^{i+j} ( - [\frac{i+j}{N} \cdot  D ] ) =
{\cal L}^{i+j-N} (- [ \frac{i+j-N}{N} \cdot  D ] ) = {\cal L}^{(i+j - N)}. \]
This implies that the multiplication of sections
\[ {\cal L}^{(i)^{-1}} \times {\cal L}^{(j)^{-1}} \>>> {\cal L}^{-i-j}
( [ \frac{i}{N} D ] + [ \frac{j}{N} D ] ) \>>> {\cal L}^{(i+j)^{-1}} \]
is well defined, and that for $i+j \geq N$ the right hand side is nothing but
${\cal L}^{(i+j-N)^{-1}}$.\\
\end{Beweis}
\Abschnitt{ }
Assume that $N$ is prime to $\Char  (k)$, $e$ a fixed primitive
$N$-th root of unit and $G = < \sigma >$ the cyclic group of order $N$.
Then $G$ acts on ${\cal A}$ by ${\cal O}_{X}$-algebra homomorphisms defined by:\\
\[ \sigma (l) = e^{i} \cdot  l \ \ \ \mbox{for a local section\ \ \ }
l \mbox{ \ \ \ of\ \ \ } {\cal L}^{(i)^{-1}} \subset {\cal A}.\] Obviously the invariants under this $G$-action are
\[ {\cal A}^{G} = {\cal O}_{X}. \]
%
\begin{defn}{Claim}  Assume that $N$
is prime to $\Char  (k)$. Then
\[ {\cal A} = \pi_{*} {\cal O}_{Y} \ \ \ \mbox{or (equivalently)}
\ \ \ \ Y = \ {\bf Spec} ({\cal A}) \ .\]
\end{defn}
%
\begin{satz}{Corollary (see \cite{Esn82})} The cyclic group $G$ acts on $Y$ and on
$\pi_{*}{\cal O}_{Y}$.
One has $Y/{G} = X$ and the decomposition $$\pi_{*}{\cal O}_{Y} =
\bigoplus_{i=0}^{N-1} {\cal L}^{(i)^{-1}}$$ is the decomposition in eigenspaces.
\end{satz}
%
\begin{Beweis}{Proof of 3.10.}
For any open subvariety $X_{0}$ in $X$ with $\codim_{X} (X-X_{0}) \geq 2$
and for $Y_{0} = \pi^{-1} (X_{0})$ consider the induced morphisms
\[
\begin{CD}
Y_{0} \> {\iota^{\prime}} >> Y\\
\V \pi_{0} VV  \V V \pi V \\
X_{0} \>\iota >> X
\end{CD}
\]
Since $Y$ is normal one has $\iota^{\prime}_{*} {\cal O}_{Y_{0}} = {\cal O}_{Y}$
and $\pi_{*}{\cal O}_{Y} = \iota_{*} \pi_{0_*} {\cal O}_{Y_{0}}$. Since
${\cal A}$ is locally free, (3.10) follows from
\[ \pi_{0_*} {\cal O}_{Y_{0}} = {\cal A}|_{X_{0}}. \]
Especially we may choose $X_{0} = X - \Sing (D_{red})$ and, by abuse of notations,
assume from now on that $D_{red}$ is non-singular.\\
\ \\
As remarked in (3.6) the equality of ${\cal A}$ and $\pi_{*}{\cal O}_{Y}$
follows from:
\begin{defn}{Claim} \ \ ${\bf Spec} \ ({\cal A}) \>>> X$ is finite and ${\bf Spec}
(\cal A)$ is normal.
\end{defn}
{\sc Proof:}
(3.12) is a local statement and to prove it we may assume that \\
$X = \Spec \ B$
and that $D$ consists of just one component, say $D = \alpha_{1} \cdot  D_{1}$.
Let us fix isomorphisms ${\cal L}^{i} \simeq {\cal O}_{X}$ for all $i$ and assume
that $D_{1}$ is the zero set of $f_{1} \in B$.
For some unit $u \in B^{\ast}$ the section $s \in H^{0}(X, {\cal L}^{N}) \simeq
B$ is identified with $f = u \cdot  f_{1}^{\alpha_{1}}$. For completeness,
we allow $D$ (or $\alpha_{1}$) to be zero.\\
The ${\cal O}_{X}$-algebra ${\cal A}^{\prime}$ is given by the
$B$-algebra $$H^{0}(X,{\cal A}^{\prime}) = \bigoplus_{i=0}^{N-1} H^{0} (X,
{\cal L}^{-i})$$ which can be identified with the quotient of the
ring of polynomials
\[ A^{\prime} = B [ t]/_{t^{N} - f} = \bigoplus_{i=0}^{N-1} \ B \cdot  t^{i}. \]
In this language
$$ A = \bigoplus_{i=0}^{N-1} \ B \cdot  t^{i} \cdot  f_{1}^{- [ \frac{i}{N}
\alpha_{1} ] }  = \bigoplus_{i=0}^{N-1} H^{0} (X, {\cal L}^{(i)^{-1}} ) =
H^{0} (X, {\cal A} ) $$
and $\phi : {\cal A}^{\prime} \rightarrow {\cal A}$ induces the natural inclusion
$A^{\prime} \hookrightarrow A$. \\
\ \\
Hence (3.12) follows from the first part of the following claim.\\
\end{Beweis}
\begin{defn}{Claim} Using the notations introduced above, assume that $N$ is
prime to $\Char  (k)$. Then one has\\
a) $\Spec \ A$ is non-singular and $\pi : \Spec \ A \>>>
\Spec \ B$ is finite.\\
b) If $\alpha_{1} = 0$, then $\Spec \ A \>>> \Spec \ B$ is non-ramified
(hence \'{e}tale).\\
c) if $\alpha_{1}$ is prime to $N$, we have a defining equation $g \in A$
for $\Delta_{1} = (\pi^{*} D_{1})_{red}$ with
\[ g^{N} = u^{a} \cdot  f_{1} \ \ \ \mbox{for some \ \ $a \in \N$.} \]
d) If $\Gamma$ is a divisor in $\Spec \ B$ such that $D + \Gamma$ has normal
crossings, then $\pi^{\ast} (D + \Gamma)$ has normal crossings as well.
\end{defn}
\begin{Beweis}{Proof} Let us first consider the case $\alpha_{1} = 0$. Then
$$A^{\prime} = A = B [t]/_{t^{N} - u}$$ for $u \in B^{\ast}$.
$A$ is non-singular, as follows, for example, from the Jacobi-criterion, and
$A$ is unramified over $B$. Hence $$\Spec \ A \>>> \Spec \ B$$ is \'{e}tale
in this case and a), b) and d) are obvious.\\
\ \\
If $\alpha_{1} = 1$ , then again
$$A^{\prime} = A = B [ t]/_{t^{N}-u\cdot  f_{1}}.$$
For $p \in \Spec \ B$, choose $f_{2}, \ldots, f_{n}$ such that $f_{1} \cdot  u, f_{2}, \ldots, f_{n}$ is a
local parameter-system in $p$. Then  $ t, f_{2}, \ldots, f_{n}$
will be a local parameter system, for $q = \pi^{-1} (p)$ .\\
\ \\
Similar, if $\alpha_{1}$ is prime to $N$,
and if c) holds true,
$g$ and $f_{2}, \ldots, f_{n}$ will be a
local parameter system in $q$ and, composing both steps, $\Spec \ A$ will always
be non-singular and d) holds true.\\
\ \\
Let us consider the ring
\[ R = B [ t_{0}, t_{1} ]/_{t_{0}^{N} - u, t_{1}^{N} - f_{1}}. \]
Identifying $t$ with $t_{0} \cdot  t_{1}^{\alpha_{1}}$ we obtain $A^{\prime}$ as
a subring of $R$.
$\Spec \ R$ is non-singular
over $p$ and $\Spec \ R \>>> \Spec \ B$ is finite.\\
\ \\
The group $H = < \sigma_{0} > \oplus < \sigma_{1} >$ with $ord \ (\sigma_{0}) =
ord \ (\sigma_{1}) = N$ operates on $R$ by
\[  \sigma_{\nu} (t_{\mu}) =
\left\{
\begin{array}{ll}
t_{\mu} & \mbox{if $ \nu \neq \mu$}\\
e \cdot  t_{\mu} & \mbox{if $\nu = \mu$}\\
\end{array}
\right.  \]
Let $H^{\prime}$ be the kernel of the map
$ \gamma : H \>>> G = < \sigma > $
given by $\gamma (\sigma_{0}) = \sigma $ and $\gamma ( \sigma_{1}) =
\sigma^{\alpha_{1}}$. The quotient
\[ \Spec \ (R)/_{H^{\prime}} = \Spec \ R^{H^{\prime}} \]
is normal and finite over $\Spec \ B$.\\
One has $(\sigma_{0}^{\mu}, \sigma_{1}^{\nu}) \ \in \ H^{\prime}$, if and only
if $\mu + \nu  \alpha_{1} \equiv 0  \mbox{ mod }  N$.
Hence $R^{H^{\prime}}$ is generated by monomials $t_{0}^{a} \cdot  t_{1}^{b}$
where $a, b \ \in \{ 0, \ldots, N-1 \} $ satisfy:\\
\ \\
$(\ast) \ \ \ \ \  a \mu + b  \nu \equiv 0 \mbox{\  mod \ }  N$
\ for all \ $(\mu, \nu)$ \ with\
$\mu + \alpha_{1}  \nu \equiv 0 \ \mbox{ mod }  N$.\\
\ \\
Obviously, $(\ast)$ holds true for $(a,b)$ if $b \equiv a \cdot  \alpha_{1}
\ \mbox{ mod } N$. On the other hand, choosing $\nu$ to be a unit in $\Z/_{N}$,
$(\ast)$ implies that $b \equiv a \cdot  \alpha_{1} \ \mbox{ mod } \ N$.\\
\ \\
Hence, for all $(a,b)$ satisfying $(\ast)$ we find some $k$ with
$b = a \cdot  \alpha_{1} + k \cdot  N$. Since \ \ $a, b \in \{ 0, \ldots, N-1 \}$
we have
\[ \frac{a \cdot  \alpha_{1}}{N} \geq -k = \frac{a \cdot  \alpha_{1}}{N} -
\frac{b}{N} > \frac{a \cdot  \alpha_{1}}{N} - 1  \]
or $k = - [ \frac{a \cdot  \alpha_{1}}{N} ]$.\\
\ \\
Therefore one obtains
\[
R^{H^{\prime}}  =  \bigoplus_{a = 0}^{N-1} t_{0}^{a} \cdot
t_{1}^{a \cdot  \alpha_{1} - N \cdot [ \frac{a \cdot  \alpha_{1}}{N} ]} \cdot  B
= \bigoplus_{a = 0}^{N-1} (t_{0} \cdot  t_{1}^{\alpha_{1}})^{a} \cdot
f_{1}^{- [ \frac{a \cdot  \alpha_{1}}{N} ]} \cdot  B\]
\[ \mbox{ and hence\ \ \ \ \ } R^{H^{\prime}}= \bigoplus_{a=0}^{N-1} t^{a} \cdot  f_{1}^{- [ \frac{a \cdot  \alpha_{1}}
{N} ]} \cdot  B = A.
\]
If $\alpha_{1}$ is prime to $N$, we can find $a \ \in \ \{ 0, \ldots, N-1 \}$
with $a \cdot  \alpha_{1} = 1 + l \cdot  N$ for $l \in \Z$. Then
\[ a \cdot  \alpha_{1} - N \cdot  [ \frac{a \cdot  \alpha_{1}}{N} ] = 1 \]
and $g = t^{a} \cdot  f_{1}^{- [\frac{a \cdot  \alpha_{1}}{N} ] }$ satisfies
\[ g^{N} = u^{a} \cdot  f_{1}^{a \cdot  \alpha_{1} - N [\frac{a \cdot
\alpha_{1}}{N} ]} = u^{a} \cdot  f_{1}. \]
\end{Beweis}
\begin{defn}{Remarks}\\
a) If $Y$ is irreducible, for example if $D$ is reduced, the local
calculation shows
$Y$ is nothing but the normalisation of $X$ in $k(X) (\sqrt[N]{f})$, where
$f$ is a rational function giving the section $s$.\\
b) $\pi^{\prime}: Y^{\prime} \>>> X$ can be as well
described in the following
way (see \cite{Har77}, p. 128-129):\\
Let ${\bf V} ({\cal L}^{-\alpha}) =
{\bf \Spec} \ (\bigoplus_{i=0}^{\infty} {\cal L}^{-\alpha})$ be the
geometric rank one vector bundle associated
to ${\cal L}^{-\alpha}$. The geometric
sections of ${\bf V} ({\cal L}^{-\alpha}) \>>> X$ correspond to
$H^{0} (X, {\cal L}^{\alpha})$. Hence $s$ gives a section $\sigma$
of ${\bf V} ({\cal L}^{-N})$ over $X$. We have a natural map
\[ \tau : {\bf V}({\cal L}^{-1}) \>>> {\bf V} ({\cal L}^{-N}) \]
and $Y^{\prime} = \tau^{-1} (\sigma (X))$. \index{Cyclic
cover!via geometric vector bundles}\\
\ \\
The local computation in (3.13) gives a little bit more information
than asked for in (3.12):\\
\end{defn}
%
\begin{satz}{Lemma} Keeping the notations and assumptions from (3.5) assume
that $N$ is prime to $\Char  (k)$. Then one has \index{cyclic
cover!singularities}\index{cyclic cover!ramification index}\\
a) $Y$ is reducible, if and only if for some $\mu > 1$, dividing $N$,
there is a section $s'$ in $H^{0} (X, {\cal L}^{\frac{N}{\mu}})$ with
$s = s'^\mu$.\\
b) $\pi : Y \rightarrow X$  is \'{e}tale over $X - D_{red}$ and $Y$
is non-singular over\\
 $X - \Sing (D_{red})$. \\
c) For $\Delta_{j} = (\pi^{\ast} D_{j})_{red}$ we have
\[ \pi^{\ast} D = \sum_{j=1}^{r} \frac{N \cdot  \alpha_{j}}{\mbox{gcd}
(N, \alpha_{j})} \cdot  \Delta_{j}. \]
d)  If $Y$ is irreducible then the components of $\Delta_{j}$ have over
$D_{j}$ the ramification index
$$e_{j} = \frac{N}{\mbox{gcd}(N, \alpha_{j})}.$$
\end{satz}
%
\begin{Beweis}{Proof} For a) we can consider the open set $\Spec \ B \subset X - D_{red}$.
Hence $\Spec \ B [t]/_{t^{N}-u}$ is in $Y$ dense and open. $Y$ is
reducible if and only if $t^{N}-u$ is reducible in $B [t]$, which is
equivalent to the existence of some $u^{\prime} \in B$ with $u =
u^{\prime^{\mu}}$.\\
b) has been obtained in (3.13) part a) and b).\\
For c) and d) we may assume that $D = \alpha_{1} \cdot  D_{1}$ and,
splitting the covering in two steps, that either $N$ divides $\alpha_{1}$ or that $N$
is prime to $\alpha_{1}$. \\
In the first case, we can as well choose $\alpha_{1}$
to be zero (by 3.3,b) and c) as well as d) follow from part b). \\
If $\alpha_{1}$ is prime to
$N$,then $\pi^{\ast} D = e_{1} \cdot  \alpha_{1} \cdot  \Delta_{1}$. Since
$\pi^{\ast} D$ is the zero locus of $f = t^{N}$, $N$ divides $e_{1} \cdot
\alpha_{1}$. On the other hand, since $e_{1}$ divides $\deg \ (Y/X) =N$, one
has $e_{1} = N$ in this case.\\
\end{Beweis}
\begin{satz}{Lemma}
Keeping the notations from (3.5) assume that $N$ is prime
to $\Char  (k)$ and that $D_{red}$ is non-singular. Then one has:\\
a) (Hurwitz's formula) $\pi^{\ast} \Omega_{X}^{b} (\log \ D) =
\Omega_{Y}^{b} (\log \ (\pi^{\ast} D))$.\index{Hurwitz's formula}\\
b) The differential $d$ on $Y$ induces a logarithmic integrable connection
\[ \pi_{*} (d): \bigoplus_{i=0}^{N-1} {\cal L}^{(i)^{-1}}
\>>> \pi_{*} \Omega_{Y}^{1}(\log \ (\pi^{\ast} D)) =
\bigoplus_{i=0}^{N-1} \Omega_{X}^{1} (\log \ D)\otimes {\cal L}^{(i)^{-1}},\]
compatible with the direct sum decomposition.\index{Cyclic
cover!induced connection}\\
c) If $\nabla^{(i)} : {\cal L}^{(i)^{-1}} \>>>
 \Omega_{X}^{1} (\log \ D)\otimes {\cal L}^{(i)^{-1}}$
denotes the $i$-th component of
$\pi_{*} (d)$ then $\nabla^{(i)}$ is a logarithmic integrable connection
with residue
\[ Res_{D_{j}} (\nabla^{(i)} ) = ( \frac{i \cdot  \alpha_{j}}{N} - [
\frac{i \cdot  \alpha_{j}}{N} ] ) \cdot  id_{{\cal O}_{D_{j}}}.
\] \index{Cyclic
cover!induced connection!their residues}
d) One has
\[ \pi_{\ast} (\Omega_{Y}^{b}) = \bigoplus_{i=0}^{N-1} \Omega_{X}^{b}
(\log \ D^{(i)})\otimes {\cal L}^{(i)^{-1}}
\mbox{ \ for \ }
D^{(i)} = \sum^{r}_{ \scriptstyle{j=1} \atop
\scriptstyle{\frac{i \cdot  \alpha_{j}}{N} \in \Q-\Z}} D_{j}. \]
e) The differential
\[ \pi_{\ast}(d) : \pi_{\ast} {\cal O}_{Y} = \bigoplus_{i=0}^{N-1}
{\cal L}^{(i)^{-1}} \>>> \pi_{\ast} (\Omega^{1}_{Y}) =
\bigoplus_{i=0}^{N-1} \Omega^{1}_{X} (\log \
D^{(i)}) \otimes {\cal L}^{(i)^{-1}}\]
decomposes into a direct sum of
\[ \nabla^{(i)} : {\cal L}^{(i)^{-1}} \>>>
\Omega_{X}^{1} (\log \ D^{(i)} ) \otimes {\cal L}^{(i)^{-1}} . \]
\end{satz}
\begin{Beweis}{Proof} Again we can argue locally and assume that $X = \Spec \ B$
and \\
$D = \alpha_{1} D_{1}$ as in (3.12). \\
If $\alpha_{1} = 0$, or if $N$
divides $\alpha_{1}$, then $f_{1}$ is a defining equation for $\Delta_{1}
= (\pi^{\ast} D_{1})_{red}$ and the generators for $\Omega_{X}^{b} (\log \ D)$
are generators for $\Omega_{Y}^{b} (\log \ \pi^{\ast} D)$ as well. \\
For $\alpha_{1}$ prime to $N$, we have by (3.13,c) a defining equation $g$ for
$\Delta_{1} =  (\pi^{\ast} D_{1})_{red}$ with $g^{N} = u ^{a} \cdot  f_{1}$. Hence
\[ N \cdot \frac{dg}{g} = \frac{df_{1}}{f_{1}} + a \cdot  \frac{du}{u} \]
and, since $N \in k^{\ast}$ and $a \cdot  \frac{du}{u} \in \Omega_{X}^{1}$,
one finds
that $\frac{df_{1}}{f_{1}}$ and $\pi^{\ast} \Omega_{X}^{1}$  generate
$\Omega_{Y}^{1} (\log \ \pi^{\ast} D)$.\\
\ \\
We can split $\pi$ in two coverings of degree $N \cdot  \mbox{ gcd }
(N, \alpha_{1})^{-1}$ and $ \mbox{ gcd } (N, \alpha_{1})$. Hence we obtain a)
for $b = 1$. The general case follows.\\
\ \\
The group $G$ acts on $\pi_{\ast} \Omega^{b}_{Y}$ and $\pi_{\ast} \Omega_{Y}^{b}
(\log \ \pi^{\ast} D))$ compatibly with the inclusion, and the action on the second sheaf
is given by $id \otimes \sigma$ if one writes $$\pi_{\ast} \Omega_{Y}^{b}
(\log \ (\pi^{\ast} D)) = \Omega_{X}^{b}
(\log \ D)\otimes \pi_{\ast} {\cal O}_{Y}. $$
Let $l$ be a local section of $\Omega_{X}^{b} ( \log \ D)\otimes {\cal L}^{(i)^{-1}}$
written as $$ l = \phi \cdot  g_{i}
\mbox{ \ for \ } \phi \in \Omega_{X}^{b}( \log \ D) \mbox{ \  and \ }
g_{i} = t^{i} \cdot  f_{1}^{- [ \frac{i \cdot  \alpha_{1}}{N} ] }.$$
Since
\[ g^{N}_{i} =u^{i} \cdot  f_{1}^{i \cdot  \alpha_{1} - N \cdot
[ \frac{i \alpha_{1}}{N} ] } \]
has a zero along $\Delta_{1}$ if and only if
$$\frac{i \cdot  \alpha_{1}}{N} \not\in \Z,$$
we find that $l$ lies in $\Omega^{b}_{Y}$
in this case.\\
On the other hand, if
$g_{i}$ is a unit, $l$
lies in $\Omega_{Y}^{b}$ if and only if $\phi$ has no pole along $D$ and
we obtain d).\\
\ \\
We have
\[ N \frac{d g_{i}}{g_{i}} = i \cdot  \frac{du}{u} + (i \cdot  \alpha_{1} - N
[\frac{i \cdot  \alpha_{1}}{N} ] ) \frac{df_{1}}{f_{1}} \]
or $$dg_{i} = (\frac{i}{N} \frac{du}{u} + (\frac{i}{N} \alpha_{1} -
[ \frac{i \cdot  \alpha_{1}}{N} ] ) \frac{df_{1}}{f_{1}} ) \cdot  g_{i} .$$
Hence, $$d(g_{i} \cdot  \phi) \in \Omega_{X}^{b+1}
(\log \ D)\otimes {\cal L}^{(i)^{-1}},$$ and $(\pi_{\ast} d)$ respects
the direct sum decomposition. Obviously,
the Leibniz rule for $d$ implies that $(\pi_{\ast} d)$ as well as the components
$\nabla^{(i)}$ are connections and b) and e) hold true.\\
\ \\
Finally, for c), let $\phi \in {\cal O}_{X}$. Then by the calculations given
above, we find $$Res_{D_{1}} (\nabla^{(i)})(g_{i} \cdot  \phi) =
(\frac{i}{N} \alpha_{1} - [ \frac{i \cdot  \alpha_{1}}{N} ] ) g_{i} \cdot
\phi \mid_{D_{1}}.$$
\end{Beweis}
\begin{Beweis}{Proof of 3.2,a}\\
If $X$ is projective and
$$D = \sum_{j=1}^{r} \alpha_{j} D_{j}$$
a normal crossing divisor we found the connection
\[ \nabla^{(i)} : {\cal L}^{(i)^{-1}} \>>>
\Omega^{1}_{X} (\log \ D^{(i)} )\otimes {\cal L}^{(i)^{-1}} \]
with the residues as given in (3.2,a) over the open submanifold
$X - \Sing (D_{red})$. Of course, $\nabla^{(i)}$ extends to $X$
since $$\codim_{X}(\Sing (D_{red})) \geq 2.$$
\end{Beweis}
Over a field $k$ of characteristic zero,
to prove the $E_{1}$-degeneration, as stated in (3.2,b) or (3.2,c)
one can apply the degeneration
of the logarithmic Hodge to de Rham spectral sequence (see (10.23)
for example) to some desingularization of $Y$. We will sketch
this approach in (3.22). One can as well
reduce (3.2,b) to the more familiar degeneration of the Hodge spectral
sequence
\[ E_{1}^{ab} = H^{b} (T, \Omega^{a}_{T}) \Longrightarrow \h ^{a+b} (T,
\Omega^{\bullet}_{T}) \]
for projective manifolds $T$ by using the following covering Lemma, due to\\
Y. Kawamata \cite{Kaw81}:\index{Covering construction!Kawamata}

\begin{satz}{Lemma} Keeping the notations from (3.5) assume that $N$ is prime
to $\Char  (k)$ and that $D$ is a normal crossing divisor. Then there
exists a manifold $T$ and a finite morphism
\[ \delta : T \>>> Y \]
such that:\\
a) The degree of $\delta$ divides a power of $N$.\\
b) If $A$ and $B$ are reduced divisors such that $D+A+B$ has at most normal
crossings and if $A+B$ has no common component with $D$, then we can choose $T$ such
that $(\pi \circ \delta)^{\ast}(D+A+B)$ is a normal crossing divisor
and $(\pi \circ \delta)^{\ast} A$ as well as $(\pi \circ \delta)^{\ast} B$
are reduced.
\end{satz}
\begin{Beweis}{Proof of (3.2) in characteristic zero, assuming the
$E_1$ degeneration of the Hodge to de Rham spectral sequence}\\
Let $X_{0} = X - \Sing (D_{red})$, $Y_{0} = \pi^{-1}(X_{0})$ and $T_{0} = \delta^{-1}
(Y_{0})$. \\
$\delta_{\ast} \Omega^{\bullet}_{T_{0}}$ contains $\Omega_{Y_{0}}^{\bullet}$
as direct summand. Since $(\pi \circ \delta)$ is flat $(\pi \circ \delta)_{\ast}
\Omega^{\bullet}_{T}$ will contain
\[ \bigoplus_{i=0}^{N-1} \Omega^{\bullet}_{X} (\log \ D^{(i)})
\otimes {\cal L}^{(i)^{-1}} \]
as a direct summand. The $E_{1}$-degeneration of the spectral sequence
\[ E_{1}^{ab} = H^{b} (T, \Omega^{a}_{T}) \Longrightarrow \h ^{a+b} (T,
\Omega^{\bullet}_{T}) \]
implies (3.2,b) for each $i \in \{ 0, \ldots, N-1 \}$. Finally, if $A$ and $B$
are the divisors considered in (3.2,c), $A^{\prime} = (\pi \circ \delta)^{\ast}
A$ and $B^{\prime} = (\pi \circ \delta)^{\ast} B$,
\[ \Omega^{\bullet}_{X} (\log \ (A+B+D^{(i)}))(-B)\otimes {\cal L}^{(i)^{-1}} \]
is a direct summand of
\[ (\pi \circ \delta)_{\ast} \Omega^{\bullet}_{T} (\log \ (A^{\prime} + B^{\prime}))
(-B^{\prime}) \]
and we can use the $E_{1}$-degeneration of
\[ E_{1}^{ab} = H^{b} (T, \Omega^{a}_{T} (\log \ (A^{\prime} + B^{\prime}))(-B^{\prime}))
\Longrightarrow
\h ^{a+b} (T, \Omega^{\bullet}_{T} (\log \ (A^{\prime} + B^{\prime}))
(-B^{\prime})). \]
\end{Beweis}
\begin{defn}{Remarks}\\
a) If $A=B=0$ the degeneration of the spectral sequence, used to get (3.2,b),
follows from classical Hodge theory. In general, i.e. for (3.2,c),
one has to use the Hodge theory for open manifolds developed by
Deligne \cite{Del72}.\\
In these lectures (see (10.23)) we will reproduce the algebraic proof
of Deligne and Illusie for the degeneration.\\
b) If $\Char \ (k) \neq 0$ and if $X, {\cal L}$ and $D$ admit a lifting
to $W_{2} (k)$ (see (8.11)), then the manifold $T$ constructed in (3.17)
will again admit a lifting to $W_{2} (k)$. Hence the proof of (3.2,b and c)
given above shows as well:\\
\ \\
{\it Assuming the degeneration of the Hodge to de Rham spectral sequence
(proved in (10.21)) theorem (3.2) holds true under the additional assumption
that ${\cal L}$ lifts to $W_{2}(k)$ as well.}\\
\ \\
c) Using (3.2,a) we will give a direct proof of (3.2,b and c) at the
end of \S 10, without using (3.17), for a field $k$ of characteristic $p \neq
0$. By reduction to characteristic $p$ one obtains a second proof of (3.2)
in characteristic zero.\\
d) In Lectures 4 - 7, we will assume (3.2) to hold true.
\end{defn}
To prove (3.17) we need:
\begin{satz}{Lemma (Kawamata \cite{Kaw81})} Let $X$ be a quasi-projective
manifold, let\index{Covering construction!Kawamata}
$$D = \sum_{j=1}^{r} D_{j}$$ be a reduced normal crossing divisor, and let
\[ N_{1}, \ldots , N_{r} \ \in \N - \{ 0 \} \]
be prime to
$\Char  (k)$. Then there exists a projective manifold $Z$ and a finite
morphism $\tau : Z \rightarrow X$ such that:\\
a) For $j=1, \ldots r$ one has $\tau^{\ast} D_{j} = N_{j} \cdot  (\tau^{\ast}
D_{j})_{red}$.\\
b) $\tau^{\ast} (D)$ is a normal crossing divisor.\\
c) The degree of $\tau$ divides some power of $\prod_{j=1}^{r} N_{j}$.\\
d) If $X$ and $D$ satisfy the lifting property
(8.11) the same holds true for $Z$.
\end{satz}
\begin{Beweis}{Proof} If we replace the condition that $D = \sum^{r}_{j=1} D_{j}$
is the decomposition of $D$ into irreducible (non-singular) components by the
condition that $D = \sum_{j=1}^{r} D_{j}$ for non-singular
divisors $D_{1} \ldots D_{r}$ we can construct $Z$ by induction and hence
assume that $N_{1}=N$ and $N_{2} = \ldots = N_{r} = 1$.\\
\ \\
Let ${\cal A}$ be an ample invertible sheaf such that ${\cal A}^{N} (-D_{1})$ is
generated by its global sections. Choose  $n = \dim \ X$ general divisors
$H_{1}, \ldots, H_{n}$ with
$${\cal O}_{X}(H_{i}) = {\cal A}^{N} (-D_{1}).$$
The divisor $D + \sum^{n}_{i=1} H_{i}$ will be a reduced normal crossing
divisor. Let
$$\tau_{i} : Z_{i} \>>> X$$
be the cyclic cover obtained
by taking the $N$-th root out of $H_{i} + D_{1}$. Then $Z_{i}$ satisfies the
properties a), c) and d) asked for in (3.19) but, $Z_{i}$ might have
singularities over $H_{i} \cap D_{1}$ and $\tau_{i}^{\ast} (D)$ might have
non-normal crossings over $H_{i} \cap D_{1}$.
Let $Z$ be the normalization of
\[ Z_{1} \times_{X} Z_{2} \times_{X} \ldots \times_{X} Z_{n}. \]
$Z$ can inductively be constructed as well in the following way:\\
Let $Z^{(\nu)}$ be the normalization of $Z_{1} \times_{X} \ldots \times_{X}Z_{\nu}$
and $\tau^{(\nu)} : Z^{(\nu)} \rightarrow X$ the induced morphism.
Then, outside of the singular locus of $Z^{(\nu)}$, the cover
$Z^{(\nu +1)}$ is obtained
from $Z^{(\nu)}$ by taking the $N$-th root out of
\[ \tau^{(\nu)^{\ast}} (H_{\nu + 1} + D_{1}) = \tau^{(\nu)^{\ast}} (H_{\nu + 1}) + N \cdot  ( \tau^{(\nu)^{\ast}} D_{1})_{red}. \]
This is the same as taking the $N$-th root out of
$\tau^{(\nu)^{\ast}} (H_{\nu + 1})$ by
(3.2,b) and (3.10). Since this divisor has no singularities, we find
by (3.15,b) that the singularities of $Z^{(\nu +  1)} $ lie over the singularities
of $Z^{(\nu)}$, hence inductively over $H_{1} \cap D_{1}$. However, as $Z$ is
independent of the numbering of the
$H_{i}$,  the singularities of $Z$ are lying  over$$\bigcap_{i=1}^{n}
(H_{i} \cap D_{1}) =  (\bigcap^{n}_{i=1} H_{i}) \cap D_{1}= \emptyset.$$
\end{Beweis}
\begin{Beweis}{Proof of (3.17)} Let $\tau : Z \rightarrow X$ be the covering
constructed in (3.19) for
$$D_{red} = \sum_{j=1}^{r} D_{j}$$ and
$N = N_{1} = \ldots = N_{r}$. Let $T$ be the normalization of $Z \times_{X} Y$.
Then $T$ is obtained again by taking the $N$-th root out of $\tau^{\ast} D$.
Since $\tau^{\ast} D = N \cdot  D^{\prime}$ for some divisor $D^{\prime}$ on $Z$,
we can use (3.3,b), (3.10) and (3.15,b) to show that $T$ is \'{e}tale
over $Z$.\\
\ \\
For part c), we apply the same construction to the manifold $Z$, given for
the divisor $D+A+B$, where the prescribed multiplicities for the
components of $A$ and $B$ are one.\\
\end{Beweis}
\ \\
\small {\bf Generalizations and variants in the analytic case}\\
\ \\
(3.17) is a special case of the more general covering lemma of Kawamata:
\begin{satz}{Lemma} Let $X$ be a projective manifold, $\Char  (k) = 0$ and
let $\pi : Y \rightarrow X$ be a finite cover such that the ramification
locus $D = \Delta (Y/X)$ in $X$ has normal crossings. Then there exists a
manifold $T$ and a finite morphism $\delta : T \rightarrow Y$. Moreover, one
can assume that $\pi \circ \delta : T \rightarrow X$ is a Galois cover.
\end{satz}
For the proof see \cite{Kaw81}. As shown in \cite{Vie82}
(3.16) can be generalized
as well:
\begin{satz}{Lemma}(Generalized Hurwitz's formula) For $\pi : Y \rightarrow X$ as
in {\rm (3.20)} let $\delta : Z \rightarrow Y$ be a desingularization such that
$(\pi \circ \delta)^{\ast} D = D^{\prime}$ is a normal crossing divisor. Then
one has an inclusion\index{Hurwitz's formula!generalized}
\[ \delta^{\ast}\pi^{\ast} \Omega^{a}_{X} (\log \ D) \>>> \Omega^{a}_{Z} (\log \
D^{\prime}) \]
giving an isomorphism over the open subscheme $U$ in $Z$ where $(\pi \circ
\delta) \mid_{Z}$ is finite.\\
\end{satz}
If $Y$ in (3.20) is normal, it has at most quotient singularities
(see (3.24) for a slightly different argument). In particular,
$Y$ has rational singularities (see \cite{Vie77} or (5.13)),
i.e.:
\[ R^{b} \delta_{\ast} {\cal O}_{Z} = 0 \ \ \mbox{for \ \ $b > 0$.} \]
One can even show (see \cite{EV81}):\\
\begin{satz}{Lemma} For $Y$ normal and $\pi : Y \rightarrow X$,
$\delta : Z \rightarrow Y$ as in (3.21) and $\tau = \pi \circ \delta$
one has:\index{Relative vanishing theorem!for log differentials}
\[ R^{b} \tau_{\ast} \Omega^{a}_{Z} (\log \ D^{\prime}) =
\left\{
\begin{array}{ll}
\Omega^{a}_{X} (\log \ D) \otimes \bigoplus_{i=0}^{N-1} {\cal L}^{(i)^{-1}}
& \mbox{for $b = 0$} \\
\ & \ \\
0 & \mbox{for $ b > 0$} \\
\end{array}
\right.  \]
\end{satz}
%
For $b = 0$ this statement follows directly from (3.21). For $b > 0$, however,
the only way we know to get (3.22) is to use GAGA and the independence
from the choosen compactification
of the mixed Hodge structure of the open manifold $Z - D^{\prime}$
(see Deligne \cite{Del70}).\\
\ \\
Using (3.21) and (3.22) one finds again (see \cite{EV86}):\\
The degeneration of the spectral sequence
\[ E_{1}^{ab} = H^{b} (Z, \Omega^{a}_{Z} (\log \ D^{\prime})) \Longrightarrow
H^{a+b} (Z, \Omega^{\bullet}_{Z} (\log \ D^{\prime})) \]
implies (3.2,b).\\
\ \\
Let us end this section with the following
\begin{satz}{Corollary}  Under the assumptions of 3.2 assume that $k = \C$.
Then\index{Cyclic cover!with quotient singularities}
\[ \dim \ (H^{b} (X, \Omega^{a}_{X} (\log \ D^{(i)})
\otimes {\cal L}^{(i)^{-1}})) =
\dim \ (H^{a} (X, \Omega^{b}_{X} (\log \ D^{(N-i)})
\otimes {\cal L}^{(N-i)^{-1}})). \]
\end{satz}
\begin{Beweis}{Proof} By GAGA we can assume that we consider the analytic sheaf of
differentials. The Hodge duality on the covering $T$ constructed in (3.17)
is given by conjugation. Since under
conjugation $e^{i}$ goes to $e^{N-i}$ for a primitive $N$-th
root of unity, we obtain (3.23).\\
\end{Beweis}
\normalsize

Let us end this section by showing that the cyclic cover $Y$ constructed in
(3.5) has at most quotient singularities. Slightly more generally one has
the following lemma which, as mentioned above, also follows from (3.20).
\begin{satz}{Lemma}  Let $X$ be a quasi-projective manifold, $Y$ a normal
variety and let $\pi : Y \>>> X$ be a separable finite cover.
Assume that the ramification divisor
\[ D = \sum^{m}_{j=1} D_j = \Delta(Y/X)\]
of $\pi$ in $X$ is a normal crossing divisor and that for all $j$ and all
components $B_j^i$ of $\pi^{-1}(D_j)$ the ramification index
$e(B^i_j)$ is prime to $\Char \ k$. \\
Then $Y$ has at most quotient singularities, i.e. each point $y \in Y$ has
a neighbourhood of the form $T/G$ where $T$ is nonsingular and $G$ a finite
group acting on $T$.
\end{satz}
\begin{Beweis}{Proof} One can assume that $X$ is affine. For $j=1,\cdots ,m$
define
\[ n_j = lcm\{ e(B^i_j) ; \ B^i_j \mbox{ component of } \pi^{-1}(D_j) \}.\]
Let $\tau : Z \>>> X$ be the cyclic cover obtained by taking
sucessively the $n_j$-th root out of $D_j$. In other terms, $Z$ is
the normalization of the fibered product of the different coverings
of $X$ obtained by taking the $n_j$ root out of $D_j$ or, equivalently,
$\tau$ is the composition of
\[
Z=Z_m \> \tau_m >> Z_{m-1} \> \tau_{m-1} >> \cdots \>>> Z_1 \> \tau_1 >> Z_0 = X
\]
where $\tau_j : Z_j \>>> Z_{j-1} $ is the cover obtained by taking
the $n_j$-th root out of
$(\tau_1 \circ \tau_2 \circ \cdots \circ \tau_{j-1})^{*}(D_j)$.
By (3.15,b) $Z_j$ is non singular. $Z$ is Galois over $X$
with Galois group
\[ G = \prod_{j=1}^m \Z / n_j \cdot \Z .\]
Let $T$ be the normalization of $Z \times_X Y$ and $\delta : T \>>> Y$
the induced morphism. Each component $T_0$ of $T$ is Galois
over $Y$ with a subgroup of $G$ as Galois group.\\
\ \\
The morphism $\delta_0=\delta |_{T_0}$ is obtained by taking sucessively the
$\frac{n_j}{\alpha_j}$-th
root out of
\[ \pi^{-1}(D_j) = \sum_{i=1}^{r_j} \frac{e(B^i_j)}{\alpha_j}\cdot B^i_j \]
for
$$\alpha_j= gcd \{ e(B^i_j) ; \ B^i_j \mbox{ component of } \pi^{-1}(D_j) \}.$$
By (3.15) all components of $\delta^{-1}(B^i_j)$ have ramification index
\[
\frac{\frac{n_j}{\alpha_j}}{gcd \{ \frac{n_j}{ \alpha_j} , \frac{e(B^i_j)}{\alpha_j}
\}} =
\frac{n_j}{e(B^i_j)}
\]
over $Y$. Hence they are ramified over $X$ with order $n_j$. In other terms,
the induced morphism $T_0 \>>> Z$ is unramified and $T_0$ is a non-singular
Galois cover of $Y$.\\
\end{Beweis}
\Paragraph{Vanishing theorems, the formal set-up.}
\ \\
Theorem 3.2 , whose proof has been reduced to the $E_{1}$-degeneration of
a Hodge to de Rham spectral-sequences, implies
immediately several vanishing theorems for the
cohomology of the sheaves ${\cal L}^{(i)}$.\\
To underline that in fact the whole information needed is hidden in
(3.2) and (2.9) we consider in this lecture a more general situation
and we state the assumptions explicitly, which are needed to obtain
the vanishing of certain cohomology groups.\\
\ \\
(4.2) and (4.8) are of special interest for applications whereas
the other variants can been skipped at the first reading.
\begin{defn}{Assumptions} Let $X$ be a projective manifold defined
over an algebraically
closed field $k$ and let $$D = \sum_{j=1}^{r} D_{j}$$ be a reduced normal
crossing divisor. Let ${\cal E}$ be a locally free sheaf on $X$ of finite
rank and let
\[ \nabla : {\cal E} \>>>  \Omega^{1}_{X} (\log \ D)
\otimes  {\cal E}\]
be an integrable connection with logarithmic poles along $D$.\\
We will assume
in the sequel that $\nabla$ {\it satisfies the $E_{1}$-degeneration}
i.e. that {\it the Hodge to de Rham spectral sequence (A.25)
\[E_{1}^{ab} = H^{b} (X, \Omega^{a}_{X} (\log \ D)\otimes  {\cal E})
\Longrightarrow \h^{a+b} (X, \Omega^{\bullet}_{X}
(\log \ D)\otimes  {\cal E}) \]
degenerates in} $E_{1}$.
\end{defn}
\begin{satz}{Lemma (Vanishing for restriction maps I)} Assume that $\nabla$
satisfies the condition (!) of (2.8), i.e. that for all $\mu \in \N$ and for
$j = 1, \ldots, r$ the map\index{General vanishing theorem!for
restriction maps}
\[ Res_{D_{j}} (\nabla) + \mu \cdot id_{{\cal O}_{D_{j}}} : {\cal E} \mid_{D_{j}}
\>>> {\cal E} \mid_{D_{j}} \]
is an isomorphism. Assume that $\nabla$ satisfies the $E_{1}$-degeneration (4.1).\\
Then for all effective divisors
$$D^{\prime} = \sum^{r}_{j=1} \mu_{j}D_{j}$$ and all $b$ the natural map
\[ H^{b} (X, {\cal O}_{X} (-D^{\prime})\otimes  {\cal E}) \>>>
H^{b} (X, {\cal E}) \]
is surjective.
\end{satz}
\begin{Beweis}{Proof} By (2.9,b) the map
$$\Omega^{\bullet}_{X}
(\log \ D)\otimes  {\cal E}(-D^{\prime}) \>>>
 \Omega^{\bullet}_{X}
(\log \ D)\otimes  {\cal E}$$
is a quasi-isomorphism and hence  induces
an isomorphism of the hypercohomology groups. Let
us consider the exact sequences of complexes
\[
\minCDarrowwidth=0.5cm
\begin{CD}
0 \>>> \Omega^{\bullet^{\geq 1}}_{X} (\log \ D)\otimes  {\cal E} \>>>
\Omega^{\bullet}_{X} (\log \ D)\otimes  {\cal E} \>>> {\cal E} \>>> 0 \\
\noarr \A A A \A A A \A A A \noarr \\
0 \>>> \Omega_{X}^{\bullet^{\geq 1}} (\log \ D)\otimes  {\cal E}(-D^{\prime})
\>>> \Omega_{X}^{\bullet}
(\log \ D)\otimes  {\cal E}(-D^{\prime})
\>>> {\cal E} (-D^{\prime}) \>>> 0 .
\end{CD}
\]
By assumption, the spectral sequence for $\Omega^{\bullet}_{X}
(\log \ D)\otimes  {\cal E}$ degenerates
in $E_{1}$, which implies that the morphism $\alpha$
in the following diagram is surjective (see (A.25)).
\[
\begin{CD}
\h^{b}(X, \Omega^{\bullet}_{X} (\log \ D)\otimes  {\cal E})
\> \alpha >>   H^{b} (X, {\cal E}) \\
\A A= A   \A A \beta A \\
\h^{b} (X, \Omega^{\bullet}_{X}
(\log \ D)\otimes  {\cal E}(-D^{\prime})) \>>>  H^{b} (X, {\cal E} (-D^{\prime}))
\end{CD}
\]
Hence $\beta$ is surjective as well.\\
\end{Beweis}
%
\begin{satz}{Variant} If in (4.2)
$$D^{\prime} = \sum_{j=1}^{s} \mu_{j} D_{j} \geq 0
\mbox{ \ for \ } s \leq r,$$
then it is enough to assume that for $j = 1, \ldots, s$ and
for $0 \leq \mu \leq \mu_{j} - 1$
\[ Res_{D_{j}} (\nabla) + \mu \cdot id_{{\cal O}_{D_{j}}} \]
is an isomorphism.
\end{satz}
\begin{Beweis}{Proof}
By (2.10) this is enough to give the quasi-isomorphism
\[ \Omega^{\bullet}_{X} (\log \ D)\otimes  {\cal E}(-D^{\prime}) \>>>
\Omega^{\bullet}_{X} (\log \ D)\otimes  {\cal E} \]
needed in the proof of (4.2).\\
\end{Beweis}
\begin{satz}{Lemma (Dual version of (4.2) and (4.3))} Assume that
\[ \nabla : {\cal E} \>>> \Omega^{1}_{X} (\log \ D)\otimes  {\cal E} \]
satisfies the $E_{1}$-degeneration and that for $j = 1, \ldots , s$ and $1 \leq
\mu \leq \mu_{j}$
\[ Res_{D_{j}} (\nabla) - \mu \cdot id_{{\cal O}_{D_{j}}} \]
is an isomorphism (for example, if $\nabla$ satisfies the condition $(\ast)$
from (2.8,a)). Then for
$$D^{\prime} = \sum^{s}_{j=1} \mu_{j} D_{j}$$ and all
$b$ the map
\[ H^{b} (X, \omega_{X} (D)\otimes  {\cal E} ) \>>> H^{b} (X,
\omega_{X} (D + D^{\prime})\otimes  {\cal E}) \]
is injective.
\end{satz}
\begin{Beweis}{Proof} Consider the diagram
\[
\begin{CD}
H^{b} (X,\omega_{X} (D + D^{\prime})\otimes  {\cal E}) \>>>
\h^{n+b} (X,\Omega^{\bullet}_{X} (\log \ D)\otimes  {\cal E}(D^{\prime})) \\
\A A \beta A \A A \gamma A \\
H^{b} (X,\omega_{X} (D)\otimes  {\cal E}) \> \alpha >>
\h^{n+b} (X,\Omega^{\bullet}_{X} (\log \ D)\otimes  {\cal E}).
\end{CD}
\]
$\alpha$ is injective by the $E_{1}$-degeneration (see (A.25))
, $\gamma$ is an isomorphism by (2.10) and hence $\beta$ is injective.\\
\end{Beweis}
The lemma (4.2) or its variant (4.3) implies
that for all $b$ the natural restriction maps
\[ H^{b}(X, {\cal E}) \>>> H^{b}
(D^{\prime}, {\cal O}_{D^{\prime}}\otimes  {\cal E}) \]
are the zero maps. For higher differential forms this remains
true, if $D^{\prime}$ is a non-singular divisor:
%
\begin{satz}{Lemma (Vanishing for restriction maps II)}
\index{General vanishing theorem!for restriction maps}Assume that
\[ \nabla : {\cal E} \>>> \Omega^{1}_{X} (\log \ D)\otimes  {\cal E} \]
satisfies $E_{1}$-degeneration. Let
$D^{\prime}$ be a non-singular subdivisor of $D$ and assume that for all
components $D_{j}$ of $D'$ the map $Res_{D_{j}} (\nabla)$ is an isomorphism.
(For example this follows from condition (!) in (2.8,b)).\\
Then the restriction (see (2.3))
\[ H^{b} (X,\Omega^{a}_{X} (\log \ (D-D^{\prime}))\otimes  {\cal E})
\>>>  H^{b} (D^{\prime}, \Omega^{a}_{D^{\prime}}
(\log \ (D - D^{\prime}) \mid_{D^{\prime}})\otimes  {\cal E}) \]
is zero for all $a$ and $b$.
\end{satz}
\begin{Beweis}{Proof} As we have seen in (2.6,b) the restriction
map factors through
\[ H^{b} (\nabla_{a}) : H^{b} (X,\Omega^{a}_{X} (\log \ D)\otimes  {\cal E})
\>>> H^{b} (X,\Omega^{a+1}_{X} (\log \ D)\otimes  {\cal E}) \]
provided $Res_{D_{j}} (\nabla)$ is an isomorphism on the different
components $D_{j}$ of $D^{\prime}$. By
$E_{1}$-degeneration, $H^{b} (\nabla_{a})$ is the zero map (see (A.25)).\\
\end{Beweis}

Before we are able to state the global vanishing for ${\cal E}$ or
$\Omega^{a}_{X} (\log \ D)\otimes  {\cal E}$ we need some more notations.
\begin{defn}{Definition} Let $U \subset X$ be an open subscheme
and let $B$ be an effective divisor with $B_{red} = X-U$.
Then we define the {\it (coherent)\index{Cohomological dimension!coherent}
cohomological dimension } of $(X,B)$ to be the least integer $\alpha$ such that
for all coherent sheaves ${\cal F}$ and all $k > \alpha$ one finds some
$\nu_0 > 0$ with $H^k (X, {\cal F}(\nu \cdot B)) = 0$ for all multiples
$\nu$ of $\nu_0$. Finally, for the reduced divisor $D = X - U$, we
write \\
\ \\ \index{$cd(X,D)$}  \index{Cohomological dimension!$cd(X,D)$}
$cd(X,D) = \mbox{Min} \{$ \parbox[t]{8.5cm}{$\alpha$ ; there exists some effective
divisor $B$ with $B_{red} = D$, such that $\alpha$ is the cohomological
dimension of $(X,B) \}.$}\\ \
\end{defn}
\begin{defn}{Examples}\\
a) For $D = X - U$ the embedding $\iota : U \rightarrow X$ is affine and
for a coherent sheaf ${\cal G}$ on $X$ we have
\[ H^{b} (U, {\cal G} \mid_{U}) = H^{b} (X, \iota_{\ast} ({\cal G} \mid_{U})) =
\lim_{\scriptstyle \rightarrow \atop \scriptstyle \alpha \in \N} H^{b} (X,
{\cal G} \otimes {\cal O}_{X} (\alpha \cdot B)), \]
where $B$ is any effective divisor with $B_{red} = D$.
In particular, if $b > cd(X,D) $ we find
$$H^{b} (U, {\cal G} \mid_{U}) = 0$$
b) By Serre duality one obtains as well that for $b < n - cd(X,D)$
we can find $B>0$  such that for a locally free sheaf ${\cal G}$ and all
multiples $\nu$ of some $\nu_{0} > 0$ one has
\[ \dim H^{b} (X, {\cal G} \otimes {\cal O}_{X}
(- \nu \cdot B)) = 0.  \]
c) If $D$ is the support of an effective ample divisor,
then Serre's vanishing theorem (see (1.1)) implies
$cd(X,D) = 0$. We are mostly interested in this case, hopefully an
excuse for the clumsy definition given in (4.6).
\end{defn}
\begin{satz}{Lemma (Vanishing for cohomology
groups)}\index{General vanishing theorem!for cohomology groups} \\
Assume that $X$ is projective and that
$$\nabla : {\cal E} \>>> \Omega^{1}_{X} (\log \ D)\otimes  {\cal E}$$
satisfies the $E_{1}$-degeneration (see (4.1)).\\
a) If $\nabla$ satisfies the condition $(\ast)$ of (2.8) and if $a + b > n +
cd(X,D)$, then
\[ H^{b}(X, \Omega^{a}_{X} (\log \ D)\otimes  {\cal E}) = 0. \]
b) If $\nabla$ satisfies the condition $(!)$ of (2.8) and if $a + b < n -
cd(X,D)$, then
\[ H^{b}(X, \Omega^{a}_{X} (\log \ D)\otimes  {\cal E}) = 0. \]
\end{satz}
\begin{Beweis}{Proof} Let us choose $\alpha \in \Z$ with $\alpha \geq 0$ in
case a) and with $\alpha \leq 0$ in case b). For $B \geq D$, (2.9) tells us that
\[ \Omega^{\bullet}_{X} (\log \ D)\otimes  {\cal E} \ \ \mbox{and \ \
$\Omega^{\bullet}_{X} (\log \ D)\otimes  {\cal E}(\alpha \cdot B)$} \]
are quasi-isomorphic.
In both cases we have a spectral sequence\\
\ \\
$ E_{1}^{ab} = H^{b} (X,\Omega^{a}_{X}
(\log \ D)\otimes  {\cal E}(\alpha \cdot B)) \Longrightarrow $
\[ \ \ \ \ \ \ \ \ \ \ \ \ \Longrightarrow \h^{a+b} (X,
\Omega^{\bullet}_{X} (\log \ D)\otimes  {\cal E}(\alpha \cdot B)) = \h^{a+b} (X,
\Omega_{X}^{\bullet} (\log \ D)\otimes  {\cal E}). \]
By assumption this spectral sequence degenerates for $\alpha = 0$ and, for
arbitrary $\alpha$ we have (see (A.16))
\[ \sum_{a+b=l} \dim \ H^{b} (X,\Omega^{a}_{X} (\log \ D)\otimes  {\cal E}) =
\dim \ \h^{l} (X,\Omega^{\bullet}_{X} (\log \ D)\otimes  {\cal E}) \]
\[ \leq \sum_{a+b=l} \dim \ H^{b} (X,\Omega^{a}_{X}
(\log \ D)\otimes  {\cal E}(\alpha \cdot B)). \]
By definition of $cd(X,D)$ we can choose $B$  such that
the right hand side is zero for $l > n + cd(X,D)$ and all $\alpha > 0$
in case a), or $l < n - cd(X,D)$ and all $\alpha < 0$ in case b).\\
\end{Beweis}
The same argument shows:
\begin{satz}{Variant} In (4.8) we can replace a) and b) by:\\
c) Let $D_{\ast}$ and $D_{!}$ be effective divisors, both
smaller than $D$, and assume that
\begin{quote}
i) For all components $D_{j}$ of $D_{\ast}$ and all
$\mu \in \N - \{ 0 \}$
\[ Res_{D_{j}} (\nabla) - \mu \cdot id_{{\cal O}_{D_{j}}} \]
is an isomorphism.\\
ii) For all components $D_{j}$ of $D_{!}$ and all $\mu \in \N$
\[ Res_{D_{j}} (\nabla) + \mu \cdot id_{{\cal O}_{D_{j}}} \]
is an isomorphism.
\end{quote}
Then
$$H^{b} (X,\Omega^{a}_{X} (\log \ D)\otimes  {\cal E}) = 0$$
for $a+b > n + cd(X,D_{\ast})$ and for $a+b < n - cd(X,D_{!})$.
\end{satz}
\ \\
\small
{\bf The analytic case}
\ \\
As we have seen in the proof of (4.8) the condition $(\ast)$
implies that
\[ \h^{l} (X,\Omega^{\bullet}_{X} (\log \ D)\otimes  {\cal E}) = 0
\ \ \mbox{ for \ \ $l > n + cd(X,D)$}. \]
For $k = \C$, this is not the best possible
result. In fact, as mentioned in (2.12), $(\ast)$ implies that over $\C$
\[ \h^{l} (X, \Omega_{X}^{\bullet} (\log \ D)\otimes  {\cal E}) = 0
\ \ \mbox{for \ \ $l > n + r (X - D)$} \]
where $r(X-D)$ is the least number $\alpha$ such that $H^{l}(X-D, V) = 0$
for all locally constant systems $V$ on $X-D$ and $l > n+\alpha$.\\
\ \\
(2.12) and the \ \ $E_{1}$-degeneration asked for in (4.8) and (4.9) imply
immediately that ``$cd( \ \ )$'' in 4.8 and 4.9 can be replaced by
``$r( \ \ )$''. \index{$r(U)$} \\
As we will see, $r(X - D)$ might be smaller than $cd(X,D)$.
For the results which follow we only know, at present, proofs by analytic
methods.
\begin{defn}{Definition} Let $U$ be an
algebraic irreducible variety and $g : U \rightarrow W$ a
morphism. Then define\index{$r(g)$}  \index{Cohomological dimension!$r(g)$}
\[
\begin{array}{ll}
r(g) =  \mbox{Max} \{& \dim \ \Gamma - \dim \ g(\Gamma) - \mbox{codim} \ \Gamma ;\\
& \Gamma \mbox{ irreducible closed subvariety of $U \ \} $}
\end{array}
\]
\end{defn}
\begin{defn}{Properties}\\
a) \[
\begin{array}{ll}
r(g) = \mbox{Max} \{& \dim \ \mbox{(generic fibre of $g \mid_{\Gamma}) - \mbox{codim} \
\Gamma$} ; \\ & \Gamma
\mbox{ irreducible closed subvariety of $U \ \} $ }
\end{array}
\]
b) If $b$ denotes the maximal fibre dimension for $g$,
then
\[ r(g) \leq \mbox{Max} \{ \dim \ U - \dim \ W ; \ \ b -1 \}. \]
c) If $U^{\prime} \subseteq U$ is open and dense, then $r(g \mid_{U^{\prime}})
\leq r (g)$.\\
d) If $\Delta \subseteq U$ is closed then $r(g \mid_{\Delta}) \leq r(g) + \mbox{codim}_{U} \
(\Delta)$.
\end{defn}
%
\begin{Beweis}{Proof} a) and c) are obvious and b) follows from a).
For d) one remarks that for $\Gamma \subset \Delta$ one has
\[ \mbox{codim}_{\Delta} \ (\Gamma) = \mbox{codim}_{U} \ (\Gamma) - \mbox{codim}_{U} \ (\Delta). \]
\end{Beweis}
\begin{satz}{Lemma} (Improvement of 4.8 using analytic methods)
\index{General vanishing theorem!with analytic methods}\\
{\it Let $X$ be a projective manifold defined over an
algebraically closed field $k$ of characteristic zero. Assume that
$$\nabla : {\cal E} \>>> \Omega_{X}^{1}
(\log \ D)\otimes  {\cal E}$$
is an integrable connection satisfying
the $E_{1}$-degeneration and let
\[ g : X - D \>>> W \]
be a proper surjective morphism to an affine variety $W$.\\
a) If $\nabla$ satisfies the condition $(\ast)$ of (2.8) then
\[ H^{b} (X,\Omega^{a}_{X} (\log \ D)\otimes  {\cal E}) = 0 \]
for $a+b > n + r (g)$.\\
b) If $\nabla$ satisfies the condition (!) of (2.8) then
\[ H^{b} (X,\Omega^{a}_{X} (\log \ D)\otimes  {\cal E}) = 0 \]
for $a+b < n - r(g). $}
\end{satz}
{\sc Proof:} By flat base chance we can replace $k$ by any other field $k^{\prime}$,
such that $X, D, {\cal E}$ are defined over $k^{\prime}$. Hence, we may assume
that $k = \C$. \\
By GAGA (see \cite{Ser56}) we may assume in (4.12) that all the
sheaves and $\nabla$ are analytic. Then, by (2.12) and by the $E_{1}$-degeneration
it is enough to show:\\
%
\begin{satz}{Lemma} Let $U$ be an analytic manifold, $W$ be an affine manifold and\\
$g: U \rightarrow W$ be a proper morphism. Then, for all local
constant systems $V$ on $U$ and $l > \dim \ (U) + r(g)$ one has
$ H^{l} (U,V) = 0 $.
\end{satz}
\begin{Beweis}{Proof (see \cite{EV89})}
The sheaves $R^{a}g_{\ast} V$ are analytically constructible sheaves
(\cite{Ver76}) and their support
$$S_{a} = {\rm Supp} (R^{a}g_{\ast} V)$$ must be a Stein space,
hence
$$H^{b} (W, R^{a}g_{\ast} V) = 0 \mbox{ \ for \ } b > \dim \ S_{a}.$$ However, the
general fibre of $g \mid_{g^{-1}(S_{a})}$ must have a dimension larger than or
equal to $\frac{a}{2}$. Hence $$2 \cdot (\dim \ g^{-1} (S_{a}) - \dim \ S_{a})
\geq a$$ and $H^{b}(W, R^{a}g_{\ast} V) = 0$ for
\[ a+b > n + r (g) \geq 2 \cdot \dim \ g^{-1} (S_{a}) - \dim \ S_{a}
\geq a+ \dim \ S_{a}. \]
By the Leray spectral sequence (A.27)
\[ E_{2}^{ba} = H^{b} (W, R^{a} g_{\ast} V) \Longrightarrow H^{a+b} (U,V) \]
one obtains (4.13).\\
\end{Beweis}
\begin{defn}{Remark}
If $W$ is affine and $g : U \rightarrow W$ obtained by blowing up a point,
then  for $X$ and $D$ as in (4.6) one has
$cd(X,D) = \dim \ U-1$, whereas $r(g) = \dim \ U-2$.
\end{defn}
\normalsize
\ \\
\Paragraph{Vanishing theorems for invertible sheaves}
\ \\
In this lecture we will deduce several known generalizations of the Kodaira-Nakano
vanishing theorem by applying the vanishing (5.1) obtained for ``integral parts of
$\Q$-divisors'' from (3.2), combined with (4.2). Needless to say that
in all those corollaries of (5.1) one loses some information and that
it might be more reasonable to try to work with (5.1) or correspondingly with
(6.2) directly, whenever it is possible.\\
\ \\
Let us remind you, that the proof of (3.2) is not yet complete. The necessary
arguments needed to show the $E_{1}$-degeneration will only be presented in
Lecture 10.\\
\ \\
Very quickly we will have to restrict ourselves to characteristic zero.
One reason is that the condition $(\ast)$ and $(!)$ are too
much to ask for in characteristic $p \neq 0$. But more substantially, most of
our proofs will start with ``blow up $B$ to get a normal crossing
divisor'', hence with an application of H. Hironaka's theorem
on the existence of desingularizations.\\

Let us start with (4.2). For simplicity, we restrict ourselves
to $i=1$ and ${\cal L} = {\cal L}^{(1)}$. By (3.3) we
are not losing any information.
\Abschnitt{\bf Vanishing for restriction maps related to
$\Q$-divisors:} \index{Vanishing theorem!for restriction maps related to
$\Q$-divisors}
{\it Let $X$ be a projective
manifold defined over an algebraically
closed field $k$, let ${\cal L}$ be an invertible sheaf, $N \in \N - \{ 0 \}$
and let $$D = \sum_{j=1}^{r} \alpha_{j}D_{j}$$ be a normal crossing divisor
with $ 0 < \alpha_{j} < N$ and ${\cal L}^N = {\cal O}_X(D)$. Let
$$D^{\prime} = \sum_{j=1}^{r} \mu_{j} D_{j}$$
be an effective divisor. Then one has:\\
{\rm a)} If $\Char \ (k) = 0$ then for all $b$ the natural morphism
\[ H^{b} (X, {\cal L}^{-1} (-D^{\prime})) \>>> H^{b}(X,
{\cal L}^{-1}) \]
is surjective and hence the morphism
\[ H^{b} (X,\omega_{X} \otimes {\cal L}) \>>> H^{b}
(X,\omega_{X} (D^{\prime}) \otimes {\cal L}) \]
injective.\\
{\rm b)} If $\Char \ (k) = 0$ and if $C$ is a reduced divisor without common component
with $D$ such that $D+C$ is a normal crossing divisor, then for all $b$
the natural morphism
\[ H^{b} (X, {\cal L}^{-1}(-C-D^{\prime})) \>>> H^{b}
(X, {\cal L}^{-1} (-C)) \]
is surjective and hence the morphism
\[ H^{b} (X, \omega_{X} (C) \otimes {\cal L}) \>>> H^{b}
(X, \omega_{X} (D^{\prime} + C) \otimes {\cal L}) \]
injective.\\
{\rm c)} If $\Char \ (k) = p \neq 0$, then a) and b)
hold true under the additional assumptions:}
\begin{quote}{\it
i) $X$ and $D$ (as well as $C$) satisfy
the lifting property (8.11) and \\
$\dim \ (X) \leq p$.\\
\ \\
ii) $N$ is prime to $p$.\\
\ \\
iii) For all $j$ and $0 \leq \mu \leq \mu_{j} -1$ one has $\alpha_{j}
+ \mu \cdot N \not\equiv 0 \ mod \ p$.}
\end{quote}
\begin{Beweis}{Proof} By (3.2,c) $ {\cal O}_{X} (-C) \otimes {\cal L}^{-1}$
has a logarithmic connection $\nabla$ with poles along $C+D_{red}$
satisfying $E_{1}$-degeneration, and
$Res_{D_{j}} (\nabla) = \frac{\alpha_{j}}{N}$. Hence (5.1) follows from
(4.2) and (4.3) or (4.4).\\
\end{Beweis}
\begin{satz}{Corollary (Kodaira \cite{Kod53}, Deligne, Illusie
\cite{DI87})} \index{Vanishing theorem!in characteristic $p>0$}
Let $X$ be a projective manifold and ${\cal L}$ an invertible sheaf.
If $\Char \ (k) = p > 0$, then assume in addition that $X$ and ${\cal L}$ admit
a lifting to $W_{2}(k)$ (8.11) and that $\dim \ X \leq p$. Then,
if ${\cal L}$ is ample,
\[ H^{b} (X, {\cal L}^{-1}) = 0 \ \ \mbox{for} \ \ b < n = \dim \ (X) \]
\end{satz}
\begin{Beweis}{Proof}
Choose $N$, prime to $p = \Char \ (k)$, such that
\[ H^{b}(X, {\cal L}^{-N-1}) = H^{n-b}(X, \omega_{X} \otimes {\cal L}^{N+1})
= 0 \]
for $b < n$, and such that ${\cal L}^{N}$ is generated by global sections.
If $D$ is a general section of ${\cal L}^{N}$, then we can
apply (5.1) and find
\[ H^{b}(X, {\cal L}^{-1} (-D)) \>>> H^{b} (X, {\cal L}^{-1}) \]
to be surjective. Since the group on the left hand side is zero, we
are done. \ \\
\end{Beweis}
If $\Char \ (k) = p$, then we will see in (11.3) that it is sufficient to assume
that $X$ lifts to $W_{2} (k)$. The condition that ${\cal L}$ lifts as
well is not necessary.
\begin{defn}{Definition} Let $X$ be a projective variety and
${\cal L}$ be an invertible sheaf on $X$. If $H^{0} (X, {\cal L}^{\nu})
\neq 0$, the sections of ${\cal L}$ define a rational map
\[ \phi_{\nu} = \phi_{{\cal L}^{\nu}} : X \>>> \P (H^{0} (X,
{\cal L}^{\nu})). \]
The {\it Iitaka-dimension} $\kappa ({\cal L})$ of ${\cal L}$ is given by
\index{Iitaka-dimension}\index{Kodaira-dimension}\index{$\kappa({\cal L})$}
\[ \kappa({\cal L}) =
\left\{
\begin{array}{ll}
- \infty  & \mbox{ if} \ H^{0} (X, {\cal L}^{\nu}) = 0  \ \mbox {for
all } \nu  \\
\ & \ \\
\mbox{Max}  \{ \dim  \phi_{\nu} (X); \ H^{0}(X, {\cal L}^{\nu}) \neq 0
\} &  \mbox{ otherwise} \\
\end{array}
\right. \]
\end{defn}
\begin{defn}{Properties} For $X$ and ${\cal L}$ as above one has:\\
\ \\
a) $\kappa({\cal L}) \in \{ - \infty, 0, 1, \ldots, \dim \ X \} $.\\
\ \\
b) If $H^{0} (X, {\cal L}^{\nu}) \neq 0$ for some $\nu > 0$
then one can find $a,b \in \R$, $a,b > 0$, such that
\[ a \cdot \mu^{\kappa({\cal L})} \leq \dim \ H^{0} (X, {\cal L}^{\nu \cdot
\mu}) \leq b \cdot \mu^{\kappa({\cal L})}
\mbox{ \ for all \ } \mu \in \N - \{ 0 \}.\]
c) If $\kappa({\cal L}) \neq - \infty$, then
\[ \kappa({\cal L}) = \mbox{ tr.deg }(\bigoplus_{\mu \geq 0} H^{0}
(X, {\cal L}^{\mu})) \ -1. \]
d) One has $\kappa({\cal L}) = \dim \ X$, if and only if for some $\nu > 0$
and some effective divisor $C$ the sheaf ${\cal L}^{\nu} (-C)$ is ample.\\
\ \\
e) If ${\cal A}$ is very ample and $A$ the zero divisor of a general
section of ${\cal A}$, then
\[ \kappa({\cal L}\mid_{A}) \geq Min \ \{ \kappa({\cal L}), \dim \ A \}. \]
\end{defn}
\begin{Beweis}{Proof} a), b) and c) are wellknown and their proof can be found,
for example, in \cite{Mor87}, \S 1.\\
For d) let $A$ be an ample effective divisor. For $n = \dim \ X$ and some \\
$\nu \in \N - \{ 0 \}$ one finds $a,b \in \R$, $a,b > 0$, with
$a \cdot \mu^{n} < \dim \ H^{0} (X, {\cal L}^{\nu \cdot \mu})$ and
$\dim \ H^{0}(X, {\cal L}^{\nu \cdot \mu}|_{A}) < b \cdot \mu^{n-1}$.
Hence, the exact sequence
\[0 \>>> H^{0} (X, {\cal L}^{\nu \cdot \mu} (-A)) \>>>
H^{0} (X, {\cal L}^{\nu \cdot \mu}) \>>> H^{0}
(A, {\cal L}^{\nu \cdot \mu}\mid_{A}) \]
shows that for some $\mu$ we have ${\cal O}_{X}(A)$ as a subsheaf of
${\cal L}^{\nu \cdot \mu}$. On the other hand, if ${\cal A} \subset
{\cal L}^{\nu}$ is ample then
\[ n = \kappa({\cal A}) \leq \kappa({\cal L}^{\nu}) = \kappa({\cal L}). \]
If $\kappa({\cal L}) = n$ in e), then, using d) for example, $\kappa({\cal L}\mid_{A})
= n-1$. \\
If $\kappa({\cal L}) < n$, then $H^{0}(X,
{\cal O}_{X} (-A) \otimes {\cal L}^{\nu}) = 0$ for all $\nu$ and b) implies
$$\kappa({\cal L} \mid_{A})
\geq \kappa({\cal L}) .$$
\end{Beweis}
For our purposes we can take (5.4,b) as definition of $\kappa({\cal L})$, and we
only need to know (5.4,d) and (5.4,e).
\begin{defn}{Definition} An invertible sheaf ${\cal L}$ on $X$ is called
\begin{quote}
a) {\it semi-ample}, if for some $\mu >0$ the sheaf ${\cal L}^{\mu}$ is
generated by global sections.\index{Semi-ample}\\
\ \\
b) {\it numerically effective (nef)} if for all curves $C$ in $X$ one has
\[ \deg \ ({\cal L} \mid_{C}) \geq 0. \]\index{Numerically effective (nef)}
\end{quote}
\end{defn}

The proof of (5.2) can be modified to give in characteristic zero a
stronger statement:
%
\begin{satz}{Corollary} Let $X$ be a projective manifold defined
over a field $k$ of characteristic zero and let ${\cal L}$ be an
invertible sheaf.\index{Vanishing
theorem!Koll\'ar}\index{Vanishing theorem!Grauert Riemenschneider}\\
\ \\
a) {\rm (Koll\'{a}r \cite{Kol86}) } \\
If ${\cal L}$ is semi-ample and $B$ an effective
divisor with
$H^{0} (X, {\cal L}^{\nu} (-B)) \neq 0$ for some
$\nu > 0$, then the natural maps
\[ H^{b} (X, {\cal L}^{-1} (-B)) \>>> H^{b} (X, {\cal L}^{-1}) \]
are surjective for all $b$, or, equivalently, the adjunction map
\[ H^{b} (X, {\cal L} \otimes \omega_{X} (B)) \>>> H^{b} (B, {\cal L}
\otimes \omega_{B}) \]
is surjective for all $b$.\\
\ \\
b) {\rm (Grauert-Riemenschneider \cite{GR70})} \\
If ${\cal L}$
is semi-ample and $\kappa({\cal L})
= n = \dim \ X$, then
$$H^{b}(X, {\cal L}^{-1}) = 0 \mbox{ \  for \ } b < n.$$
\end{satz}
\begin{Beweis}{Proof} Obviously a) and b) are compatible with blowing ups
$\tau : X^{\prime} \rightarrow X $. In fact, using the Leray spectral
sequence (A.27) we just
have to remember that
\[
R^{b} \tau_{\ast} \tau^{\ast} {\cal L} = {\cal L} \otimes R^{b}
\tau_{\ast} {\cal O}_{X^{\prime}} =
\left\{
\begin{array}{ll}
{\cal L} & \ \ \mbox{for \ \ $ b = 0$}\\
0 & \ \ \mbox{for \ \ $ b \neq 0$}\\
\end{array}
\right. \]
(See (5.13) for a generalization).\\
Hence we may assume in a) that ${\cal L}^{\nu} = {\cal O}_{X} (B+C)$ for an
effective normal crossing divisor $B+C$. We can choose some $\mu$,
with
$$[ \frac{B+C}{\mu} ]= 0$$ and such that ${\cal L}^{\mu}$ is generated
by its global sections. If $D_{1}$ is a general divisor of
${\cal L}^{\mu}$, i.e. the zero set of a general $s \in H^{0}(X,
{\cal L}^{\mu})$, then $D = D_{1} + B + C$ has normal crossings and
$[ \frac{D}{\mu} ] = 0$. Hence, for $N = \nu + \mu$ and
$D^{\prime} = B$ the assumptions of (5.1,a) hold true and we obtain a).\\
\ \\
For b), let us choose some divisor $C$ and some $\nu$ such that ${\cal L}^{\nu}
(-C) = {\cal A}$ is ample. Replacing ${\cal A}$ by some multiple,
we may assume by Serre's vanishing theorem (1.1) that
\[ H^{b} (X, {\cal L}^{-1} \otimes {\cal A}^{-1}) = H^{n-b} (X, \omega_{X}
\otimes {\cal L} \otimes {\cal A} ) = 0 \]
for $b < n$, and that ${\cal A} = {\cal O}_{X}(B)$ for some
divisor $B$. By a)
\[ H^{b} (X, {\cal O}_{X} (-B) \otimes {\cal L}^{-1}) \>>> H^{b} (X,
{\cal L}^{-1}) \]
is surjective. One obtains b), since the left hand side is zero.
\ \\ \end{Beweis}

It is not difficult to modify both parts of this proof to include in b)
the case that ${\cal L}$ is nef and $\kappa({\cal L}) = \dim \ X$.
Moreover, considering very ample divisors on $X$ and using induction on
$\dim(X)$, one can as well remove the assumption
``$\kappa({\cal L}) = \dim(X)$'' and obtain the vanishing for
$b< \kappa ({\cal L})$.\\
We leave the details to the reader. Those techniques will appear in
(5.12) anyway, when we prove a more general statement.
\begin{satz}{Lemma} For an invertible sheaf ${\cal L}$ on a projective
manifold $X$ the following two conditions are equivalent:
\begin{quote}
{\rm a)} ${\cal L}$ is numerically effective.\\
\ \\
{\rm b)} For an ample sheaf ${\cal A}$ and all $\nu > 0$ the sheaf
${\cal L}^{\nu} \otimes {\cal A}$ is ample.
\end{quote}
\end{satz}
\begin{Beweis}{Proof}
By Seshadri's criterion ${\cal A}^{\prime}$ is ample if and only if
for some $\epsilon > 0$ and all curves $C$ in $X$
$$\deg \ ({\cal A}^{\prime} \mid_{C}) \geq \epsilon \cdot m (C)$$
where $m(C)$ is the maximal multiplicity of points on $C$.
\ \\ \end{Beweis}
\begin{satz}{Lemma} For $X, {\cal L}$ as in (5.7), assume that ${\cal L}$
is numerically effective (and, if $\Char \ (k) = p \neq 0$,
that $X$ and ${\cal L}$ satisfy the lifting property
(8.11) and that $\dim \ X \leq p$). Then one has:\\
{\rm a)} $\kappa({\cal L}) = n = \dim \ X$, if and only if $c_{1} ({\cal L})^{n} > 0$
(where $c_{1} ({\cal L})$ is the Chern class of ${\cal L}$).\\
\ \\
{\rm b)} For $b \geq 0$ and for all invertible sheaves ${\cal F}$ one has
a constant $c_{b} > 0$ with
\[ \dim \ H^{b} (X, {\cal F} \otimes {\cal L}^{\nu}) \leq c_{b} \cdot
\nu^{n-b} \mbox{ \ \ for all \ \ } \nu \in \N. \]
\end{satz}
\begin{Beweis}{Proof} a) follows from b) and from the Hirzebruch-Riemann-Roch
theorem which tells us that $\chi (X, {\cal L}^{\nu})$ is a polynomial of
$\deg \ n$ with highest coefficient
\[ \frac{1}{n!} \cdot c_{1} ({\cal L})^{n}. \]
For b) we assume by induction on $\dim \ X$, that it holds true for all
hypersurfaces $H$ in $X$. We can choose an $H$, which satisfies
\[ H^{b} (X, {\cal O}_{X} (H) \otimes {\cal L}^{\nu} \otimes {\cal F}) = 0. \]
In fact, we just have to choose $H$ such that ${\cal F} \otimes
\omega_{X}^{-1} \otimes {\cal O}_{X} (H)$ is ample. Then by (5.7)
\[ {\cal F} \otimes \omega_{X}^{-1} \otimes {\cal O}_{X} (H) \otimes
{\cal L}^{\nu} \]
will be ample for all $\nu \geq 0$ and the vanishing required holds true
by (5.2). From the exact sequence
\[ 0 \>>> {\cal F} \otimes {\cal L}^{\nu} \>>> {\cal F} \otimes
{\cal L}^{\nu} \otimes {\cal O}_{X} (H) \>>> {\cal F} \otimes
{\cal O}_{H} (H) \otimes {\cal L}^{\nu} \>>> 0 \]
we obtain an isomorphism
\[ H^{b-1} (H, {\cal F} \otimes {\cal O}_{H} (H) \otimes {\cal L}^{\nu} )
\simeq H^{b} (X, {\cal F} \otimes {\cal L}^{\nu}) \]
for $b > 1$ and a surjection
\[ H^{0} (H, {\cal F} \otimes {\cal O}_{H} (H) \otimes {\cal L}^{\nu})
\>>> H^{1} (X, {\cal F} \otimes {\cal L}^{\nu}). \]
By induction we find $c_{b}$ for $b \geq 1$ and, since $H^{0} (X,
{\cal F} \otimes {\cal L}^{\nu})$ is bounded above by a polynomial
of $\deg \ \nu$, for $b = 0$ as well.
\ \\ \end{Beweis}
Even if ${\cal L}$ is nef, there is in general no numerical characterisation
of $\kappa({\cal L})$. For example, there are numerically effective
invertible sheaves $\cal L$ with $\kappa ({\cal L}) = -\infty$.
Following Kawamata \cite{Kaw85}, one defines:
\begin{defn}{Definition} Let ${\cal L}$ be a numerically effective
invertible sheaf. Then the {\it numerical Iitaka-dimension} is
defined as \index{Iitaka-dimension!numerical}
\index{Kodaira-dimension!numerical}\index{$\nu ({\cal L})$}
\[ \nu ({\cal L}) = \mbox{Min} \ \{ \nu \in \N - \{ 0 \}; \ \ c_{1}({\cal L})^{\nu}
\mbox{ numerically trivial } \} -1. \]
\end{defn}
\pagebreak[2]
\begin{defn}{Properties} Let $X, {\cal L}$ be as in (5.8). Then one has:
\begin{quote}
a) $\nu ({\cal L}) \geq \kappa({\cal L})$.\\
\ \\
b) If ${\cal L}$ is semi-ample then
\[ \nu ({\cal L}) = \kappa({\cal L}). \]
\end{quote}
\end{defn}
\begin{Beweis}{Proof} If $\nu ({\cal L}) = \dim \ X$ or $\kappa({\cal L}) = \dim \ X$,
then (5.8,a) gives $$\nu ({\cal L}) = \kappa ({\cal L}) = n = \dim \ X.$$
Hence we can assume both to be smaller than $n$. By (5.4,e)
we have for a general
hyperplane section $H$ of $X$
\[ \kappa({\cal L}\mid_{H}) \geq \kappa({\cal L}) \]
and obviously
\[ \nu ({\cal L}\mid_{H}) = \nu ({\cal L}). \]
By induction on $\dim \ (X)$ one obtains a). \\
For b) we may assume that
${\cal L} = \tau^{\ast} {\cal M}$ for a morphism $\tau : X \rightarrow Z$
with ${\cal M}$ ample and with $\dim \ (Z) = \kappa({\cal L})$. Then
\[ c_{1} ({\cal L})^{\nu} = \tau^{\ast} c_{1} ({\cal M})^{\nu} = 0 \]
if and only if $\nu > \dim \ Z$.
\ \\ \end{Beweis}
The following lemma, due to Y. Kawamata \cite{Kaw85},
is more difficult to prove,
and we postpone its proof  to the end of this lecture.
\begin{satz}{Lemma} For an invertible sheaf ${\cal N}$ on a projective manifold
$X$, defined over a field $k$ of characteristic zero, the following two
conditions are equivalent:\\
{\rm a)} ${\cal N}$ is numerically effective and $\nu ({\cal N}) =
\kappa({\cal N})$\\
{\rm b)} There exist a blowing up $\tau : Z \rightarrow X$, some $\mu_{0}
\in \N - \{ 0 \}$ and an effective divisor $C$ on $Z$ such that
\[ \tau^{\ast} {\cal N}^{\mu} \otimes {\cal O}_{Z}(-C) \]
is semi-ample for all
$\mu \in \N - \{ 0 \}$ divisible by $\mu_{0}$.
\end{satz}
\begin{satz}{Corollary} Let $X$ be a projective manifold defined over a
field $k$ of characteristic zero, let ${\cal L}$ be an invertible sheaf on $X$,
let $$D = \sum_{j=1}^{r} \alpha_{j} D_{j}$$ be a normal crossing divisor
and $N \in \N$. Assume that $$0 < \alpha_{j} < N \mbox{ \ for \ }
j = 1, \ldots, r.$$
Then one has:\\
{\rm a)} \index{Vanishing theorem!for restriction maps related
to $\Q$-divisors}
If ${\cal L}^{N}(-D)$ is semi-ample and $B$ an effective
divisor such that
\[ H^{0} (X, ({\cal L}^{N} (-D))^{\nu} \otimes {\cal O}_{X} (-B)) \neq 0 \]
for some $\nu > 0$, then for all $b$ the maps
\[ H^{b} (X, {\cal L}^{-1} (-B)) \>>> H^{b}(X, {\cal L}^{-1}) \]
are surjective.\\
{\rm b)} In a) one can replace ``semi-ample'' by the assumption that ${\cal L}^{N}
(-D)$ is numerically effective and
$$\kappa({\cal L}^{N}(-D)) = \nu ({\cal L}^{N}(-D)).$$
{\rm c) \ (Kawamata \cite{Kaw82} - Viehweg
\cite{Vie82})}\index{Vanishing theorem!Kawamata Viehweg}\\
If ${\cal L}^{N}(-D)$ is numerically effective
and
\[
c_{1} ({\cal L}^{N}(-D))^{n} > 0 , \]
then $$H^{b}(X, {\cal L}^{-1}) = 0 \mbox{ \ for \ }b < n.$$
{\rm d) \ (Kawamata \cite{Kaw82} - Viehweg \cite{Vie82})} \\
If ${\cal L}^{N} (-D)$ is numerically effective,
then $$H^{b}(X, {\cal L}^{-1}) = 0 \mbox{ \ for \ } b < \kappa({\cal L}).$$
{\rm e)} Part d) remains true if one replaces $\kappa ({\cal L})$ by
$\kappa ({\cal L} \otimes {\cal N}^{-1})$
for a numerically effective invertible sheaf
${\cal N}$.
\end{satz}
Again, the assumptions are compatible with blowing ups, except for\\
``$0 < \alpha_{j} < N$''. We need:
\begin{defn}{Claim} Let $\tau : X^{\prime} \rightarrow X$ be a proper
birational morphism and ${\cal M} = \tau^{\ast} {\cal L}.$
Assume that $ \Delta = \tau^{\ast} D $ has normal crossings.
Then for\index{Relative vanishing theorem!for \Q-divisors}
\[ {\cal M}^{(i)} = {\cal M} (- [\frac{i \cdot \Delta}{N} ]) \]
one has
\[ R^{b} \tau_{\ast} {\cal M}^{(i)^{-1}} =
\left\{
\begin{array}{ll}
{\cal L}^{(i)^{-1}} & \ \ \mbox{for \ \ $b = 0$} \\
0 & \ \ \mbox{for \ \ $b \neq 0$.}\\
\end{array}
\right. \]
\end{defn}
\begin{Beweis}{Proof}
We may assume that $X$ is affine and that ${\cal L} = {\cal O}_X$.
For $b=0$ claim (5.13) follows from the inequality
\[ [\frac{i\cdot \Delta}{N}] = [\frac{i\cdot \tau^* D}{N}] \geq
\tau^* [\frac{i\cdot D}{N}]. \]
In general, for $b\geq 0$, (5.13) follows from the
rationality of the singularities of the cyclic covers $Y$ and $Y'$ obtained
by taking the $N$-th rooth out of $D$ and $\Delta$. \\
In fact, let $Y''$ be a desingularization of $Y'$ and let
\[
\begin{CD}
Y'' \> \sigma >> Y' \> \delta >> Y \\
\noarr \V \pi' VV \V \pi VV \\
\noarr X' \> \tau >> X
\end{CD}
\]
be the induced morphisms. If $Y'$ has rational singularities, one has
by definition $R^a \sigma_* {\cal O}_{Y''} = 0$ for $a>0$. Hence,
if $Y$ has rational singularities as well,
\[
R^a (\delta \circ \sigma)_* {\cal O}_{Y''} = R^a \delta_* {\cal O}_{Y'} =0
\]
and
\[
R^a \tau_* (\pi'_* {\cal O}_{Y'}) =0,
\]
which implies (5.13).\\
\ \\
By (3.24) we know that $Y$ and $Y'$ have quotient singularities.
This implies that $Y$ and $Y'$ have rational singularities
(see for example \cite{Vie77}). Let us recall the proof:\\
\ \\
Let $Y$ be any normal variety with quotient singularities and
$\varphi: Z \>>> Y$ the corresponding Galois cover with $Z$ non singular.
Let $\delta : Y' \>>> Y $ be a desingularization such that
$D' = \delta^*( \Delta (Z/Y)) $ is a normal crossing divisor, where
$\Delta (Z/Y)$ denotes the set of ramified points in $Y$.
If $Z'$ is the normalization of $Y'$ in the function field of $Z$,
(3.24) tells us that $Z'$ has at most quotient singularities.
Let finally $\gamma : Z'' \>>> Z'$ be a desingularization.
Altogether we obtain
\[
\begin{CD}
Z'' \> \gamma >> Z' \> \delta' >> Z \\
\noarr \V \varphi' VV \V \varphi VV \\
\noarr Y' \> \delta >> Y
\end{CD}
\]
where $Z''$, $Z$ and $Y'$ are nonsingular.\\
\ \\
Let us assume that for all quotient singularities and for all
$a$ with $a_0 > a > 0$ we know that the $a$-th higher direct image of
the structure sheaf of the desingularization is zero.
Then the Leray spectral sequence gives an injection
\[
R^{a_0} \delta'_* {\cal O}_{Z'} = R^{a_0} \delta'_* (\gamma_* {\cal O}_{Z''})
\hookrightarrow R^{a_0} (\delta' \circ \gamma)_* {\cal O}_{Z''}.
\]
Since $\delta' \circ \gamma$ is a birational proper morphism of
non singular varieties
$$R^{a_0} (\delta' \circ \gamma)_* {\cal O}_{Z''} = 0.$$
Since the finite morphisms $\varphi$ and $\varphi'$ have no
higher direct images one obtains
\[
R^{a_0} \delta_* (\varphi'_* {\cal O}_{Z'}) = \varphi_*
(R^{a_0} \delta'_* {\cal O}_{Z'}) = 0.
\]
However, ${\cal O}_{Y'}$ is a direct summand of $\varphi'_* {\cal O}_{Z'}$
and hence $R^{a_0}\delta_*{\cal O}_{Y'}=0$.
\ \\ \end{Beweis}
\begin{Beweis}{Proof of 5.12} let us first reduce b) to a):\\
Applying (5.13) and replacing ${\cal M}$ by
${\cal M}^{(1)}$, we can assume
that the morphism $\tau : Z \rightarrow X$
in (5.11,b), applied to ${\cal N} = {\cal L}^{N} (-D)$, is an
isomorphism and that for the divisor $C$ in (5.11,b) $D+C$ is a normal
crossing divisor. ${\cal L}^{N \cdot \mu} (- \mu \cdot D - C)$ is semi-ample
for all $\mu$ divisible by $\mu_{0}$.
Choosing $\mu$ large enough, the multiplicities of $\mu \cdot D + C$
will be bounded above by $N \cdot \mu$.
Moreover, we can assume that
\[ H^{0} (X, {\cal L}^{N \cdot \mu} ( - \mu \cdot D - C)) \neq 0 \]
and hence, replacing $\mu$ again by some multiple, that
\[ H^{0} (X, ({\cal L}^{N \cdot \mu} ( - \mu {\cdot} D - C))^{\nu} \otimes
{\cal O}_{X} (-B)) \neq 0 \]
for some $\nu > 0$. Hence a) implies b).\\
\ \\
To prove a), let us write
\[ {\cal L}^{\nu {\cdot} N} (- \nu \cdot D) = {\cal O}_{X} (B + B^{\prime}) \]
or
\[ {\cal L}^{ \nu \cdot N} = {\cal O}_X ( \nu \cdot D + B + B^{\prime} ) \]
Blowing up, again, we can assume $D+B+B^{\prime}$ to be a normal crossing
divisor. For $\mu$ sufficiently large, we can assume that
$({\cal L}^{N} (-D))^{\mu}$ is generated by global sections. If
$H$ is zero set of a general section, then
\[ {\cal L}^{(\nu + \mu) \cdot N} = {\cal O}_{X} (H + (\nu + \mu) \cdot
D + B + B^{\prime}). \]
If $\mu$ is large enough, the multiplicities of the components of
\[ D^{\prime} = H + (\nu + \mu ) \cdot D + B + B^{\prime} \]
are smaller than $(\nu + \mu) \cdot N$ and, applying
(5.1,a) for $N^{\prime} = ( \nu + \mu) \cdot N$ and
${\cal L}^{N^{\prime}} = {\cal O}_{X} (D^{\prime})$, we obtain (5.12,a).\\
\ \\
Let us remark next, that
d), under the additional condition that
$\kappa({\cal L}) = n$, implies c):\\
In fact, $c_{1}({\cal L}^{N}(-D))^{n} > 0$ implies by (5.8,a) that
\[ n = \kappa({\cal L}^{N}(-D)) \leq \kappa({\cal L}^{N}) = \kappa({\cal L }). \]
To prove d), for $\kappa({\cal L}) = n$, we can apply (5.4,d). Hence we find
a divisor $C > 0$ and $\mu > 0$ with ${\cal L}^{\mu}(-C)$ ample.
Then by (5.7) $${\cal L}^{N \cdot \nu + \mu} ( - \nu \cdot D - C)$$
is ample for all $\nu$ and, by Serre's vanishing theorem (1.1)
\[ H^{b} (X, {\cal L}^{-1} \otimes ({\cal L}^{-N {\cdot} \nu - \mu} (\nu
{\cdot} D + C))^{\eta}) = 0 \ \ \mbox{for} \ \ b < n \]
and for $\eta$ sufficiently large. As in the proof of (5.6) or by (5.13)
this condition is compatible with blowing ups. Hence we may assume
$D + C$ to be a normal crossing divisor and, choosing $\nu$ large enough,
we may again assume that the multiplicities of $D' = \nu {\cdot} D + C$ are
smaller than $N' = N {\cdot} \nu + \mu$. Replacing $D'$ and $N'$ by some
high multiple we can assume in addition that ${\cal L}^{N'} (- D')$
is generated by global section and that
$$H^{b} (X, {\cal L}^{-N' - 1} (D'))
= 0 \mbox{ \ for \ } b < n.$$
For an effective divisor $B$ with
${\cal O}_{X} (B) = {\cal L}^{'N}(-D')$ we can apply a) and find
\[ 0 = H^{0} (X, {\cal L}^{-1} (-B)) \>>>  H^{0}
(X, {\cal L}^{-1}) \]
to be surjective.\\
For $\kappa({\cal L}) < \dim \ X$ part d) is finally reduced to the case $\kappa({\cal L})
= \dim \ X$ by induction:\\
Let $H$ be a general hyperplane such that
$$H^{b} (X, {\cal O}_{X} (-H) \otimes
{\cal L}^{-1}) = 0 \mbox{ \ for \ } b < n.$$ The exact sequence
\[ 0 \>>> {\cal O}_{X} (-H) \otimes {\cal L}^{-1} \>>>
{\cal L}^{-1} \>>> {\cal L}^{-1} \mid_{H} \>>> 0 \]
give isomorphisms
\[ H^{b} (X, {\cal L}^{-1}) \simeq H^{b} (H,{\cal L}^{-1} \mid_{H}) \]
for $b < n-1$. Since $\kappa({\cal L}) \leq n - 1$ we have
$\kappa({\cal L} \mid_{H}) \geq \kappa({\cal L})$ and both groups vanish for
$b < \kappa({\cal L})$ by induction on $\dim X$.\\
e) follows by the same argument: If $\kappa ({\cal L} \otimes {\cal N}^{-1}) =
\dim X$, then (5.4,c) and (5.7) imply that $\kappa({\cal L}) = \dim X$ as well.
For $\kappa ({\cal L} \otimes {\cal N}^{-1}) < \dim X$ again
$$\kappa ({\cal L}|_{H} \otimes {\cal N}^{-1}|_{H}) \geq
\kappa ({\cal L} \otimes {\cal N}^{-1})$$
and by induction one obtains e).\\
\end{Beweis}

Let us end this section by proving Kawamata's lemma (5.11) which was needed to reduce
(5.12,b) to (5.12,a):
\begin{Beweis}{Proof of (5.11)}
Let us assume b). Since $\tau^{\ast} {\cal N}^{\mu}$ is
nef, if and only if ${\cal N}$ is nef, and since $\nu ({\cal N}) = \nu
(\tau^{\ast} {\cal N})$, we can assume that $\tau$ is an isomorphism.
Moreover, we can assume $\mu_{0} = 1$.\\
``${\cal N}^{\mu}(-C)$ semi-ample for all $\mu > 0$'' implies that
${\cal N}$ is nef. One obtains from (5.10)
\[ \nu ({\cal N}) \geq \kappa({\cal N}) \geq \kappa ({\cal N}^{\mu}(-C)) =
\nu ({\cal N}^{\mu} (-C)). \]
For $\nu = \nu ({\cal N})$ the leading term in $\mu$ of
\[ c_{1} ({\cal N}^{\mu}(-C))^{\nu} = (\mu \cdot c_{1} ({\cal N}) - C)^{\nu} \]
is $\mu^{\nu} \cdot c_{1} ({\cal N})^{\nu}$. Since this term intersects
$H_{1} \cdot \ldots \cdot H_{n - \nu}$ strictly positively, for general hyperplanes
$H_{1}, \ldots, H_{n- \nu}$, we find $\nu \leq \nu({\cal N}^{\mu} (-C))$.\\
\ \\
To show the other direction, let $\phi_{\mu_{0}} : X \rightarrow Y$ be the
rational map
\[ X \>>> \phi_{\mu_{0}} (X) = Y \subset \P (H^{0}
(X, {\cal N}^{\mu_{0}})). \]
We can and we will assume that $\phi_{\mu_{0}}$ has a connected general fibre, that
$\dim \ (Y) = \kappa({\cal N})$ and, blowing $X$ up if necessary, that
$\phi_{\mu_{0}}$ is a morphism. For some effective divisor $D$ we have
\[ {\cal N}^{\mu_{0}} (-D) = \phi^*_{\mu_{0}} {\cal L}, \ \ \mbox{for
\ ${\cal L}$ \ ample on \ $Y$.} \]
If $F$ is a general fibre of $\phi_{\mu_{0}}$,
then $D\mid_{F}$ is nef.
\begin{defn}{Claim} $D\mid_{F}$ is zero.
\end{defn}
Assuming (5.14) we can blow up $Y$ and $X$ and assume that $D = \phi^*_{\mu_{0}}
\Delta$ for some divisor $\Delta$ on $Y$. For example, blowing up $Y$
one can assume that $\phi_{\mu_{0}}$ factors over a flat morphism $\phi ' :
X' \rightarrow Y$ and that ${\cal N}$ is the pullback of a sheaf
${\cal N}^{' \mu_{0}} ( - D') = \phi^{' \ast} {\cal L}$ for some semi-ample
sheaf ${\cal L}$ and by (5.14) $D' \subseteq \phi^{' \ast} \Delta$ for
some divisor $\Delta$ on $Y$. Since ${\cal N}'$ is numerically effective
$D' {\cdot} C \geq 0$ for all curves $C$ in $X'$ contained in a fibre
of $\phi '$. This is only possible if $D' = \phi^{' \ast} \Delta$.
Let us denote by $\tau : X \rightarrow Y$ the morphism obtained. We have
${\cal N}^{\mu_{0}} = \tau^{\ast} {\cal M}$ for some sheaf ${\cal M}$ on
$Y$. Of course $\kappa({\cal M}) = \dim Y$ and (5.12,b)
holds true for ${\cal M}$
on $Y$, i.e. ${\cal M}^{\mu}(- \Gamma)$ is ample for
some divisor $\Gamma > 0$ and all $\mu > > 0$. Then
\[ {\cal N}^{\mu \cdot \mu_{0}} (- \tau^{\ast} \Gamma ) \]
is semi-ample for all $ \mu > > 0$.
\ \\ \end{Beweis}
\begin{Beweis}{Proof of (5.14)} We may assume that $\mu_{0} = 1$. For $\phi =
\phi_{1}$, one has
\[ c_{1} ({\cal N}) = c_{1} (\phi^{\ast} {\cal L}) + D = \phi^{\ast}
c_{1} ({\cal L}) + D. \]
$D$ is effective, hence $c_{1} ({\cal N})^{\nu_{1}} \cdot c_{1} (\phi^{\ast}
{\cal L})^{\nu_{2}} \cdot D$ are semi-positive cycles, i.e. for
$ n = \nu_{1} + \nu_{2} + 1 + r $ one has
\[ H_{1} \cdot \ldots \cdot H_{r} \cdot c_{1}({\cal N})^{\nu_{1}}
\cdot c_{1} (\phi^{\ast}{\cal L})^{\nu_{2}} \cdot D \geq 0. \]
By definition
\[ 0 \equiv c_{1} ({\cal N})^{\nu + 1} = c_{1}({\cal N})^{\nu}
\cdot (c_{1} (\phi^{\ast} {\cal L}) + D)  \mbox{ \ for \ } \nu = \nu
({\cal N}) . \]
Since $c_{1}(\phi^{\ast} {\cal L})$
is also represented by an effective divisor, this is only possible if
\[ 0 \equiv c_{1}({\cal N})^{\nu} \cdot c_{1}(\phi^{\ast} {\cal L})
= c_{1} ({\cal N})^{\nu - 1} \cdot c_{1}(\phi^{\ast}{\cal L}) \cdot
(c_{1}(\phi^{\ast}{\cal L}) + D). \]
The same argument shows that $c_{1}(\phi^{\ast} {\cal L})^{2} \cdot c_{1}
({\cal N})^{\nu - 1} \equiv 0$ and after $\nu$ steps we get
\[ c_{1}(\phi^{\ast}{\cal L})^{\nu} \cdot c_{1} (\phi^{\ast} {\cal L})
+ c_{1} (\phi^{\ast} {\cal L})^{\nu} {\cdot} D \equiv 0 \]
and hence
\[ c_{1} (\phi^{\ast} {\cal L})^{\nu} \cdot D = F \cdot D = 0. \]
\end{Beweis}
\Paragraph{Differential forms and higher direct images }
\ \\
The title of this lecture is a little bit misleading. We want to apply
the vanishing theorems for differential forms with values in invertible
sheaves of integral parts of $\Q$-divisors (which follow
directly from (3.2), (4.8) and (4.13)) to some more concrete situations.\\
\ \\
For invertible sheaves themselves one obtains thereby different
proofs of (5.2), (5.6,b), (5.12,c) and (5.12,d) but, as far as
we can see, nothing more. For $\Omega^{a}_{X} \otimes {\cal L}^{-1}$ we obtain
the Kodaira-Nakano vanishing theorem and some generalizations.
Finally we consider the vanishing for higher direct images,
which can be reduced, as usually, to the global vanishing theorems. As
a straightforward application one obtains vanishing theorems for
certain non compact manifolds.\\
\ \\
In Lecture 5 we could at least point out some of the intermediate
steps which remain true in characteristic $p \neq 0$.
However, since $(\ast)$ and $(!)$ only make sense
in characteristic $0$,
we can as well assume throughout this chapter:
\begin{defn}{Assumptions} $X$ is a projective manifold defined
over an algebraically closed field $k$ of characteristic zero
and ${\cal L}$ is an invertible sheaf on $X$.
\end{defn}
\ \\
Global vanishing theorems in characteristic $p > 0$ will appear,
as far as it is possible, in Lecture 11.
\Abschnitt{Global vanishing theorem for integral parts of
$\Q$-divisors.} \index{Vanishing theorem!for logarithmic
differential forms with values in Kodaira
integral parts of $\Q$-divisors}
{\it For $X,{\cal L}$ as in (6.1) let
$$D = \sum_{j=1}^{r}\alpha_{j}D_{i}$$
be a normal crossing divisor, $N \in \N$ with
$0 < a_{j} < N$ for $j = 1 \ldots r$ and \\
${\cal L}^{N} = {\cal O}_{X} (D)$.
Then one has:\\
{\rm a)} $$H^{b} (X,\Omega^{a}_{X} (\log \ D) \otimes {\cal L}^{-1}) = 0,$$
for $a + b < n - cd (X,D)$ and for $a + b > n + cd (X,D)$.\\
{\rm b)} Let $A$ and $B$ be reduced divisors such that $D + A + B$ has normal
crossings and such that $A$, $B$ and $D$ have pairwise no common component.
Then
\[ H^{b} (X, \Omega_{X}^{a} (\log \ (A + B + D))(-B) \otimes {\cal L}^{-1})
= 0 \]
for $a + b < n -cd (X,D+B)$ and for $a + b > n + cd (X,D+A)$.\\
{\rm c)} If there exists a proper morphism
\[ g: X - D \>>> W \]
for an affine variety $W$, then one can replace $ cd(X,D)$ by $r(g)$
in a) (see (4.10)).}
\begin{Beweis}{Proof} By 3.2 ${\cal L}^{-1} (-B)$ has a logarithmic integrable
connection $\nabla$ with poles along $A + B + D$, such that the
$E_{1}$-degeneration holds true. Moreover, $Res_{D_{j}} (\nabla) \not\in \Z$ for
$j = 1, \ldots, r$. For a component $A_{j}$ of $A$ we have $Res_{A_{j}}
(\nabla) = 0$, and for a component $B_{j}$ of $B$ we have
$Res_{B_{j}} (\nabla) = 1$.
Hence a) follows from (4.8), b) from (4.9) and finally c) from (4.13).
\ \\ \end{Beweis}
\begin{satz}{Corollary} For $X, {\cal L}$ as in (6.1), assume that
${\cal L}^{N} = {\cal O}_{X}(D)$ for a normal crossing divisor
$D = \sum_{j=1}^{r} \alpha_{j} D_{i}$ with $ 0 < \alpha_{j} < N$ and
assume that there exists an ample effective
divisor $B$ with $B_{red} = D_{red}$.
Then $H^{b} (X, {\cal L}^{-1}) = 0$ for $b < n$.
\end{satz}
\begin{Beweis}{Proof} Apply (6.2,a), and (4.7,c).\\
\end{Beweis}
\begin{Beweis}{2${}^{\mbox{nd}}$ proof of (5.12,c), (5.12,d) and
(5.12.E).} As we have
seen in Lecture 5, it is enough to proof (5.12,d) for $\kappa({\cal L}^{N}
(-{D})) = \dim \ X$. Moreover, we may blow up, whenever we like.\\
\ \\
We can write (replacing $N$ and $D$ by some multiple)
\[ {\cal L}^{N}(-D) = \ \ {\cal A}(\Gamma) \]
for some effective divisor $\Gamma$ and some ample sheaf ${\cal A}$.
Blowing up, we can replace everything by some high multiple and
subtract some effective divisor $E$ from the pullback of ${\cal A}$
such that the sheaf obtained remains ample. Hence
one can assume $D + \Gamma$ to be a normal crossing divisor.
Since ${\cal L}^N(-D)$ is numerically effective,
we can replace ${\cal A}$ by ${\cal A} \otimes {\cal L}^N(-D)$
and repeating this we can assume that the multiplicities of $D+\Gamma$
are bounded by $N$. Finally, replacing again everything by some multiple
we are reduced to the case that ${\cal A}$ is very ample. Writing
\[ {\cal L}^{N} = {\cal O}_{X} (D^{\prime}) \ \ \mbox{for} \ \ D^{\prime}
= D + \Gamma + H, \]
$H$ a general divisor for ${\cal A}$, we can apply (6.3).\\
\end{Beweis}
\begin{satz}{Corollary (Akizuki-Kodaira-Nakano \cite{AN54})}
\index{Vanishing theorem!Akizuki Kodaira Nakano}
For $X, {\cal L}$ as in (6.1) assume
that ${\cal L}$ is ample. Then
$$H^{b} (X, \Omega_{X}^{a} \otimes {\cal L}^{-1})
= 0 \mbox{ \ for \ } a + b < n.$$ Moreover, if $A + B$ is a reduced normal crossing divisor,
the same holds true for
\[ H^{b} (X, \Omega_{X}^{a} (\log \ (A + B ))(-B) \otimes {\cal L}^{-1}). \]
\end{satz}
\begin{Beweis}{Proof} We can write ${\cal L}^{N} = {\cal O}_{X} (D)$ for
a non-singular divisor $D$ and we may even assume that $D + A + B$ is a reduced
normal crossing divisor. Moreover, for $N$ large enough,
$D+B$ and $D+A$ will both be ample and $$cd(X,D+B)=cd(D+A)=0.$$
By (2.3,b) one has an exact sequence
\[ \ldots \>>> H^{b-1} (D,\Omega_{D}^{a-1}
(\log \ (A+ B)\mid_{D})(-B\mid_{D}) \otimes {\cal L}^{-1})
\>>> \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \]
\[ \ \ \ \ \ \ \ \ \ \ \>>> H^{b} (X,
\Omega_{X}^{a} (\log \ (A+B))(-B) \otimes {\cal L}^{-1}) \>>> \]
\[ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \>>> H^{b} (X,
\Omega_{X}^{a} (\log \ (A + B + D))(-B) \otimes {\cal L}^{-1})
\>>> \ldots \]
By induction on $\dim \ X$ we can assume that the first group
 is zero for \\
 $a + b < n + 1 $ and by (6.2,b) the last group
is zero for $a + b \neq n$.
\ \\ \end{Beweis}
If one tries to weaken ``${\cal L}$ ample'' in this proof, one has to replace
it by some condition compatible with the induction step.
%
\begin{defn}{Definition} An invertible sheaf
${\cal L}$ is called {\it $l$-ample} if the following two
conditions hold true:\index{$l$-ample}
\begin{quote}
a) ${\cal L}^{N}$ is generated by global
sections, for some $N \in \N - \{ 0 \}$, and hence
$\phi_{N} : X \>>> \P (H^{0} (X, {\cal L}^{N}))$ a morphism. \\
\ \\
b) For $N$ as in a) \ $l \geq \mbox{Max} \{ \dim \ \phi_{N}^{-1} (z);
z \in \phi_{N} (X) \} .$
\end{quote}
\end{defn}
\begin{satz}{Corollary (A. Sommese \cite{SS85}, generalized in
\cite{EV89})} \index{Vanishing theorem!for differential forms with
values in $l$-ample sheaves}\\
For $X, {\cal L}$ as in (6.1) assume ${\cal L}$ to be $l$-ample. Then
\[ H^{b} (X, \Omega_{X}^{a} \otimes {\cal L}^{-1}) = 0 \]
for $a + b < \mbox{Min} \ \{\kappa ({\cal L}), \ \dim \ X - l + 1 \}$.
\end{satz}
\begin{Beweis}{Proof} Using the notation from (6.5), we have seen
in (4.11,b) that
$$r (\phi_{N}) \leq \mbox{Max} \ \{ \dim \ X - \kappa({\cal L}), \ l-1 \ \}.$$
Hence (6.6) is a special case of the following more technical statement .
\ \\ \end{Beweis}
\begin{satz}{Corollary} For $X, {\cal L}$ as in (6.1) let
$\tau : X \rightarrow Y$ be a morphism and let $E$ be an effective normal
crossing divisor with $\tau^{-1} (\tau (X - E)) = X - E$. If for
some ample sheaf ${\cal A}$ on $Y$ and for some $\nu > 0$ one has
${\cal L}^{\nu} = \tau^{*}{\cal A}$, then
\[ H^{b} (X, \Omega_{X}^{a} (\log \ E) \otimes {\cal L}^{-1}) = 0 \]
for $a + b  < \dim \ X - r (\tau \mid_{X - E} )$.
\end{satz}
\begin{Beweis}{Proof} If $I$ is the ideal sheaf of $\tau (E)$, then
${\cal A}^{\mu} \otimes I$ will be generated by global
sections for some $\mu \gg 0$. Hence, we may assume
that ${\cal L}^{N} (-E)$ is  generated by global section for $N = \nu
\cdot \mu$. Moreover, we can assume that $N$ is larger than
the multiplicities of the components of $E$.
If $D$ is the divisor of a general section of ${\cal L}^{N} (-E)$, then $D+ E$
is a normal crossing divisor.
We have for
$$\tau : D \>>> \tau (D), \ {\cal L}\mid_{D} \mbox{ \ and \ }
E\mid_{D}$$ the same assumptions as those asked for in 6.7. Moreover,
by (4.11,d) we have
\[ r( \tau \mid_{X- E}) + 1 \geq r( \tau \mid_{D- E}). \]
By induction on $\dim \ X$ we may assume that
\[ H^{b-1} (D, \Omega_{D}^{a-1} (\log \ E\mid_{D}) \otimes {\cal L}^{-1})
= 0 \]
for
\[ a + b < n - r(\tau \mid_{X- E}) \ \leq \ n + 1 - r (\tau \mid_{D- E}). \]
The exact sequence (see (2.3,b))
\[ 0 \rightarrow  \Omega_{X}^{a} (\log \ E)  \otimes {\cal L}^{-1} \rightarrow
\Omega_{X}^{a} ( \log \ (D+E))  \otimes {\cal L}^{-1} \rightarrow
\Omega_{D}^{a-1} (\log \ E \mid_{D})  \otimes {\cal L}^{-1} \rightarrow 0 \]
implies that for those $a,b$ the map
\[ H^{b} (X,\Omega_{X}^{a} (\log \ E) \otimes {\cal L}^{-1}) \>>>
H^{b} (X,\Omega_{X}^{a} (\log \ (D + E)) \otimes {\cal L}^{-1}) \]
is injective. However, since
\[ r( \tau \mid_{X - E} ) \geq r(\tau \mid_{X- (D+E)}) \]
(6.2,c) tells us that
\[ H^{b} (X,\Omega_{X}^{a} (\log \ (D+E)) \otimes {\cal L}^{-1}) = 0 \]
for $a + b  < \dim \ X - r (\tau \mid_{X - E} )$.
\end{Beweis}
\begin{defn}{Remarks}\\
a) The reader will have noticed that (6.7) does not use the full strength
of (6.2). Several other applications and extensions of (6.2) can be found
in the literature, (see for example \cite{Ara91} \cite{AJ89} \cite{Mae90}
\cite{Mae91}).\\
b) Sometimes it is nicer to use the dual version of (6.7). Since
$$\bigwedge^{n} \Omega_{X}^{1} (\log \ E) = \omega_{X} \otimes {\cal O}
(E_{red})$$ we find the dual of $\Omega_{X}^{a} (\log \ E)$ to
be $$\omega_{X}^{-1} \otimes \Omega_{X}^{n-a} ( \log \ E) ( -E_{red})$$ and
by Serre duality (6.7) is equivalent to the vanishing of
\[ H^{b} (X,\Omega_{X}^{a} (\log \ E)(-E_{red}) \otimes {\cal L}) \]
for
\[ a + b > \dim \ X + r (\tau \mid_{X - E}). \]
c) For (6.7) we used the invariant $r(g)$ and lemma (4.12), the
latter being proved by analytic methods. However, playing
around with de Rham complexes and their hypercohomology, one
should be able to find an algebraic analogue of those arguments.
\end{defn}
\ \\

In \cite{Vie82} the second author used the Hodge duality (as in (3.23))
to reduce vanishing for $H^{b} (X, {\cal L}^{-1})$ to the Bogomolov-Sommese
vanishing theorem. The latter fits nicely into the scheme explained in this
lecture, (see the proof of (13.10,a)).
\begin{satz}{Corollary (F. Bogomolov, A. Sommese)}
\index{Vanishing theorem!Bogomolov Sommese}
For $X, {\cal L}$ as in (6.1) and
for a normal crossing divisor $B$ one has
\[ H^{0} (X, \Omega_{X}^{a} (\log \ B) \otimes {\cal L}^{-1}) = 0 \]
for $a < \kappa ({\cal L})$.
\end{satz}
\begin{Beweis}{Proof} (6.9) is compatible with blowing ups and
we can assume that
$$\phi_{N} : X \>>>\P (H^{0}(X, {\cal L}^{N}))$$
is a morphism. For $N$ large
enough, we can choose $D$ such that $\phi_{N} \mid_{X-D}$ has equidimensional
fibres of dimension $n - \kappa({\cal L})$ and
such that ${\cal L}^{N} = {\cal O}_{X}(D)$. Moreover we may assume $B+D$ to
be a normal crossing divisor. By (6.2,c)
\[ H^{0} (X,\Omega_{X}^{a} (\log \ (B + D)) \otimes {\cal L}^{{(1)}^{-1}})
= 0 \]
for $a < \kappa ({\cal L})$. As
$ \Omega_{X}^{a} (\log \ B)  \otimes {\cal L}^{-1} $
is a subsheaf of $\Omega_X^a (\log \ (B + D)) \otimes {\cal L}^{(1)^{-1}}$,
one obtains (6.8). \\
\end{Beweis}

Global vanishing theorems always give rise to the vanishing of certain direct
image sheaves.
\begin{defn}{Notations} Let $X$ be a manifold, defined over an algebraically
closed field of characteristic zero and let $f: X \rightarrow Z$ be a
proper surjective morphism. Let ${\cal L}$ be an invertible sheaf on $X$.
We will call ${\cal L}$\\
a) $f$-{\it numerically effective} if for all curves $C$ in $X$ with
$\dim \ f(C) = 0$ one has $\deg \ ({\cal L} \mid_{C}) \geq 0$
\index{$f$-numerically effective}
\\
b) $f$-{\it semi-ample} if for some $N > 0$ the natural map
$f^{\ast}f_{\ast}{\cal L}^{N} \>>> {\cal L}^{N}$ is surjective.
\index{$f$-{\it semi-ample}}
\end{defn}
\ \\
The relative Grauert-Riemenschneider vanishing theorem says, that for
a birational morphism
$f: X \rightarrow Z$ one has $R^{b} f_{\ast} \omega_{X} = 0$ for $b > 0$.
As a generalization one obtains:
\begin{satz}{Corollary}\index{Relative vanishing theorem!for
$f$-numerically effective $\Q$-divisors}\\
{\rm a)} For $f: X \rightarrow Z$ as in (6.10)
let ${\cal L}$ be an invertible sheaf such that ${\cal L}^{N} (-D)$ is
$f$-numerically effective for a normal crossing divisor
$$D = \sum_{j=1}^{r} \alpha_{j} D_{j}.$$
Then $$R^{b} f_{\ast} ({\cal L}^{(1)} \otimes \omega_{X}) = 0 \mbox{ \ for \ }
b > \dim \ X - \dim \ Z - \kappa ({\cal L} \mid_{F})$$
where $F$ is a general fibre of $f$.\\
{\rm b)} In particular,
if $f: X \rightarrow Z$ is birational and if $D = f^{\ast}
\Delta$ is a normal crossing divisor for some effective Cartier
divisor $\Delta$ on $Z$, then
$$R^{b} f_{\ast} (\omega_{X} \otimes {\cal O}_{X} ( - [ \frac{D}{N} ])) = 0
\mbox{ \ for \ } b > 0.$$
\end{satz}
\begin{Beweis}{Proof} Obviously, since ${\cal O}_{X} (-D) = f^{\ast}
{\cal O}_{Z} (- \Delta)$ is $f$ numerically effective, b) is a special
case of a). \\
As usual, in a), we can add the assumption $0 < \alpha_{j} <
N$ for $j = 1, \ldots r$, and we will have ${\cal L}^{(1)} = {\cal L}$.\\
\ \\
The statement being local in $Z$, we can assume $Z$ to be affine or,
compactifying $X$ and $Z$, we can assume $Z$ to be projective. By (5.13)
we are allowed to blow $X$ up and we can assume that $X$ is projective
as well.\\
The assumptions made imply that for ${\cal A}$ ample invertible on $Z$
$$f^{\ast} {\cal A}^{\nu} \otimes {\cal L}^{N}(-D)$$ will be numerically
effective for $\nu
>> 0$ and that
\[ \kappa(f^{\ast} {\cal A}^{\nu}
\otimes {\cal L}) \geq \kappa({\cal L}\mid_{F})
+ \dim \ (Z). \]
Using Serre's vanishing (1.1) we can assume that for all $c > 0$
and $b \geq 0$
\[ H^{c} (Z, {\cal A}^{\nu} \otimes R^{b} f_{\ast} ({\cal L} \otimes
\omega_{X})) = 0 \]
and that $H^{0} (Z, {\cal A}^{\nu} \otimes R^{b} f_{\ast} ({\cal L} \otimes
\omega_{X}))$ generates the sheaf $ {\cal A}^{\nu} \otimes R^{b}f_{\ast}
({\cal L} \otimes \omega_{X})$.
By the Leray spectral sequence (A.27) we obtain
\[ H^{b} (X, f^{\ast} {\cal A}^{\nu} \otimes {\cal L} \otimes \omega_{X})
= H^{0} (Z, {\cal A}^{\nu} \otimes R^{b} f_{\ast} ({\cal L} \otimes
\omega_{X}) ) \]
and by (5.12,d) this group is zero for
\[ b > \dim \ X - \dim \ (Z) - \kappa ({\cal L} \mid_{F}) \geq
\dim \ X - \kappa (f^{\ast} {\cal A}^{\nu} \otimes {\cal L}). \]
\end{Beweis}
In the special case for which  ${\cal L}^{N} (-D)$ is $f$-semi-ample the vanishing
of
\[ R^{b} f_{\ast}({\cal L} \otimes \omega_{X}) \ \ \mbox{for \ \ $b >
\dim \ X - \dim Z$} \]
follows as well from the next statement, due to J. Koll\'{a}r, \cite{Kol86}.
\begin{satz}{Corollary of 5.12,a) (J. Koll\'{a}r)}\index{Torsion
freeness! Koll\'ar}
In addition to the assumption of (6.11,a)
we even assume that ${\cal L}^{N} (-D)$ is $f$-semi-ample. Then
\[ R^{b} f_{\ast} ({\cal L}^{(1)} \otimes \omega_{X}) \]
has no torsion for $b \geq 0$.
\end{satz}
\begin{Beweis}{Proof} As above we can assume $X$ and $Z$ to be projective
and ${\cal L}^{N} (-D)$ to be semi-ample. Moreover, we may assume
${\cal L}^{N}(-D)$ to contain $f^{\ast} {\cal A}$ for a very ample sheaf
${\cal A}$ on $Z$, that ${\cal L} = {\cal L}^{(1)}$, that
\[ H^{c} (X, R^{b} f_{\ast} ({\cal L} \otimes \omega_{X}) ) = 0 \ \ \mbox{for
\ \ $c > 0$} \]
and that $R^{b}f_{\ast} ({\cal L} \otimes \omega_{X})$ is generated by
its global sections. If $R^{b} f_{\ast} ({\cal L} \otimes \omega_{X})$
has torsion for some $b$, then the map
\[ R^{b} f_{\ast}( {\cal L} \otimes \omega_{X}) \>>> R^{b}
f_{\ast} ({\cal L} \otimes \omega_{X}) \otimes {\cal O}_{Z} (A) \]
has a non-trivial kernel $\cal K$ for some effective ample
divisor $A$ on $Z$. We may assume that ${\cal O}_{Z} (A) = {\cal A}$.
Replacing ${\cal L}$ by ${\cal L} \otimes f^{\ast} {\cal A}^{\nu}$
again, we can assume that $H^{0} (Z, {\cal K}) \neq 0$. For
$B = f^{\ast} A$ ,this implies that
$H^{0} (Z, {\cal K})$ lies in the kernel of
\[ H^{b} (X, {\cal L} \otimes \omega_{X}) \>>> H^{b} (X, {\cal L}
\otimes \omega_{X} (B)) \]
and hence that
\[ H^{n-b} (X, {\cal L}^{-1} (-B)) \>>> H^{n-b} (X, {\cal L}^{-1}) \]
is not surjective, contradicting (5.12,a).
\ \\ \end{Beweis}

Some of the vanishing theorems mentioned and a partial degeneration of the
Hodge to de Rham spectral sequence remain true for certain non-compact manifolds.
One explanation for those results, obtained by I. Bauer and S. Kosarew in
\cite{BK89} and \cite{BK90} by different methods,
is the following lemma.
\begin{satz}{Lemma} Let $Z$ be a projective variety in characteristic zero,
$U \subseteq Z$ be an open non-singular subvariety, $\delta : X \rightarrow Z$
be a desingularization such that $ \iota : U \simeq \delta^{-1} (U) \rightarrow
X$. Assume that $X - \iota (U) = E$ for a reduced normal crossing
divisor $E$. Then, for $ a + b < \dim \ X - \dim \ \delta (E) - 1$ and all
invertible sheaves ${\cal M}$ on $Z$ one has
\[ H^{b} (X, \Omega_{X}^{a} (\log \ E)
\otimes \delta^{\ast} {\cal M} ) = H^{b} (U,
\Omega_{X}^{a} \otimes {\cal M} \mid_{U} ). \]
\end{satz}
\begin{Beweis}{Proof}
Let ${\cal A}$ be an ample invertible sheaf on $Z$ and let $E^{\prime}$
be an effective exceptional divisor, such that ${\cal O}_{X} (-E^{\prime})$
is relatively ample for $\delta$.\\
For fixed $\nu \geq 0$ we can choose ${\cal A}$ large enough,
such that
for all $a,b$
\[ R^{b} \delta_{*} (\Omega_{X}^{a} (\log \ E) \otimes {\cal O}_{X}
(-E - \nu \cdot E^{\prime})) \otimes {\cal A} \]
is generated by global sections and
\[ H^{c} (Z, R^{b} \delta_{*} (\Omega_{X}^{a} (\log \ E) \otimes
{\cal O}_{X} (-E - \nu \cdot E^{\prime})) \otimes {\cal A}) = 0 \]
for $c > 0$. Moreover, for $\nu > 0$, we may assume that $\tau^{*}
{\cal A} ( - \nu \cdot E')$ is ample.
Using Serre duality (as explained in (6.8,b)) and the Leray spectral
sequence (A.27) one finds
\[ H^{n-b} (X, \Omega_{X}^{n-a} (\log \ E) \otimes \delta^{\ast}
{\cal A}^{-1} (\nu \cdot E^{\prime}))^* =  \]
\[ H^{b} (X, \Omega_{X}^{a} (\log \ E)(-E) \otimes
\delta^{\ast}{\cal A} (- \nu \cdot E^{\prime}) ) = \]
\[ H^{0} (Z, R^{b} \delta_{\ast} (\Omega_{X}^{a}(\log \ E) \otimes
{\cal O}_{X} (-E - \nu \cdot E^{\prime})) \otimes {\cal A}). \]
By (6.4), for $\nu > 0$, or by (6.7), for $\nu = 0$, we find
\[ R^{b} \delta_{\ast} (\Omega_{X}^{a} (\log \ E) \otimes
{\cal O}_{X} (- E - \nu \cdot E^{\prime} )) = 0 \]
for $a+b > \dim \ X$. For those $a$ and $b$ and for
\[ {\cal K}_{\nu} = {\cal O}_{X} / {\cal O}_{X} (- \nu \cdot E^{\prime}) \]
we have
\[ R^{b} \delta_{*} (\Omega_{X}^{a} (\log \ E) (-E) \otimes
{\cal K}_{\nu}) = 0. \]
One obtains for $a + b > \dim \ X + \dim \ \tau (E)$
from the Leray spectral sequence (A.27) that
\[ H^{b}(X, \delta^{\ast} {\cal M}^{-1} \otimes \Omega_{X}^{a} (\log \ E)
(-E) \otimes {\cal K}_{\nu}) = 0. \]
Hence for all $\nu \geq 0$ and
$a+b > \dim \ X + \dim \ \tau (E) + 1$ the map
\[ H^{b}(X, \delta^{\ast} {\cal M}^{-1} \otimes \Omega_{X}^{a} (\log \ E)
\otimes {\cal O}_{X}(-E - \nu \cdot E^{\prime})) \>>>
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \]
\[  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\>>> H^{b} (X, \delta^{\ast} {\cal M}^{-1} \otimes \Omega_{X}^{a}
(\log \ E) (-E)) \]
is bijective. By Serre duality again,
\[ H^{b}(X, \delta^{\ast} {\cal M} \otimes \Omega_{X}^{a}
(\log \ E)) \>>> H^{b} (X, \delta^{\ast} {\cal M} \otimes \Omega_{X}^{a}
(\log \ E) \otimes {\cal O}_{X} (\nu \cdot E^{\prime})) \]
is an isomorphism for $a + b < \dim \ X - \dim \ \tau (E) - 1$ and, taking
the limit for $\nu \in \N$, we obtain (6.13).
\ \\ \end{Beweis}
\begin{satz}{Corollary (I. Bauer, S. Kosarew \cite{BK89})} Let $Z$
be a projective variety in characteristic zero,
$U \subseteq Z$ be an open non-singular subvariety. Then, for \\
$k < n - \dim \ (Z - U) - 1$ one has
\[ \dim \h ^{k}(U, \Omega^{\bullet}_{U}) = \sum_{a+b=k} \dim H^{b}
(U, \Omega_{U}^{a}). \]
\end{satz}
\begin{Beweis}{Proof} We can choose a desingularisation $\delta : X
\rightarrow Z$ and $E$ as in (6.13). Then we have a natural map of
spectral sequences
\[
\minCDarrowwidth=0.1cm
\begin{CD}
E_{1}^{ab} = H^{b} (U, \Omega^{a}_{U}) \noarr \Longrightarrow \noarr \h ^{a+b}
(U, \Omega^{\bullet}_{U}) \\
\A A \varphi_{a,b} A  \noarr \A A \varphi A \\
E_{1}^{\prime ab} = H^{b} (X, \Omega_{X}^{a} (\log \ E)) \ \
\noarr \Longrightarrow \noarr
\ \ \h ^{a+b} (X, \Omega^{\bullet}_{X} (\log \ E)).
\end{CD}
\]
Since $E^{\prime ab}_{1}$ degenerates in $E_{1}$ and since $\varphi_{a,b}$
are isomorphisms for $$a+b < n - \dim \ (Z-U) - 1$$ the second spectral sequence
has to degenerate for those $a,b$.
\ \\ \end{Beweis}
\begin{satz}{Corollary (see also I. Kosarew, S. Kosarew
\cite{BK90})} \index{Vanishing theorem!Bauer Kosarew}\\
For $Z$ and $U$ as in (6.14) let ${\cal L}$ be an
$l$-ample invertible sheaf on $Z$. Then
\[ H^{b} (U, \Omega_{U}^{a} \otimes  {\cal L}^{-1} \mid_{U} ) = 0 \]
for
\[ a + b  < \mbox{Min} \ \{ \kappa({\cal L}), \dim \ X - l + 1, \ \dim \ X - \dim \
(Z - U) - 1  \ \} \]
\end{satz}
\begin{Beweis}{Proof} For $X,E$ as in (6.13) and ${\cal M} =
\delta^{\ast} {\cal L}$ we have by (6.7)
\[ H^{b} (X, {\cal M}^{-1} \otimes \Omega_{X}^{a} (\log \ E)) = 0 \]
for $a + b <  \dim \ X - r(\tau \mid_{U})$ where
\[ \tau : X \> \delta >> Z \> \phi_{N} >> \phi_{N} (Z) \]
is the composition of $\delta$ with the map given by global sections of
${\cal L}^{N}$.\\
However,
\[ r( \tau \mid_{U}) \leq r (\phi_{N}) \leq \mbox{Max} \ \{ \dim \ X - \kappa({\cal L}),
l - 1 \}. \]
\end{Beweis}
\begin{defn}{Remark} For $k = \C$, the reason for which certain
coherent sheaves ${\cal F}$ on $Z$ satisfy $H^{b} (Z, {\cal F} \otimes
{\cal H} ) = 0$ for ${\cal H}$ ample,
seems to be related to the existence of connections. This point of
view, which is exploited in J. Koll\'{a}r's work on vanishing theorems
\cite{Kol86}, \cite{Kol89} and extended by M. Saito (see \cite{Sai89} and
the references given there), should
imply that sheaves arising as natural subquotients of ${\cal O}_{Z} \otimes
R^{k} f_{\ast} V$ for a morphism $f : X \rightarrow Z$ of
manifolds and a locally constant system $V$,
sometimes have vanishing properties as the one
stated above.\\
J. Koll\'{a}r proved, for example, that for a morphism
$f: X \rightarrow Z$, where $X$ and $Z$ are projective varieties and $X$
non-singular, one has
\[ H^{c} (Z, R^{b} f_{\ast} \omega_{X} \otimes {\cal H}) = 0 \]
for $c > 0$ and ${\cal H}$ ample on $Z$.
\end{defn}
Slightly more generally one has
%
\begin{satz}{Corollary (of (5.12,b))}
\index{Vanishing theorem!for multiplier ideals}
\index{Vanishing theorem!for direct images}
Let $f: X \rightarrow Z$ be a
surjective morphism of projective varieties defined over an algebraically
closed field of characteristic zero, with $X$ non-singular. Let ${\cal L}$
be an invertible sheaf on $X$,
$$D = \sum_{j=1}^{r} \alpha_{j} D_{j}$$ a normal
crossing divisor and $N \in \N$ with
\[ 0 < \alpha_{j} < N \ \ \mbox{for} \ \ j = 1, \ldots, r . \]
{\rm a) } If ${\cal L}^{N}(-D)$ is semi-ample and ${\cal K}$ a numerically
effective invertible sheaf on $Z$ with $\kappa({\cal K}) = \dim \ Z$, then
for $c > 0$ and $b \geq 0$
\[ H^{c} (Z, {\cal K} \otimes R^{b} f_{\ast} (\omega_{X} \otimes
{\cal L})) = 0. \]
b) If ${\cal L}^{N} (-D)$ is numerically effective,
$\kappa({\cal L}^{N} (-D)) = \nu ({\cal L}^{N} (-D))$, and if
$({\cal L}^{N}(-D))^{\mu}$ contains $f^{\ast} {\cal H}$ for some ample sheaf
${\cal H}$ on $Z$ and some $\mu > 0$, then for $c > 0$ and all $b$
\[ H^{c} (Z, R^{b} f_{\ast} (\omega_{X} \otimes {\cal L})) = 0. \]
\end{satz}
\begin{Beweis}{Proof} By (5.10,b), replacing ${\cal L}$ by ${\cal L} \otimes
f^{\ast} {\cal K}$, a) follows from b).\\
If $H$ is the zero divisor of a general section of ${\cal H}^\mu$
for $\mu \gg 0$ and if $B = f^{\ast} H$ then $B$ is a
non-singular divisor and the assumptions of (5.12,b) hold true. Hence
\[ H^{b} (X,\omega_{X} \otimes  {\cal L} ) \>>> H^{b} (X,
\omega_{X} (B) \otimes {\cal L}) \]
is injective for all $b$. Since $H$ is in general position we have exact
sequences
\small
\[
\minCDarrowwidth=0.45cm
\begin{CD}
0 \>>> R^{b} f_{\ast} ({\cal L} \otimes \omega_{X}) \>>> R^{b} f_{\ast}
({\cal L} \otimes \omega_{X} (B)) \>>> { R^{b} f_{\ast} ({\cal L} \otimes
\omega_{B}) \rightarrow 0} \\
\noarr \| \noarr \| \noarr \| \\
\noarr R^{b} f_{\ast} ({\cal L} \otimes \omega_{X}) \>>> {\cal O}_{Z}
(H) \otimes R^{b} f_{\ast} ({\cal L} \otimes \omega_{X}) \>>> {\cal O}_{Z}
(H) \otimes R^{b} f_{\ast} ({\cal L} \otimes \omega_{X}) \mid_{H}
\end{CD}
\]
\normalsize
By induction on $\dim \ Z$ we may assume that
$$H^{c} (H, R^{b} f_{\ast}
({\cal L} \otimes \omega_{B})) = 0 \mbox{ \ for \ } c > 0$$
and, if we choose $\mu$ large enough, we find
by Serre's vanishing theorem
$$H^{c} (Z, R^{b} f_{\ast} ({\cal L} \otimes \omega_{X})) = 0
\mbox{ \ for \ } c \geq 2.$$
In the Leray-spectral sequence (see A.27) all the differentials
are zero, since $E_2^{ab} \neq 0$ just for $a = 0$ or $a = 1$, and hence
the upper line in the following diagram is exact.
\small
\[
\minCDarrowwidth=0.37cm
\begin{CD}
0 \>>> H^{1} (Z, R^{b-1}f_{\ast} ({\cal L} \otimes \omega_{X})) \>>>
H^{b} (X, {\cal L} \otimes \omega_{X}) \>>> H^{0} (Z, R^{b} f_{\ast}
({\cal L} \otimes \omega_{X}))  \\
\noarr \noarr \V\alpha VV \V V V \\
\noarr \noarr H^{b} (X,  {\cal L} \otimes \omega_{X}
(B)) \>>> H^{0} (Z, R^{b} f_{\ast} ({\cal L} \otimes \omega_{X} (B)))
\end{CD}
\]
\normalsize
Since $\alpha$ is injective and since
$$H^{b} (X,  {\cal L} \otimes \omega_{X}(B)) = H^{0} (Z, R^{b} f_{\ast}
({\cal L} \otimes \omega_{X} (B)))$$ we find
$H^{1} (Z, R^{b-1} f_{\ast} ({\cal L} \otimes \omega_{X})) = 0$ for
all $b$.\\
\end{Beweis}
\Paragraph{Some applications of vanishing theorems }
\ \\
The vanishing theorems for integral parts of $\Q$-divisors and for
numerically effective sheaves (5.12,c) and (5.12,d), as well as
(5.6,a) turned out to be useful for applications in
higherdimensional complex projective geometry. We will not be able
in these notes to include an outline of the Iitaka-Mori classification of
threefolds, and the reader interested in this direction is invited
to regard S. Mori's beautiful survey \cite{Mor87}.\\
\ \\
In this lecture we just want to give a flavour as to  how one should try to
use vanishing theorems to attack  certain types of questions. The
choice made is obviously influenced by our personal taste.\\
\\
We will assume in this lecture:\\
\ \\
{\it All varieties are defined over an algebraically closed field
$k$ of characteristic zero.}
\begin{defn}{Example: Surfaces of general type}
\index{Surfaces of general type!semi-ampleness of the canonical sheaf}
For a projective surface $S^{\prime}$ of general type, i.e. for a
non-singular $S^{\prime}$ with $\kappa (\omega_{S^{\prime}}) = \dim
S^{\prime} = 2$, one can blow down exceptional curves
$E \simeq \P^{1}$ with $E \cdot E = -1$ (\cite{Har77}, p. 414).
After finitely many steps one obtains a surface $S$ without any exceptional
curve, a minimal model of $S^{\prime}$ (\cite{Har77}, p. 418). $S$ is
characterised by
\end{defn}
\begin{defn}{Claim} $\omega_{S}$ is $nef$.
\end{defn}
\begin{Beweis}{Proof} $\kappa (S) \geq 0$ implies that
$\omega_{S}^{N} = {\cal O}_{S} (D)$ for $D$ effective. A curve $C$ with
$\deg \ (\omega_{S}\mid_{C}) < 0$ must be a component of $D$
and $C^{2} < 0$. However, the adjunction formula gives
\[ -2 \leq 2g (C) - 2 = \deg \ (\omega_{S} \mid_{C}) + C^{2}. \]
Hence the only solution is $C^{2} = -1$ and
$\deg (\omega_{S} \mid_{C}) = -1$, which forces $C$ to be exceptional.\\
\end{Beweis}
D. Mumford in his appendix to \cite{Zar62}
used the contraction of $(-2)$ curves to show:\\
\begin{satz}{Theorem} If $S$ is a minimal model and $\kappa (\omega_{S}) = 2$,
then
$\omega_{S}$ is semi-ample.
\end{satz}
\ \\
X. Benveniste and Y. Kawamata ($\dim X = 3$) and Y. Kawamata and\\
V. Shokurov (see \cite{Mor87} for the references) generalised (7.3) to
the higher dimensional case. Their ideas, cut back to the surface case, give a
simple proof of (7.3).
\\
\begin{Beweis}{Proof of 7.3 (from the Diplom-thesis of T. Nakovich,
Essen)}\\
{\it Step 1.:} If $p \in S$ does not lie on any curve $C$ with $\deg \
(\omega_{S} \mid_{C}) = 0$, then for some $\nu \gg  0$ there is $s \in H^{0}
( S , \omega_{S}^{\nu })$ with $s(p) \neq 0$.
\begin{Beweis}{Proof} Let $\tau : S^{\prime} \rightarrow S$ be the blowing up
of $p$ and $E$ the exceptional curve. One has
\[ \deg \ ( \tau^{\ast} \omega_{S}^{\mu}(-E) \mid_{C^{\prime}}) =
\mu \cdot \deg \ ( \omega_{S} \mid_{C}) - E \cdot C^{\prime} \]
for curves $C^{\prime}$ in $S^{\prime}$ with $C = \tau (C^{\prime}) \neq p$.
Hence, for some $\mu \gg  0$ the sheaf
${\cal L} = \tau^{\ast} \omega_{S}^{\mu} (-E)$
will be nef. By (5.12,c) we find
\[ H^{1} (S^{\prime}, {\cal L}^{-2}) \cong H^{1} (S^{\prime},
\omega_{S^{\prime}} \otimes {\cal L}^{2}) =
H^{1} (S^{\prime}, \tau^{\ast} \omega_{S}^{2 \mu + 1} (-E)) = 0 \]
and hence
\[ H^{0} (S^{\prime}, \tau^{\ast} \omega_{X}^{2 \mu + 1})
\>>> H^{0} (E, {\cal O}_{E}) \]
is surjective.\\
\end{Beweis}
{\it Step 2.:} For $\nu \gg  0$ let $$D = \sum_{j=1}^{r} \alpha_{j} C_{j}$$
be the base locus of $\omega_{S}^{\nu}$ (i.e. $\omega_{S}^{\nu}(-D)$ is
generated by $H^{0} (S, \omega_{S}^{\nu})$ outside of a finite number of
points). Then $D$ is a normal crossing divisor, $C_{j}^{2} = -2$ for $j = 1,
\ldots, r$ and $\omega_{S}^{\nu}$ is generated by $H^{0} (S, \omega_{S}^{\nu})$
outside of $D$.
\begin{Beweis}{Proof} By step 1, if for some $p \in S$ there is no section $s$
of $\omega_S^\nu$ with $s(p) \neq 0$, then $p$
lies on some curve $C$ with $\deg \ (\omega_{S} \mid_{C}) = 0$
and necessarily $C$ is contained in the base locus. We know
thereby that $\deg \ ( \omega_{S} \mid_{C_{j}}) = 0$ for the components $C_j$
of $D$.
By the Hodge-index theorem (\cite{Har77}, p. 364) one finds for any reduced
subdivisor $C$ of $D$ that $C \cdot C < 0$. If we take
$C = C_{j}$, then the adjunction formula shows that
$$C \cdot C = -2\mbox{ \ and \ }C \simeq \P^{1}.$$
For $ C = (C_{1} + C_{2})$ we get
$$C_{1} \cdot C_{2} < 2 $$ and $C_{1}$ and $C_{2}$ intersect transversally.\\
\end{Beweis}
{\it Step 3.} For $D$ as  in Step 2, $\omega_{S}^{\nu} (-D)$ is nef, hence
$\omega_{S}^{N}(-D)$ for $N \geq \nu$ is nef as well.
We can choose $N > \alpha_{j}$
for $j = 1, \ldots r$. For some $i > 0$
\[ D^{\prime} = [ \frac{i \cdot D}{N} ] = \sum_{j=1}^{r} [ \frac{i \cdot
\alpha_{j}}{N} ] \cdot C_{j} \]
will be reduced and non zero. By (5.12,c)
again, we have for ${\cal L} = \omega_{S}$
\[ H^{1} (S, {\cal L}^{(i)^{-1}}) \cong H^{1} (S, \omega_{S} \otimes
{\cal L}^{(i)} ) = H^{1} (S, \omega_{S}^{i+1} (-D^{\prime})) = 0 \]
and
\[ H^{0} (S, \omega_{S}^{i+1}) \>>> H^{0} (D^{\prime},
\omega_{S}^{i+1} \mid_{D^{\prime}}) \]
is surjective. Since the right hand side is nontrivial (in fact its
dimension is just the number of connected components of $D^{\prime}$),
for $i + 1$ the base locus does not contain $D^{\prime}$. After finitely
many steps we are done.\\
\end{Beweis}

The proof of (7.1) is a quite typical example in two respects. First of all, vanishing
of $H^{1}$ or more general by the surjectivity of the adjunction map in (5.6,a)
allows to pull back sections of
invertible sheaves on divisors. Secondly it shows
again how to play around with integral parts, a method
which already appeared in the proof of (5.12).

Corollary (5.12), as stated, has the disadvantage that $D$ has to be a normal crossing
divisor. Let us try next to study some weaker conditions.
\begin{defn}{Definition} Let $X$ be a normal variety and $D$ be an effective
Cartier divisor on $X$. Let $\tau : X^{\prime} \rightarrow X$ be a
blowing up, such that $X^{\prime}$ is non singular and $D^{\prime} =
\tau^{\ast} D$ is a normal crossing divisor. We define:\\
a) $$\omega_{X} \{ \frac{-D}{N} \} = \tau_{\ast} \omega_{X^{\prime}}
(- [ \frac{D'}{N} ]).$$
\index{Multiplier ideals}\index{$\omega_{X} \{ \frac{-D}{N} \}$}
b) ${\cal C}_{X}(D,N) = \mbox{Coker} \ (\omega_{X} \{ \frac{-D}{N} \} \>>>
\omega_{X})$ where $\omega_{X}$ is the reflexive hull of $\omega_{X_{0}}$ for
$X_{0} = X - \mbox{Sing} \ (X)$.\index{${\cal C}_{X}(D,N)$}\\
c) (see \cite{EV90}) $$e(D) = \mbox{Min} \ \{ N > 0; {\cal C}_{X}
(D,N) = 0 \}.$$\index{$e(D)$}
d) If $X$ is compact and ${\cal L}$ invertible, $H^{0} (X, {\cal L}) \neq 0$,
then $$e({\cal L}) = \mbox{Max} \ \{ e(D); \ D \geq 0 \mbox{ \ and  \ }
{\cal O}_{X} (D) = {\cal L} \}.$$\index{$e({\cal L})$}
\end{defn}
\begin{defn}{Properties (see \cite{EV90})} Let $X$ and $D$ be as in (7.4).\\
a) If $X$ has at most rational singularities, then $e(D)$ is finite.\\
b) If $X$ is non-singular and $D$ a normal crossing divisor then
$$\omega_{X} \{ \frac{-D}{N} \} = \omega_{X} ( - [ \frac{D}{N} ] ).$$
c) $\omega_{X} \{ \frac{-D}{N} \}, {\cal C}_{X} (D,N)$ and $e(D)$ are
independent of the blowing up $\tau : X^{\prime} \rightarrow X$ choosen.\\
d) Let $H$ be a prime Cartier divisor on $X$, not contained in $D$, such
that $H$ is normal. Then one has a natural inclusion
\[ \omega_{H} \{ \frac{-D\mid_{H}}{N} \} \>>> \omega_{X} \{
\frac{-D}{N} \} \otimes {\cal O}_{X} (H) \mid_{H}. \]
e) If in d) $X$ and $H$ have rational sigularities, then for $N \geq
e(D \mid_{H})$, $H$ does not meet the support of ${\cal C}_{X} (D,N)$.
\end{defn}
\begin{Beweis}{Proof} a) is obvious since for $N \gg  0$
\[ \tau_{\ast} \omega_{X^{\prime}} ( - [ \frac{D}{N} ]) = \tau_{\ast}
\omega_{X^{\prime}} = \omega_{X}. \]
Similar to (5.13), part b) can be deduced from (3.24) and from
the fact, that quotient singularities are rational singularities.
A more direct argument is as follows. We have an inclusion $$\omega_{X}
\{ \frac{-D}{N} \} \>>> \omega_{X} ( - [ \frac{D}{N} ] )$$ and
it is enough to prove b) for some blowing up dominating $\tau$. Hence, it is
enough to consider the case that $\tau$ is a sequence of blowings with
non-singular centers. Let us write
$$\omega_{X'} = \tau^* \omega_X \otimes {\cal O}_{X'}
(\sum^t_{i=1} \alpha_i \cdot E_i).$$
For $m_i = \codim_X(\tau(E_i))$ one has $\alpha_i \geq m_i -1$.\\
\ \\
In fact, assume this to hold true for $\tau_1 : X_1 \>>> X$ and
$$\omega_{X_1} = \tau_1^* \omega_X \otimes {\cal O}_{X_1}
(\sum^{t-1}_{i=1} \alpha_i \cdot E'_i).$$
If $\delta:X'\>>> X_1$ is the blowing up with center $S$ and $E_t$
the exceptional divisor then, for $m= \codim_{X_1} (S)$, one has
$$\omega_{X'} = \delta^* \omega_{X_1} \otimes {\cal O}_{X'}
((m-1) \cdot  E_t)$$
(see \cite{Har77}, p. 188). If $m_t > m$ then $S$ lies on some $E'_\nu$
with $m_t-m \leq m_\nu-1$. Hence
$$\alpha_t \geq m-1+\alpha_\nu \geq m_t-1.$$
On the other hand, assume that $\tau(E_\mu)$ lies on $s$ different
components of $D$, let us say on $D_1, \cdots ,D_s$ but not in
$D_{j}$ for $j > s$. Then $m_\mu \geq s$ and, if
$$D = \sum_{j=1}^{r} \alpha_{j}D_{j}$$
one has
$$ [ \sum_{j=1}^{s} \frac{\alpha_{j}}{N} ] \leq \sum_{j=1}^{s}
[ \frac{\alpha_{j}}{N} ] + s -1 \leq \sum_{j=1}^{s}
[ \frac{\alpha_{j}}{N} ] + \alpha_\mu. $$
One obtains
$$ [\frac{D'}{N}] \leq \tau^*[\frac{D}{N}] + \sum_{i=1}^t
\alpha_i \cdot E_i $$
and hence
\[ \tau^{\ast} \omega_{X} ( - [ \frac{D}{N} ] ) \subset \omega_{X^{\prime}}
( - [ \frac{D'}{N} ] ). \]
c) follows from b). Hence in d) we may assume that $D^{\prime}$ intersects
the proper transform $H^{\prime}$ on $H$ transversally and, of course, that
$H^{\prime}$ is non-singular. Then
$$ [ \frac{D^{\prime}}{N} ] \mid_{H^{\prime}}
= [ \frac{D^{\prime}\mid_{H'}}{N} ].$$
One has a commutative diagram
\[
\begin{CD}
\tau_{\ast} \omega_{X^{\prime}} ( - [ \frac{D^{\prime}}{N} ] + H^{\prime})
\> \alpha >> \tau_{\ast} \omega_{H^{\prime}} (-
[ \frac{D^{\prime}\mid_{H^{\prime}}}{N} ] ) \>>> \omega_{H} \\
\V VV \noarr \V V = V \\
\tau_{\ast} \omega_{X^{\prime}} ( - [ \frac{D^{\prime}}{N} ] ) \otimes
{\cal O}_{X} (H) \>>> \tau_{\ast} \omega_{X^{\prime}} ( - [
\frac{D^{\prime}}{N} ] ) \otimes {\cal O}_{X} (H) \mid_{H} \> \gamma >>
\omega_{H}
\end{CD}
\]
The cokernel of $\alpha$ lies in
$R^{1} \tau_{\ast} \omega_{X^{\prime}} ( - [ \frac{D^{\prime}}{N} ])$, and
(6.11) shows that $\alpha$ is surjective. We obtain therefore a non-trivial
morphism
\[ \alpha^{\prime} : \omega_{H} \{ \frac{-D \mid_{H}}{N} \} \>>>
\omega_{X} \{ \frac{-D}{N} \} \otimes {\cal O}_{X} (H) \mid_{H}. \]
Since $\omega_{H} \{ \frac{-D \mid_{H}}{N} \}$ is torsion free d) holds true.
\\
In e) we know that $\omega_{H} \{ \frac{-D|_H}{N} \}$ is
isomorphic to $\omega_{H}$. Hence $\gamma$ is surjective.
Therefore $\omega_{X} \{ \frac{-D}{N} \}
\otimes {\cal O}_{X} (H)$ must be isomorphic to $\omega_{X} \otimes
{\cal O}_{X} (H)$ in a neighbourhood of $H$.\\
\end{Beweis}
\begin{defn}{Remark} The diagram used to prove d) gives slightly more.
Instead of assuming that $D^{\prime} = \tau^{\ast} D$ it is enough to take
any normal crossing divisor $D^{\prime}$ on $X^{\prime}$ not containing
$H^{\prime}$. Then the inclusion
\[ \tau_{\ast} \omega_{H^{\prime}} ( - [
\frac{D^{\prime} \mid_{H^{\prime}}}{N} ] ) \hookrightarrow
\tau_{\ast} \omega_{X^{\prime}} ( - [\frac{D}{N} ]) \otimes
{\cal O}_{X} (H) \mid_{H} \]
exists whenever $H^{\prime} + D^{\prime}$ is a normal crossing divisor
and ${\cal O}_{X^{\prime}} (-D^{\prime})$ is \\
$\tau$-numerically effective (see (6.10)).
\end{defn}
\ \\

Up to now, we do not even know that $e({\cal L})$ is finite.
This however follows from the first part of the next lemma, since every
sheaf ${\cal L}$ lies in some ample invertible sheaf.
\begin{satz}{Lemma} Let $X$ be a projective manifold and let
${\cal L}$ be an invertible sheaf.
\index{Bounds for $e({\cal L})$}\\
{\rm a)} If ${\cal L}$ is very ample and $\nu > 0$, then
\[ e ({\cal L}^{\nu}) \leq \nu \cdot c_{1} ({\cal L})^{\dim X} + 1 \]
{\rm b)} For $s \in H^{0} (X, {\cal L})$ with zero-locus $D$
assume that for some
$p \in X$ the section $s$ has the multiplicity $\mu$ i.e.:
$$s \in {m}_{p}^{\mu} \otimes {\cal L} \mbox{ \ but \ }
s \notin {m}_{p}^{\mu + 1} \otimes {\cal L}.$$
Then
$$\omega_{X} \{ \frac{-D}{N} \} \>>> \omega_{X}$$ is an isomorphism in
a neighbourhood of $p$ for $N > \mu$. \\
{\rm c)} If under the assumption of \ {\rm b)}
$$\mu^{\prime} = [ \frac{\mu}{N} ] -
\dim X + 1 \geq 0$$ then $\omega_{X} \{ \frac{-D}{N} \}$ is contained
in ${m}_{p}^{\mu^{\prime}} \otimes \omega_{X}$.
\end{satz}
\begin{Beweis}{Proof} a) Let $D \geq 0$ be a divisor, ${\cal O}_{X} (D) =
{\cal L}^{\nu}$. \\
\ \\
If $X$ is a curve then $ [ \frac{D}{N} ] = 0 $ for $N >
\deg \ D + 1 = \nu \cdot c_{1} ({\cal L}) + 1$.\\
\ \\
In general, let $H$ be the divisor of a general section of ${\cal L}$. By
induction
\[ e ({\cal L}^{\nu} \mid_{H}) \leq \nu \cdot c_{1} ({\cal L}
\mid_{H})^{\dim  H} + 1 = \nu \cdot c_{1} ({\cal L})^{\dim  X} + 1. \]
(7.5,e) tells us that ${\cal C}_{X} (D, N)$ is supported outside of $H$
for $$N \geq \nu \cdot c_{1} ({\cal L})^{\dim  X} + 1$$ and moving
$H$ we find ${\cal C}_{X} (D,N) = 0$.\\
\ \\
For b) and c) we may assume that the blowing up
$\tau : X^{\prime} \rightarrow X$
factors through the blowing up $\varrho : X_{p} \rightarrow X$ of $p$.
For $D_{p} = \varrho^{\ast} D$ and for the exceptional divisor $E$ of $\varrho$
we have $$\Delta = D_{p} - \mu \cdot E \geq 0$$ and $\Delta$ does not
contain $E$.
Assume $N > \mu$. One has $${\cal O}_{E} (\Delta \mid_{E}) = {\cal O}_{\P^{n-1}}
(\mu)$$ and, by part a), one obtains
$$\omega_{E} \{ \frac{- \Delta \mid_{E}}{N} \} = \omega_{E}.$$
From (7.5,e) one knows that $$\omega_{X_{p}} \{ \frac{- \Delta}{N}
\} \>>> \omega_{X_{p}}$$ is an isomorphism in a neighbourhood
of $E$. Hence
\[ \omega_{X_{p}} (-E) = \omega_{X_{p}} \{ \frac{ - \Delta}{N} \} \otimes
{\cal O}_{X_{p}} (-E) = \omega_{X_{p}} \{ \frac{ - \Delta - N \cdot E}{N} \} \]
is contained in $\omega_{X_{p}} \{ \frac{ - D_{p}}{N} \}$ which implies that
$\omega_{X} = \omega_{X} \{ \frac{-D}{N} \}$ near $p$.\\
\ \\
If $\mu^{\prime} = [ \frac{\mu}{N} ] - n + 1 \geq 0$ then
\[ \omega_{X_{p}} \{ \frac{-D_{p}}{N} \} \subset \omega_{X_{p}}
\{ \frac{- \mu \cdot E}{N} \} = \omega_{X_{p}} ( - [ \frac{\mu}{N}] \cdot E) \]
and
\[ \omega_{X} \{ \frac{-D}{N} \} = \varrho_{*} \omega_{X_{p}} \{
\frac{-D_{p}}{N} \} \subset \varrho_{\ast} \varrho^{\ast} \omega_{X}
((n-1- [ \frac{\mu}{N} ]) \cdot E) = {m}_p^{\mu^{\prime}} \otimes \omega_{X}. \]
\end{Beweis}
\ \\
The sheaves $\omega_{X} \{ \frac{-D}{N} \}$ are describing the correction terms
needed if one wants to generalize the vanishing theorems (5.12,c) or
(5.12,d) to non normal crossing divisors. For example one obtains:
\begin{satz}{Proposition} \index{Vanishing theorem!for
multiplier ideals}
Let $X$ be a projective manifold, ${\cal L}$ be an
invertible sheaf and $D$ be a divisor such that ${\cal L}^{N}(-D)$ is
numerically effective and $$c_{1} ({\cal L}^{N}(-D))^{n} > 0$$ for $n = \dim
X$. Then $$H^{b} (X, \omega_{X} \{ \frac{-D}{N} \} \otimes {\cal L}) = 0
\mbox{ \ for \ } b > 0.$$
\end{satz}
\begin{Beweis}{Proof} This follows from (5.12,c) and (6.11,b) by using the
Leray spectral sequence (A.27).\\
\end{Beweis}
\begin{defn}{Remark} Demailly proved in \cite{Dem89} an analytic
improvement of Kodaira's vanishing theorem. It would be nice to
understand the relation of his positivity condition with the one arising
from (7.8), i.e. with the condition that $$\omega_{X} \{ \frac{-D}{N} \} =
\omega_{X}.$$

One of the reasons for the interest in vanishing theorems as (7.8) is
implication that certain sheaves are generated by  global
sections. For example one has:
\end{defn}
\begin{satz}{Corollary} Under the assumptions of (7.8) let ${\cal H}$ be
a very ample sheaf. Then $${\cal H}^{\dim X} \otimes {\cal L}
\otimes \omega_{X} \{ \frac{-D}{N} \}$$ is generated by global sections.
\end{satz}
\begin{Beweis}{Proof.} For $${\cal F} = {\cal L} \otimes \omega_{X}
\{ \frac{-D}{N} \} \otimes \omega_{X}^{-1}$$ we have $$H^{b} (X, {\cal F} \otimes
{\cal H}^{\nu} \otimes \omega_{X}) = 0 \mbox{ \ for \ } b > 0
\mbox{ \ and \ }
\nu \geq 0.$$
For general sections $H_{1}, \ldots, H_{n}$ of ${\cal H}$ passing
through a given point $p$ and for $$Y_{r}= \bigcap_{i=1}^{r} H_{i}$$ we obtain
$$H^{b} (Y_{r}, {\cal F} \otimes {\cal H}^{\nu} \otimes \omega_{Y_{r}}) = 0,
\mbox{ \ for \ } b > 0 \mbox{ \ and \ } \nu \geq 0,$$ by regarding the
cohomology sequence given by the short exact sequence
\[ 0 \>>> {\cal F} \otimes {\cal H}^{\nu} \otimes \omega_{Y_{r}}
\>>> {\cal F} \otimes {\cal H}^{\nu + 1} \otimes \omega_{Y_{r}}
\>>> {\cal F} \otimes {\cal H}^{\nu} \otimes \omega_{Y_{r+1}}
\>>> 0. \]
By induction we may assume that $${\cal F} \otimes {\cal H}^{\dim (Y_{r+1})}
\otimes \omega_{Y_{r+1}}$$ is generated by global sections
in $p$, and, using the cohomology sequence again one finds
the same for
$${\cal F} \otimes {\cal H}^{\dim (Y_{r})} \otimes \omega_{Y_{r}}.$$
\end{Beweis}

Let us apply (7.10) for $X = \P^{n}$ to study the behaviour
of zeros of homogeneous polynomials:
\begin{defn}{Example: Zeros of polynomials }\index{Zeros of polynomials}
Let $S$ be a finite set
of points in $\P^{n},$ for $ n \geq 2$,
and
$$\begin{array}{ll}
\omega_{\mu} (S) = \mbox{Min} \{& d > 0; \ \mbox{there exists }
s \in H^{0} (\P^{n}, {\cal O}_{\P^{n}} (d))\\
& \mbox{ with multiplicity at least } \mu \mbox{ in each } p \in S\}.
\end{array}$$
\end{defn}
\begin{defn}{Claim} For $\mu^{\prime} < \mu$ one has
\[ \frac{\omega_{\mu^{\prime}}(s)}{\mu^{\prime} + n -1} \leq
\frac{\omega_{\mu}(s)}{\mu}. \]
\end{defn}
\begin{Beweis}{Proof.} For $d = \omega_{\mu}(S)$ we have a section $s \in
H^{0} (\P^{n}, {\cal O}_{\P^{n}} (d))$ with divisor $D^{\prime}$ such
that $s$ has multiplicity at least $\mu$ in each $p \in S$. Choose
$$d^{\prime} = [ \frac{d}{\mu} (\mu^{\prime} + n - 1) ].$$ Since $d^{\prime}$
does not change if we replace $d$ by $\nu \cdot d + 1$ and $\mu$ by
$\nu \cdot \mu$ for $\nu \gg  0$, we can assume that $D^{\prime} = D + H$ for
a hyperplane $H$ not meeting $S$.\\
\ \\
For ${\cal L} = {\cal O}_{\P^{n}} (d^{\prime} + 1)$, for the divisor
$(d^{\prime} + 1) \cdot D$, and for
$N = d$, the assumptions of (7.8) hold true and
(7.10) tells us that
\[ {\cal O}_{\P^{n}} (n + d^{\prime} + 1) \otimes \omega_{\P^{n}}
\{ \frac{- (d^{\prime} + 1) \cdot D}{d} \} \]
is globally generated.
Since $d^{\prime} + 1 > \frac{d}{\mu} (\mu^{\prime} + n - 1)$ and hence
$ \mu^{\prime} + n - 1 \leq \frac{(d^{\prime} + 1) \cdot \mu}{d}$
we can apply (7.7,c) and find
\[ {\cal O}_{\P^{n}} (n + d^{\prime} + 1) \otimes \omega_{\P^{n}}
\{ \frac{- (d^{\prime} + 1) \cdot D}{d} \} \]
to be a subsheaf of
\[ {\cal O}_{\P^{n}} (d^{\prime}) \otimes \bigotimes_{p \in S}
{m}_{p}^{\mu^{\prime}}. \]
\end{Beweis}
\begin{defn}{Remark} For some generalizations and improvements
and for the history of this kind of problem
see \cite{EV83}.\\
\ \\

Up to this point, the applications discussed are
based on the global vanishing theorems
for invertible sheaves. J. Koll\'{a}r's vanishing theorem (5.6,a) for
restriction maps or, equivalently, vanishing theorems for the cohomology
of higher direct image sheaves ((6.16) and (6.17)) are nice tools
to study families of projective varieties:
\end{defn}
\begin{defn}{Example: Families of varieties over curves }\\
Let $X$ be
a projective manifold, $Z$ a non-singular curve and $f: X \rightarrow
Z$ a surjective morphism.
We call a locally free sheaf ${\cal F}$ on $Z$ {\it semi-positive},
if for some (or equivalently: all)
ample invertible sheaf ${\cal A}$ on $Z$ and for all
$\eta > 0$ the sheaf
\[ S^{\eta} ({\cal F}) \otimes {\cal A} \]
is ample. One has
\end{defn}
\begin{satz}{Theorem (Fujita \cite{Fuj78})} For $f: X \rightarrow Z$ as
above $f_{\ast} \omega_{X/Z}$ is
semi-positive.\index{Semipositivity theorem!Fujita}
\end{satz}
Here $\omega_{X/Z} = \omega_{X} \otimes f^{\ast} \omega_{Z}^{-1}$ is the
dualizing sheaf of $X$ over $Z$.
In \cite{Kol86} J. Koll\'{a}r used his vanishing theorem (5.6,a)
to give a simple proof of (7.15) and of its generalization to higher
dimensional $Z$, obtained beforehand by Y. Kawamata \cite{Kaw81}.
As usually, one obtains similar results adding the
${\cal L}^{(i)}$. For example, using the notations introduced in
(7.4) one has:
\begin{satz}{Variant} Assume in addition that ${\cal L}$ is an invertible
sheaf, $D$ an effective divisor and that ${\cal L}^{N}(-D)$ is semi-ample.
Then one has\\
{\rm a)} The sheaf $f_{\ast} ({\cal L}
\otimes \omega_{X/Z} \{ \frac{-D}{N} \})$ is semi-positive.\\
{\rm b)} If for a general fibre $F$ of $f$ one has $N \geq e(D \mid_{F})$,
then $f_{\ast} ({\cal L} \otimes \omega_{X/Z})$ is
semi-positive.
\end{satz}
\begin{satz}{Corollary} Under the assumption of (7.16) assume that $D$
contains a smooth fibre of $f$. Then, if $N \geq e (D \mid_{F})$, the sheaf
$f_{\ast} ({\cal L} \otimes \omega_{X/Z})$ is ample.
\end{satz}
\begin{Beweis}{Proof of (7.17)} Recall that a vectorbundle ${\cal F}$ on $Z$
is ample, if and only if $\tau^{\ast} {\cal F}$ is ample for some finite
cover
\[ \tau : Z^{\prime} \>>> Z. \]
Hence, if ${\cal L}^{\prime}$, $D^{\prime}, X^{\prime}$ and $f'$
are obtained by
pullback from the corresponding objects over $Z$, it is enough to show
that
\[ \tau^{\ast} f_{\ast} ({\cal L} \otimes \omega_{X/Z}) = f^{\prime}_{\ast}
({\cal L}^{\prime} \otimes \omega_{X^{\prime}/Z^{\prime}} ) \]
is ample. If, for the ramification locus $\Delta(Z^{\prime}/Z)$ of $Z^{\prime}$
over $Z$, the morphism $f$ is smooth in a neighbourhood of $f^{-1} (\Delta
(Z^{\prime}/Z))$ then $f^{\prime} : X^{\prime} \>>>
Z^{\prime}$ satisfies again the assumption made in (7.14).
\\
Choosing $Z^{\prime}$ to be ramified of order $N$ over the point $p \in Z$ with
$f^{-1} (p) \subseteq D$, we can reduce (7.17) to the case for which
$D$ contains the $N$-th multiple
of a fibre, say $N \cdot f^{-1} (p)$. We have
\[ f_{\ast} ({\cal L} \otimes \omega_{X/Z} \{ - \frac{D}{N} \} ) \subset
f_{\ast} ({\cal L} \otimes \omega_{X/Z}) \otimes {\cal O}_{Z} (-p) \]
and this inclusion is an isomorphism over some open set. In fact,
by (7.5,e) the assumption $N \geq e(D \mid_F)$ implies that
$\omega_{X/Z}$ and $\omega_{X/Z} \{ - \frac{D}{N} \}$ are the same
in a neighbourhood of a general fibre $F$. Hence
\[ f_{\ast} ({\cal L} \otimes \omega_{X/Z}) \otimes {\cal O}_{Z} (-p) \]
is semi-positive.\\
\end{Beweis}
\begin{Beweis}{Proof of (7.16)} As in the proof of (7.17) the
assumption made in part b) implies that
\[ f_{\ast} ({\cal L} \otimes \omega_{X/Z} \{ \frac{-D}{N} \}) \subset
f_{\ast} ({\cal L} \otimes \omega_{X/Z}) \]
is an isomorphism over some non-empty open subvariety and b) follows from a).\\
\ \\
By definition of $\omega_{X} \{ \frac{-D}{N} \}$ we can assume $D$ to
be a normal crossing divisor. Moreover, we can assume that the
multiplicities in $D$ are strictly smaller than $N$ and hence ${\cal L} =
{\cal L} (- [ \frac{D}{N} ] )$.
Let $p \in Z$ be a point in general position and $F = f^{-1} (p)$.
By (5.12,a) applied to the semiample sheaf ${\cal L} (F)$ we have a surjection
\[ H^{0} (X, {\cal L} (F) \otimes \omega_{X} (F)) \>>> H^{0} (F,
{\cal L} (F) \otimes \omega_{F}) \]
and
$ f_{\ast} ({\cal L} \otimes \omega_{X} ) \otimes {\cal O}_{Z} (2 \cdot p) $
is generated by global sections in a neighbourhood of $p$.
\begin{defn}{Claim} $\omega_{Z} (2 \cdot p) \otimes
\bigotimes^{\eta} f_{\ast} ({\cal L} \otimes \omega_{X/Z})$
is semi-positive for all $\eta > 0$.
\end{defn}
\begin{Beweis}{Proof} For $\eta = 1$ (7.18) holds true as  we just
found a trivial subsheaf
$ \bigoplus^{r} {\cal O}_{Z} $
of \[ f_{\ast} ({\cal L} \otimes
\omega_{X}) \otimes {\cal O}_{Z} (2 \cdot p) \]
of full rank.\\
In general, let $Y^{\prime} = X \times_{Z} \ldots \times_{Z} X$ ($\eta$-times)
be the fibre product and $\delta : Y \rightarrow Y^{\prime}$ be a
desingularization.
The induced morphisms $g^{\prime} : Y^{\prime}
\rightarrow Z$ and $g: Y \rightarrow Z$ satisfy:\\
\ \\
i) $Y^{\prime}$ is flat and Gorenstein over $Z$ and
\[ \omega_{Y^{\prime}/Z} =
\bigotimes_{j=1}^{\eta} pr_{j}^{\ast} \omega_{X/Z} .\]
ii) For
$${\cal M}^{\prime} = \bigotimes_{j=1}^{\eta} pr_{j}^{\ast} {\cal L}
\mbox{ \ and \ }
{\cal M} = \delta^{\ast} {\cal M}^{\prime}$$ one has an inclusion,
surjective at the general point of $Z$,
\[ g_{\ast} ({\cal M} \otimes \omega_{Y/Z}) \subset g^{\prime}_{\ast}
({\cal M}^{\prime} \otimes \omega_{Y^{\prime}/Z}) = \bigotimes^{\eta} f_{\ast}
({\cal L} \otimes \omega_{X/Z}) \]
iii) On a general fibre $\tilde{F}$ of $g$
the divisor
$$\Delta = \sum_{j=1}^{\eta} \delta^{\ast} pr_{j}^{\ast} D$$ has
normal crossings,
$ [ \frac{\Delta}{N} ]|_{\tilde{F}} = 0$, and ${\cal M}^{N} (-\Delta)$ is semi-ample.\\
\ \\
In fact, i) is the compatibility of relative dualizing sheaves
with pullback, ii) follows from flat base change and the inclusion
$\delta_{*} \omega_Y \subset \omega_{Y'}$ and iii) is obvious.
Using those three properties,
(7.18) follows from the case ``$ \eta = 1$'' applied to $g : Y \rightarrow
Z$.\\
\end{Beweis}
Since $S^{\eta}( \ \ )$ is a quotient of $\bigotimes^{\eta} ( \ \ )$
and since the quotient of a semi-positive sheaf is again semipositive,
one obtains (7.16).\\
\end{Beweis}
\begin{defn}{Remarks}\\
a) If $\dim Z > 1$, then the arguments used in the proof of (7.16,a)
show that for all $\eta > 0$ and ${\cal H}$ very ample the reflexive
hull ${\cal G}$ of
\[ S^{\eta} (f_{\ast} ({\cal L} \otimes \omega_{X/Z})) \otimes
{\cal H}^{\dim (X) + 1} \otimes \omega_{Z} \]
is generated by $H^{0}(Z, {\cal G})$ over some open set.
This led the
second author to the definition ``weakly-positive''
(see \cite{Vie86}).\\
b) One can make (7.17) more explicit and measure the degree of ampleness
by giving lower bound for the degree of invertible quotient sheaves of
$f_{\ast}({\cal L} \otimes \omega_{X/Z})$. Details have been worked out in
\cite{EV90}. These explicit bounds, together with the Kodaira-Spencer
map can be used for families of curves over $Z$ to give another proof
of the Theorem of Manin saying that the Mordell conjecture holds
true for curves over function fields over $\C$.
\end{defn}

The vanishing statements for higher direct images (6.11) and its
corollaries (7.5,d) and (7.6) are useful to study singularities.
\begin{defn}{Example: Deformation of quotient singularities}\\
\index{Deformation!of quotient singularities}
Let $X$ be a normal variety and $f: X \rightarrow S$ a flat morphism from
$X$ to a non-singular curve $S$.
\end{defn}
\begin{satz}{Theorem} Assume that $X$ is normal and that for some $s_{0}
\in S$ the variety $X_{0} = f^{-1} (s_{0})$ is a reduced normal surface
with quotient singularities. Then the general fibre $f^{-1} (\eta) = X_{\eta}$
has at most quotient singularities.
\end{satz}
{\sc Proof (see \cite{EV85}):}\\
If $Y$ is a normal surface with rational singularities then Mumford
\cite{Mum61} has shown
that for $p \in Y$, $$\Spec \ ({\cal O}_{Y,p}) - p = U$$ has only
finitely many non-isomorphic invertible sheaves. Hence for some $N > 0$
one has $\omega_{U}^{N} = {\cal O}_{U}$ and
for some $N > 0$ the reflexive hull $\omega_{X}^{[N]}$ of
$\omega_{Y}^{N}$ is invertible. Let
\[ \delta : Y^{\prime} \>>> Y \]
be a desingularization. Since $Y$ has rational singularities we have
\[ \delta_{\ast} \omega_{Y^{\prime}} = \omega_{Y} \]
and
\[ \delta^{\ast} \omega_{Y}/torsion \ \subset \omega_{Y^{\prime}}. \]
We may assume that $\delta^{\ast} \omega_{Y}/torsion = {\cal K}$ is invertible
and we write
\[ \omega_{Y^{\prime}} = {\cal K} \otimes {\cal O}_{Y^{\prime}} (F). \]
With this notation we have for some effective divisor $D$
\[ \delta^{\ast} \omega_{Y}^{[N]} = {\cal K}^{N} \otimes {\cal O}_{Y^{\prime}}
(D). \]
The divisors $D$ and $F$ can be used to characterize the quotient
singularities among the rational singularities:
\begin{defn}{Claim} $Y$ has quotient singularities if and only if
$ [ \frac{D}{N} ] \leq F$.
\end{defn}
\begin{Beweis}{Proof.} If one replaces $D$ and $N$ by some common multiple,
the inequality in (7.22) is not affected.
The question being local we may hence assume that $N$ is
the smallest integer with $\omega_{Y}^{[N]}$ invertible and
$\omega_{Y}^{[N]} \simeq {\cal O}_{Y}$.\\
\ \\
For ${\cal K}^{-1} = {\cal L}$ one has ${\cal L}^{N} = {\cal O}_{Y^{\prime}}
(D)$ and, as in (3.5), we can consider the cyclic cover $Z^{\prime}$ obtained
by taking the $N$-th root out of $D$. Let $Z$ be the normalization of $Y$ in
$k(Z^{\prime})$ and
\[
\begin{CD}
Z^{\prime} \> \delta^{\prime} >> Z \\
\V \pi^{\prime} VV \V V \pi V \\
Y^{\prime} \>> \delta > Y
\end{CD}
\]
the induced morphism ($Z$ is usually called {\it the canonical covering} of
 $Y$).
One has
\[ \pi^{\prime}_{*} \omega_{Z^{\prime}} = \bigoplus_{i = 0}^{N-1}
\omega_{Y^{\prime}} \otimes {\cal L}^{(i)}. \]
In fact, this follows from (3.11) by duality for finite morphisms
(see \cite{Har77}, p. 239) or, since
\[ {\cal L}^{-i} ([ \frac{i \cdot D}{N} ] + D^{(i)} ) =
{\cal L}^{-i} (D - [ \frac{(N-i) \cdot D}{N} ] ) = {\cal L}^{(N-i)} \]
from (3.16,d).\\
Recall that $Z$ has rational singularities if and only if
$\delta^{\prime}_{\ast} \omega_{Z^{\prime}} = \omega_{Z}$.\\
Assume that $Y$ has a quotient singularity in the point $p$.
If $\tilde{U}$ is the universal cover of $Y-p$ then the normalization
$\tilde{Z}$ of $Y$ in $k(\tilde{U})$ is non singular and,
by construction it dominates $Z$. Hence $Z$
has quotient singularities and $$\pi_{\ast} \delta^{\prime}_{\ast}
\omega_{Z^{\prime}} = \delta_{\ast} \pi^{\prime}_{\ast} \omega_{Z^{\prime}}$$
is reflexive. In particular $\delta_{\ast} \omega_{Y^{\prime}} \otimes
{\cal L}^{(1)}$ is reflexive. One has
$$\omega_{Y^{\prime}} \otimes {\cal L}^{(1)} =
{\cal K} \otimes {\cal L} \otimes {\cal O}_{Y^{\prime}} (F - [ \frac{D}{N} ])=
{\cal O}_{Y^{\prime}} (F - [ \frac{D}{N} ])$$
and the reflexivity of  $\delta_{\ast} \omega_{Y^{\prime}} \otimes
{\cal L}^{(1)}$ is equivalent to $F \geq [ \frac{D}{N} ]$.\\
\ \\
On the other hand, $F \geq [ \frac{D}{N} ]$ implies that
the summand $\delta_*{\cal O}_{Y^{\prime}} (F - [ \frac{D}{N} ])$
of $\delta_{\ast}
\pi^{\prime}_{\ast} \omega_{Z^{\prime}}$ has one section without zero
on $Y$.
Hence $\delta^{\prime}_{\ast} \omega_{Z^{\prime}}$ has a section
without zero on $Z-\pi^{-1}(p)$, which implies that
$\delta^{\prime}_{\ast} \omega_{Z^{\prime}}$ is
invertible and coincides with $\omega_Z$.
So $Z$ has a rational singularity and is Gorenstein.
Those singularities are called rational double points, and they are known to be
quotient singularities. Therefore $Y$ has a quotient singularity as well.\\
\end{Beweis}
\begin{Beweis}{Proof of (7.21)}
Let $\delta : X^{\prime} \rightarrow X$ be a desingularization. We assume
that the proper transform $X^{\prime}_{0}$ of $X_{0}$ is non-singular and
write $\delta_{0} = \delta \mid_{X^{\prime}_{0}}$.\\
By (7.5,d), applied in the case ``$D = 0$ '', we have a natural inclusion $$\delta_{0 \ast}
\omega_{X^{\prime}_{0}} \>>> \delta_{\ast} \omega_{X^{\prime}}
\otimes {\cal O}_{X} (X_{0}) \mid_{X_{0}}.$$
Since $X_0$ has rational singularities one has
$$\delta_{0 \ast} \omega_{X^{\prime}_{0}} = \omega_{X_{0}} = (\omega_X
\otimes {\cal O}_X(X_0))|_{X_0}.$$
One obtains $\delta_{\ast} \omega_{X^{\prime}} = \omega_{X}$, at least if
one replaces $S$ by a neighbourhood of $s_{0}$. Hence $X$ and $X_{\eta}$ have
at most rational singularities.\\
\ \\
Let us choose $N > 1$, such that both, $\omega^{[N]}_{X_{\eta}}$ and
$\omega^{[N]}_{X_{0}}$ are invertible (It might happen, nevertheless,
that $\omega_{X}^{[N]}$ is not invertible).
We may assume that we have choosen $X$ such that
\[ {\cal K} = \delta^{\ast} \omega_{X} / torsion \]
is an invertible sheaf. Hence
\[ {\cal K}_{0} = {\cal K} \otimes {\cal O}_{X'} (\delta^{\ast} X_{0})
\mid_{X^{\prime}_{0}} \]
is invertible and generated by global sections.
Moreover, one has maps
$$\delta^*_0\omega_{X_0} = \delta^*(\omega_X \otimes
{\cal O}_X(X_0))|_{X_0^{\prime}} \>>> {\cal K}_0$$ and ${\cal K}_0$
contains $\delta_{0}^{\ast} \omega_{X_{0}}/ torsion.$ Hence both sheaves
must be the same.\\
\ \\
Blowing up again, we can assume ${\cal M} = \delta^{\ast} \omega_{X}^{[N]}/
torsion$ to be locally free and isomorphic to ${\cal K}^{N}(D)$ where
$D$ is a divisor in the exceptional locus of $\delta$ such that $X^{\prime}_{0}
+ D$ has at most normal crossings.\\
\ \\
We can choose an embedding $\omega_{X} \hookrightarrow {\cal O}_{X}$ such that
the zero-set does not contain $X_{0}$. If correspondingly
${\cal K} = {\cal O}_{X^{\prime}} (- \Delta)$, for some $\Delta \geq 0$
we can choose the inclusion $\omega_{X} \hookrightarrow {\cal O}_{X}$
such that
$$D^{\prime} = N \cdot \Delta - D \geq 0.$$
Blowing up we can assume that $D^{\prime} + X^{\prime}_{0}$
is a normal crossing divisor. By definition
\[ {\cal O}_{X} (-D^{\prime}) = {\cal O}_{X} (-N \cdot \Delta + D) =
{\cal M} \]
and ${\cal O}_{X} (-D^{\prime})$ is $\delta$-numerically effective.\\
\ \\
It is our aim to use (7.6) in order to compare the sheaves
${\cal K}_0$ and $\delta_0^* \omega_{X_0}^{[N]}$ with $\cal K$
and with $\cal M$. Some unpleasant but elementary calculations
will show that the inequality (7.22), applied to $X_0$,
gives a similar inequality for the general fibre $X_\eta$.\\
\ \\
Let us write $$\omega_{X^{\prime}_{0}} = {\cal K}_{0} (F_{0})
\mbox{ \ and \ } \delta_0^*\omega_{X_0}^{[N]} = {\cal K}_0^N \otimes
{\cal O}_{X_0^{\prime}}(D_0).$$ By (7.22) one has
\[ F_{0} \geq [ \frac{1}{N} D_{0} ]. \]
Since
\[ {\cal K}_{0}^{N} \otimes {\cal O}_{X'_0}(D \mid_{X^{\prime}_{0}})
\simeq ({\cal K}^{N}  \otimes {\cal O}_{X'}(D)
\otimes {\cal O}_{X^{\prime}} (+ N \cdot
\delta^{\ast} X_{0})) \mid_{X^{\prime}_{0}} \]
is a subsheaf of $\delta_{0}^{\ast} \omega_{X_{0}}^{[N]}$ one obtains that
\[ (*) \hspace{4cm} F_{0} \geq [ \frac{1}{N} D \mid_{X^{\prime}_{0}} ] \ .
\hspace{4.5cm}\]
On the other hand, since
\[{\cal K}_0^{N-1} = {\cal O}_{X^{\prime}_{0}}(-(N-1)\Delta
\mid_{X^{\prime}_{0}}) \otimes {\cal O}_{X^{\prime}}((N-1)\delta^*X_0)
\mid_{X^{\prime}_{0}}\]
and
\[ [\frac{N-1}{N}D^{\prime}] = (N-1)\Delta - D + [\frac{1}{N}D] \]
one has
\[
\omega_{X^{\prime}_{0}} ( - [ \frac{(N-1)}{N} D^{\prime}
\mid_{X^{\prime}_{0}} ] ) =
{\cal K}_{0} (F_{0} - (N-1) \Delta \mid_{X^{\prime}_{0}} +
D \mid_{X^{\prime}_{0}} - [ \frac{1}{N} D \mid_{X^{\prime}_{0}} ] ) = \]
\[{\cal K}^{N}_{0} (D \mid_{X^{\prime}_{0}} + F_{0} - [
\frac{1}{N} D \mid_{X^{\prime}_{0}} ] ) \otimes {\cal O}_{X^{\prime}}
(-(N-1) \delta^{\ast} X_{0})\mid_{X'_{0}} = \hspace{2.5cm} \]
\[{\cal M} \mid_{X^{\prime}_{0}} (F_{0} - [ \frac{1}{N}
D\mid_{X^{\prime}_{0}} ] ) \otimes {\cal O}_{X^{\prime}} (\delta^{\ast}
X_{0}) \mid_{X'_{0}}.
\hspace{4.7cm} \]
By the inequality \ $(*)$ \ the sheaf
\[ \delta_{0 \ast} ( \omega_{X^{\prime}_{0}} ( - [ \frac{(N-1)}{N} D^{\prime}
\mid_{X^{\prime}_{0}} ] ) \otimes {\cal O}_{X^{\prime}} (- \delta^{\ast} X_{0})
\mid_{X'_{0}}) \]
contains $\delta_{0 \ast} ({\cal M} \mid_{X^{\prime}_{0}})$.
If $F$ is the divisor with $\omega_{X^{\prime}} =
{\cal K} \otimes {\cal O}_{X'}(F)$ we get from
(7.6) $\delta_{0 \ast} ({\cal M} \mid_{X^{\prime}_{0}})$ as a subsheaf of
\[ \delta_{\ast} \omega_{X} ( - [ \frac{(N-1)}{N} D^{\prime} ]) \mid_{X_{0}}
= \delta_*({\cal K} \otimes {\cal O}_{X'}(F) \otimes {\cal O}_{X'}
(-(N-1)\Delta+D-[\frac{1}{N}D]))\mid_{X_{0}}
\]
\[=
\delta_*({\cal K}^N \otimes {\cal O}_{X'}(D+F-[\frac{1}{N}D]))\mid_{X_{0}}
= \delta_{\ast} {\cal M} (F - [ \frac{1}{N} D ]) \mid_{X_{0}} . \hspace{2.5cm} \]
Of course, we have a natural morphism $\delta_{\ast}{\cal M} \>>>
\delta_{0 \ast} ({\cal M} \mid_{ X'_{0}})$
and the induced map
$$\delta_{\ast} {\cal M} \>>> \delta_{\ast} {\cal M}
(F - [ \frac{1}{N} D ] ) \mid_{X_{0}}$$ is surjective outside of the
singular locus of $X_0$.
We have natural maps
\[
\omega_X^{[N]} \>>> \delta_*\delta^*\omega_X^{[N]} \>>> \delta_*{\cal M}
\>>> \delta_*{\cal M}(F-[\frac{1}{N}D])|_{X_0} \>>> \omega_X^{[N]}|_{X_0}.
\]
The sheaf $\delta_*{\cal M}(F-[\frac{1}{N}D])$ is torsionfree and, since $X_0$
is a Cartier divisor, $\delta_*{\cal M}(F-[\frac{1}{N}D])|_{X_0}$
has no torsion as well. Therefore
\[
\omega_X^{[N]}|_{X_0}= \delta_*{\cal M}(F-[\frac{1}{N}D])|_{X_0}.
\]
Since ${\cal M} = \delta^{\ast} \omega_{X}^{[N]} / torsion$, this is
only possible if $F \geq [ \frac{1}{N} \cdot D]$. Hence
$F_{\eta} \geq [ \frac{D_{\eta}}{N}]$ where ``$\eta$'' denotes the restriction
to the general fibre and the theorem follows from (7.22).\\
\end{Beweis}
\begin{defn}{Example: Adjoint linear systems on surfaces}\\
\index{Adjoint linear systems on surfaces}
\index{Reider's theorem}
Studying adjoint linear systems on higher dimensional manifolds,
L. Ein and R. Lazarsfeld \cite{EL92} realized, that (7.7, b and c) and
(7.8) can be used to reprove part of I. Reider's theorem \cite{Rei88}
and to obtain similar results for threefolds.\\
We cordially thank them for allowing us to add their argument in the surface
case to the final version of these notes.
\end{defn}
\begin{satz}{Theorem (I. Reider, \cite{Rei88})}
Let $S$ be a non-singular projective surface, defined over an algebraically
closed field of characteristic zero, let $p \in S$
be a closed point and let $\cal L$ be a numerically
effective invertible sheaf on $S$.
Assume that $c_1({\cal L})^2 > 4$ and that for all curves $C$ with
$p \in C \subset S$ one has $c_1({\cal L}) \cdot C > 1$.\\
Then there is a section $\sigma \in H^0(S,{\cal L}\otimes \omega_S)$ with
$\sigma (p) \neq 0$.
\end{satz}
\begin{Beweis}{Proof, following \S 1 of \cite{EL92}}\\
Let $\cal H$ be an ample invertible sheaf on $S$ and let $m_p$ be the ideal
sheaf of $p$. For $\nu \gg  0$, one has
$H^2(S,{\cal L}^\nu \otimes {\cal H}^{-1}) = 0$ and by the Riemann-Roch formula
one finds $a,b \in \N$ with
$$ h^0(S,{\cal H}^{-1} \otimes {\cal L}^\nu \otimes m_p^{2\cdot \nu})
\geq h^0(S,{\cal H}^{-1} \otimes {\cal L}^\nu ) -
h^0(S,{\cal O}_S / m_p^{2\cdot \nu})$$
$$ \geq \frac{1}{2} \cdot c_1({\cal L})^2 \cdot \nu^2 + a \cdot \nu + b
- h^0(S,{\cal O}_S / m_p^{2\cdot \nu}).$$
Since
$$h^0(S,{\cal O}_S / m_p^{2\cdot \nu}) = \frac{1}{2} \cdot ( 4\cdot \nu^2
+ 2 \cdot \nu)$$
one finds for $\nu \gg 0$ a section $s$ of ${\cal H}^{-1} \otimes
{\cal L}^\nu $ with multiplicity $\mu_p \geq 2\cdot \nu$ in $p$ (see (7.7,b).
Let
$$D=\Delta + \sum_{i=1}^{r} \nu_i \cdot D_i $$
be the zero-divisor of $s$, where $\Delta$ is an effective divisor not
containing $p$ and $p \in D_i$ for $i=1 \cdots r$.
If $D'$ is any effective divisor the ${\cal H}^\eta (-D')$
will be ample for $\eta \gg 0$. Replacing $D$ by $\eta \cdot D + D'$ and
$\nu$ by $\nu \cdot \eta$ for a suitably choosen divisor
$D'$, we may assume that $r > 1$ and that
$\nu_1 > \nu_2 > \cdots > \nu_r$. Of course we can also assume that
$\mu_p$ is even.
\begin{defn}{Claim} If $\mu_p > 2 \cdot \nu_1$ then (7.24) holds true.
\end{defn}
\begin{Beweis}{Proof}
Let $N=\frac{\mu_p}{2}$ By the choice of $s$ one has $N \geq \nu$ and
$${\cal L}^N(-D) = {\cal L}^{N-\nu} \otimes {\cal H}$$ is ample.
By (7.8) one has
$$ H^1(S,\omega_S\{\frac{-D}{N}\}\otimes{\cal L}) = 0.$$
By (7.7,b), or just by definition of $\omega_S\{\frac{-D}{N}\}$,
one can find some open neighbourhood $U$ of $p$ such that
$$\omega_S\{\frac{-D}{N}\} \>>> \omega_S$$
is an isomorphism on $U - p$. Moreover, by (7.7,c), the inclusion factors
like
$$\omega_S\{\frac{-D}{N}\} \>>> \omega_S \otimes m_p \>>> \omega_S.$$
Let $\cal M$ be the sheaf $j_*j^*(\omega_S\{\frac{-D}{N}\}\otimes {\cal L})$
where $j:S-p \>>> S$ is the inclusion. For some nontrivial skyscraper sheaf
$\cal C$ supported in $p$, one has an exact sequence
$$0\>>> \omega_S\{\frac{-D}{N}\}\otimes {\cal L} \>>> {\cal M} \>>> {\cal C}
\>>> 0 . $$
Hence, $\cal M$ has a section $\sigma$ with $\sigma(p) \neq 0$.\\
\end{Beweis}
It remains to consider the case where $\mu_p \leq 2 \cdot \nu_1$.
If $\mu_p(D_i)$ denotes the multiplicity of $D_i$ in $p$, then
$$\mu_p = \sum_{i=1}^{r} \nu_i \cdot \mu_p(D_i) .$$
Since $r \geq 2$ this implies that $\mu_p(D_1)=1$.\\
\ \\
Let us take $N=\nu_1$. Again, $N \geq \nu$ and by (7.8)
$$ H^1(S,\omega_S\{\frac{-D}{N}\}\otimes{\cal L}) = 0.$$
One has an inclusion
$$ \omega_S\{\frac{-D}{N}\}\otimes{\cal L}\>>> \omega_S\otimes {\cal L}
(-D_1-[\frac{\Delta}{N}])$$
whose cokernel is a skyscraper sheaf.
Hence
$$ H^1(S,\omega_S\otimes{\cal L}(-D_1-[\frac{\Delta}{N}])) = 0$$
and the restriction map
$$H^0(S,\omega_S\otimes{\cal L}(-[\frac{\Delta}{N}])) \>>>
H^0(D_1,\omega_{D_1}\otimes{\cal L}(-D_1-[\frac{\Delta}{N}])|_{D_1})$$
is surjective.\\
\ \\
The right hand side contains a section $\sigma$ with $\sigma(p) \neq 0$,
since $$\deg ({\cal L}(-D_1-[\frac{\Delta}{N}])|_{D_1}) \geq 2.$$
In fact one has:
$$N \cdot \deg ({\cal L}(-D_1-[\frac{\Delta}{N}])|_{D_1}) =
\deg ({\cal L}^{N}(-D)|_{D_1}) + \sum^r_{i=2} \nu_i \cdot D_i \cdot D_1
+ (\Delta -[\frac{\Delta}{N}]) \cdot {D_1}$$
$$ \geq (N-\nu) \cdot c_1({\cal L}) \cdot D_1 + c_1({\cal H}) \cdot D_1 +
\sum^r_{i=2} \nu_i \cdot D_i \cdot D_1 $$
$$> (N-\nu) \cdot c_1({\cal L}) \cdot D_1 + \sum^r_{i=2} \nu_i \cdot \mu_p(D_i)
=(N-\nu) \cdot c_1({\cal L}) \cdot D_1 + (\mu_p - \nu_1).$$
Since $ c_1({\cal L}) \cdot D_1 \geq 2 $ and $\mu_p \geq 2 \cdot \nu$,
one obtains
$$N \cdot \deg ({\cal L}(-D_1-[\frac{\Delta}{N}])|_{D_1}) >
2\cdot N - 2 \cdot \nu + \mu_p - N \geq N. $$
\end{Beweis}
\begin{defn}{Remark}
It is likely that the other parts of I. Reider's theorem \cite{Rei88},
i.e. the lower bounds for $c_1({\cal L})^2$ and for
$c_1({\cal L}) \cdot C$ which imply that
$H^0(S,\omega_S\otimes {\cal L})$ separates
points and tangent directions, can be obtained in a similar way.
\end{defn}
\ \\
\Paragraph{Characteristic $p$ methods: Lifting of schemes}
\ \\
Up to this point we did not prove the degeneration of the Hodge spectral
sequence used in (3.2). Before doing so in Lecture 10 let us first recall
what we want to prove.

\Abschnitt{ } Let $X$ be a proper smooth variety (or a scheme) over a
field $k$. One introduces the {\it de Rham cohomology}
\index{de Rham cohomology}\index{$H^{b}_{DR} (X/k)$}
\[ H^{b}_{DR} (X/k) : = {\h}^{b} (X, \Omega^{\bullet}_{X/k}) \]
where $\Omega_{X/k}^{\bullet}$ is the complex of regular differential
forms, defined over $k$, the so called {\it de Rham complex}.\\
\ \\
In order to compute it, one introduces the {\it ``Hodge to de Rham'' spectral
sequence} associated to the Hodge filtration $\Omega_{X/k}^{\geq a}$
(see (A.25):\index{Spectral sequence!Hodge to de Rham}
\[ E_{1}^{ab} = H^{b} (X, \Omega_{X/k}^{a}) \Longrightarrow H_{DR}^{a+b}
(X /k). \]
If $k = \C$, the field of complex numbers, the classical Hodge
theory tells us that the Hodge spectral sequence
\[ E_{1 \ an}^{ab} = H^{b} (X_{an}, \Omega_{X_{an}}^{a})
\Longrightarrow \h^{a+b}(X_{an}, \Omega_{X_{an}}^{\bullet}) \]
degenerates in $E_{1}$, where $\Omega_{X_{an}}^{\bullet}$ is the de Rham
complex of holomorphic differential forms (see (A.25)).\\
\ \\
In fact, one has
\[ \h^{a+b} (X_{an}, \Omega_{X_{an}}^{\bullet}) =
H^{a+b} (X_{an}, \C) \]
and by Hodge theory
\[ \dim \ H^{l} (X_{an}, \C) = \sum_{a+b = l} \dim \ H^{b} (X_{an},
\Omega_{X_{an}}^{a}). \]
As explained in (A.22), this equality is equivalent to the degeneration of
$E_{1 \ an}$.\\
\ \\
As by Serre's GAGA theorems \cite{Ser56},
\[ H^{b} (X_{an}, \Omega_{X_{an}}^{a}) = H^{b} (X, \Omega_{X/ \C}^{a}), \]
the Hodge spectral sequence and the ``Hodge to de Rham'' spectral sequence
coincide and therefore the second one
degenerates in $E_{1}$ as well.\\

If $k$ is any field of characteristic zero, one obtains the same result
by flat base change:

\begin{satz}{Theorem} Let $X$ be a proper smooth variety over a field $k$
of characteristic zero. Then the Hodge to de Rham spectral sequence
degenerates in $E_{1}$ or, equivalently,
\[ \dim \ H^{l}_{DR} (X/k) = \sum_{a+b = l} \dim \ H^{b}
(X, \Omega_{X/k}^{a}). \]
\end{satz}
As we have already seen in Lecture 1 and 6, theorem (8.2) implies the Akizuki -
Kodaira - Nakano vanishing theorem:\\
\ \index{AKNV} \\
{\bf AKNV:} {\it If ${\cal L}$ is ample invertible, then
\[ H^{b} (X, \Omega_{X/k}^{a} \otimes {\cal L}^{-1}) = 0
\ \ \mbox{for \ \ $a + b < \dim \ X$} \]
(where, of course, $\Char \ k = 0$).}\\

Mumford \cite{Mum61} has shown that over a field $k$ of characteristic
$p > 0$ the $E_{1}$ degeneration for the Hodge to de Rham spectral sequence
fails and, finally, Raynaud \cite{Ray78} gave a counterexample to AKNV
in characteristic $p > 0$.\\
\ \\
The aim of this and of the next three lectures is to present Deligne-Illusie's
answer to those counterexamples:

\begin{satz}{Theorem (Deligne - Illusie \cite{DI87})}
\index{Vanishing theorem!Deligne Illusie Raynaud}Let $X$ be a
proper smooth variety over a perfect field $k$ of characteristic
$p \geq \dim \ X$ lifting to the ring $W_{2}(k)$ of the second Witt
vectors (see (8.11)). Then both, the $E_{1}$-degeneration of the
Hodge to de Rham spectral sequence and AKNV hold true.
\end{satz}
%
Actually they prove a slightly stronger version of (8.3), as will be
explained later.
Unfortunately one cannot derive from their methods the stronger vanishing
theorems mentioned in Lecture 5 such as Grauert-Riemenschneider or
Kawamata-Viehweg directly. As indicated, the geometric methods
of the first part of these Lecture Notes fail as well. It is still
an open problem which of those statements remains true under the
assumptions of (8.3).\\
\ \\
Finally, by standard techniques of reduction to characteristic $p > 0$,
Deligne - Illusie show:
\begin{satz}{Proposition}
Theorem (8.2) and AKNV over a field $k$ of
characteristic zero are consequences of theorem (8.3).
\end{satz}
\ \\

In the rest of this lecture, we will try to discuss to some extend elementary
properties and examples of liftings to $W_{2}(k)$.

\begin{defn}{Liftings of a scheme}\index{Liftings of a scheme}\\
Let $S$ be a scheme defined over $\F_{p}$ the field with $p$ elements.
\end{defn}
\begin{defn}{Definition} A {\it lifting} of $S$ to $\Z / p^{2}$ is a scheme
$\tilde{S}$, defined and flat over $\Z /p^{2}$, such that $S =
\tilde{S} \times_{\Z /p^{2}} \F_{p}$.
\end{defn}
\begin{defn}{Properties}\\
a) $S$ is defined by a nilpotent ideal sheaf (of square zero) in $\tilde{S}$.
In particular the inclusion $S \subset \tilde{S}$ or, if one prefers,
the projection
\[ {\cal O}_{\tilde{S}} \>>> {\cal O}_{S} \]
induces the identity on the {\it underlying topological spaces}
$(S)_{top}$ and $(\tilde{S})_{top}$.\\
b) From the exact sequence of $\Z/p^{2}$-modules
\[ 0 \>>> p \cdot \Z /p^{2} \>>> \Z / p^{2} \>>>
\Z /p \>>> 0 \]
one obtains, since ${\cal O}_{\tilde{S}}$ is flat over $\Z / p^{2}$,
the exact sequence of ${\cal O}_{\tilde{S}}$-modules
\[ 0 \>>> p \cdot {\cal O}_{\tilde{S}} \>>>
{\cal O}_{\tilde{S}} \>>> {\cal O}_{S} \>>> 0 \]
and, from the isomorphism of $\Z / p^{2}$-modules
\[ p : \Z /p \>>> p \cdot \Z/p^{2} ,\]
one obtains the isomorphism of ${\cal O}_{\tilde{S}}$-modules
\[ p : {\cal O}_{S} \>>> p \cdot {\cal O}_{\tilde{S}}.  \]
\end{defn}
\begin{defn}{Example} Let $k$ be a perfect field of characteristic $p$ and
$S = \Spec \ k$. Then $\tilde{S}$ exists and is uniquely determined
(up to isomorphism) by (8.7,b):\\
\ \\
$\tilde{S} = \Spec \ W_{2} (k)$,
where $W_{2} (k)$ is called the {\it ring of the second Witt vectors}
of $k$. \index{Second Witt vectors}\index{$W_{2} (k)$}\\
\ \\
In concrete terms, $W_{2} (k) = k \oplus k \cdot p$ as additive
group and the multiplication is defined by
\[ (x + y \cdot p) (x' + y' \cdot p) = x \cdot x' + (x \cdot y' + x' \cdot
y) \cdot p. \]
\end{defn}
\begin{defn}{Assumptions} Throughout Lectures 8 to 11 $S$ will
be a {\it noetherian scheme over $\F_{p}$ with a lifting
$\tilde{S}$ to }$\Z / p^{2}$.\\
$X$ will denote a noetherian $S$-scheme, $D \subset X$ will be a reduced Cartier
divisor. $X$ will be supposed to be {\it smooth over} $S$, which means that
locally $X$ is \'{e}tale over the affine space $\AF^{n}_{S}$ over $S$
(here $n = \dim_{S} \ X$). $D$ {\it will be a normal crossing divisor over
$S$}, i.e.:\\
$D$ is the union of smooth divisors $D_i$ over $S$ and
one can choose the previous \'{e}tale cover such that the coordinates of
$\AF_{S}^{n}$ pull back to a parameter system $(t_{1}, \ldots, t_{n})$ on
$X$, for which $D$ is defined by
$$t_{1} \cdot \ldots \cdot t_{r}, \mbox{ \ for some \ }
r \leq n.$$ We allow $D$ to be empty.
\end{defn}
\begin{defn}{Definition} For $X$ smooth and $D$ a normal crossing
divisor over $S$, we define the sheaf $\Omega_{X/S}^{1} \ (\log \ D)$ of
{\it one forms with logarithmic poles along} $D$ as the ${\cal O}_{X}$-sheaf
generated locally by
$$\frac{dt_{i}}{t_{i}} \mbox{ , for \ } i \leq r \mbox{ , and by \ } dt_{i}
\mbox{ , for \ } i > r$$ (where we use the notation from (8.9)).
$\Omega_{X/S}^{1} (\log \ D)$
is locally free of rank $n$ and the definition coincides with
the one given in (2.1) for $S = \Spec \ k$. Finally we define
\[ \Omega_{X/S}^{a} (\log \ D) = \bigwedge^{a} \Omega_{X/S}^{1} (\log \ D). \]
\end{defn}
\begin{defn}{Definition} A {\it lifting of}
$$D = \sum_{j=1}^{r} D_{j} \ \subset \ X$$
to $\tilde{S}$ consists of a scheme $\tilde{X}$ and subschemes
$\tilde{D}_{j}$ of $\tilde{X}$, all defined and flat over $\tilde{S}$
such that $X = \tilde{X} \times_{\tilde{S}} S$ and $D_{j} = \tilde{D}_{j}
\times_{\tilde{S}}S$. We write
$$\tilde{D} = \sum_{j=r}^{r} \tilde{D}_{j}.$$
If $k$ is a perfect field of characteristic $p$ and $S = \Spec \ k$, we say
that $(X,D)$ {\it admits a lifting to} $W_{2}(k)$ if liftings $\tilde{X}$ and
$\tilde{D}_{j}$ exist over $\tilde{S} = \Spec \ W_{2}(k)$.\\
\ \\
If ${\cal L}$ is an invertible sheaf on $X$, we say that $X$ and $\cal L$
{\it admit a  lifting to} $W_2(k)$ if there is a lifting $\tilde{X}$ of $X$
over $\tilde{S} = \Spec \ W_2(k)$ and an invertible sheaf $\tilde{\cal L}$ on
$\tilde{X}$ with $\tilde{\cal L}|_X = {\cal L}$.
\end{defn}
\begin{defn}{Remark} Of course, $\tilde{X}$ is also a lifting of the
$\F_{p}$-scheme $X$ to a scheme $\tilde{X}$ over $\Z / p^{2}$. In particular,
(8.7.a) remains true and we have
\[ (\tilde{X})_{top} = (X)_{top} \ \ \mbox{and \ \ $(\tilde{D})_{top} =
(D)_{top}$}. \]
One can make (8.7,b) more precise:
\end{defn}
\begin{satz}{Lemma} Let $X$ be smooth over $S$ and let $\tilde{X}$ be a scheme
over $\tilde{S}$ with $\tilde{X} \times_{\tilde{S}} S = X$. Then the following
conditions are equivalent.\\
{\rm a)} $\tilde{X}$ is smooth over $\tilde{S}$.\\
\ \\
{\rm b)} $\tilde{X}$ is a lifting of $X$ to $\tilde{S}$.\\
\ \\
{\rm c)} There is an exact sequence of ${\cal O}_{\tilde{X}}$-modules
\[ 0 \>>> p \cdot {\cal O}_{\tilde{X}} \>>>
{\cal O}_{\tilde{X}} \> r >> {\cal O}_{X} \>>> 0 \]
together with an ${\cal O}_{\tilde{X}}$-isomorphism
\[ p: {\cal O}_{X} \>>> p \cdot {\cal O}_{\tilde{X}} \]
satisfying $$p(x) = p \cdot \tilde{x} \mbox{ , for \ }
\tilde{x} \in {\cal O}_{\tilde{X}} \mbox{ , and \ } x = r (\tilde{x}).$$
{\rm d)} If $\tilde{U} \subseteq \tilde{X}$ is an open subscheme, $U$ its
image in $X$,
\[ \pi : U \>>> \AF^{n}_{S} = {\bf Spec} \ {\cal O}_{S} [t_{1},
\ldots, t_{n}] \]
an \'{e}tale morphism and if $\tilde{\varphi}_{1}, \ldots, \tilde{\varphi}_{n}
\in {\cal O}_{\tilde{U}}$ satisfy $r(\tilde{\varphi}_{i}) = \varphi_{i} =
\pi^{*} t_{i}$, then $\pi$ extends to
an \'{e}tale morphism
\[ \tilde{\pi} : \tilde{U} \>>> \AF_{\tilde{S}}^{n} = {\bf Spec} \
{\cal O}_{\tilde{S}} [t_{1}, \ldots, t_{n}] \]
with $\tilde{\pi}^{*} (t_{i}) = \tilde{\varphi}_{i}$ for $i = 1, \ldots, n$.\\
\ \\
{\rm e)} For each $a \geq 0$ one has an exact sequence of
${\cal O}_{\tilde{X}}$-modules
\[ 0 \>>> p \cdot \Omega^{a}_{\tilde{X} / \tilde{S}} \>>>
\Omega^{a}_{\tilde{X} / \tilde{S}} \> r >> \Omega^{a}_{X/S}
\>>> 0 \]
and an ${\cal O}_{\tilde{X}}$-isomorphism
\[ p : \Omega^{a}_{X/S} \>>> p \cdot
\Omega^{a}_{\tilde{X} / \tilde{S}} \]
satisfying $$p(\omega) = p \cdot \tilde{\omega} \mbox{ , for \ }
\tilde{\omega} \in \Omega^a_{\tilde{X}/\tilde{S}}
\mbox{ , and \ } \omega = r (\tilde{\omega}).$$
\end{satz}
\begin{Beweis}{Proof} A smooth morphism is flat, and flatness implies c).
Obviously d) implies e) and a). Hence the only part to prove is that c)
implies d).\\
Using the notations from d) (for $\tilde{X} = \tilde{U}$) we can, of course,
define
\[ \tilde{\pi} : \tilde{X} \>>> \AF_{\tilde{S}}^{n}
\mbox{ \ with \ } \tilde{\pi}^{*} (t_{i}) = \tilde{\varphi}_{i}.\]
Given a relation $\sum \tilde{\lambda}_{\nu} \tilde{m}_{\nu} = 0$ in
${\cal O}_{\tilde{X}}$ between different monomials $\tilde{m}_{\nu}$ in
$\tilde{\varphi}_{1}, \ldots, \tilde{\varphi}_{n}$ the exact sequence in c)
implies $\tilde{\lambda}_{\nu} = p \cdot \tilde{\mu}_{\nu}$ for
$\tilde{\mu}_{\nu} \in {\cal O}_{\tilde{X}}$ and the isomorphism
in c) shows that one has $$\sum \mu_{\nu} \cdot m_{\nu} = 0 \mbox{ \ for \ }
\mu_{\nu} = r(\tilde{\mu}_{\nu}) \mbox{ \ and \ } m_{\nu} = r(\tilde{m}_{\nu}).$$
Hence $\mu_{\nu} = 0$ as well as $\tilde{\mu}_{\nu} = 0$.\\
\ \\
If $g_{1}, \ldots, g_{r}$ are locally independent generators of ${\cal O}_{X}$
as a ${\cal O}_{\AF^{n}_{S}}$-module, and if $\tilde{g}_{1}, \ldots,
\tilde{g}_{r}$ are liftings to ${\cal O}_{\tilde{X}}$, then each $\tilde{x} \in
{\cal O}_{\tilde{X}}$ verifies $$x = r (\tilde{x}) = \sum^r_{i=1} \lambda_{i} g_{i}$$
for some $\lambda_{i} \in {\cal O}_{\AF_{S}^{n}}$. If $\tilde{\lambda}_{1}, \ldots,
\tilde{\lambda}_{r}$ are liftings of $\lambda_{1}, \ldots, \lambda_{r}$ to
${\cal O}_{\AF^{n}_{\tilde{S}}}$, then
$$\tilde{x} - \sum^r_{i=1} \tilde{\lambda}_{i} \tilde{g}_{i}
\in p \cdot {\cal O}_{\tilde{X}}$$ and one can find
$ \tilde{\mu}_{i} \in {\cal O}_{\AF^{n}_{\tilde{S}}} $ with
\[ \tilde{x} - \sum^r_{i=1} \tilde{\lambda}_{i} \tilde{g}_{i} = p ( \sum^r_{i=1} \mu_{i} g_{i})
= \sum^r_{i=1} p \cdot \tilde{\mu}_{i} \tilde{g}_{i}, \]
and
$$\tilde{x} = \sum^r_{i=1} (\tilde{\lambda}_{i} + p \cdot \tilde{\mu}_{i})
\cdot \tilde{g}_{i}.$$
In other terms, $\tilde{g}_{1}, \ldots \tilde{g}_{r}$
are generators of ${\cal O}_{\tilde{X}}$ as a
${\cal O}_{\AF_{\tilde{S}}^{n}}$-module. They are independent by the same
argument which gave the independence of the $m_{\mu}$ above.
${\cal O}_{\tilde{X}}$ as a free ${\cal O}_{\AF^{n}_{\tilde{S}}}$-module
is flat. \\
\ \\
Finally, (locally in $X$)
$$\Omega^{1}_{X/S} = \pi^{*} \Omega^{1}_{\AF^{n}_{S}} = \bigoplus^n_{1=1}
{\cal O}_{X} d \varphi_{i}$$
and
$$\tilde{\pi}^{*} \Omega^{1}_{\AF^{n}_{\tilde{S}}} = \bigoplus^n_{i=1}
{\cal O}_{\tilde{X}} d \tilde{\varphi}_{i}$$
surjects to $\Omega^{1}_{\tilde{X}/ \tilde{S}}$. In fact, if
$\tilde{\omega} \in \Omega^{1}_{\tilde{X} / \tilde{S}}$,
\[ \tilde{\omega} - \sum^n_{i=1} \tilde{\lambda}_{i} d
\tilde{\varphi}_{i} \in \im
(\Omega^{1}_{X/S} \> \cdot p >> \Omega^{1}_{\tilde{X} / \tilde{S}}) \]
for
some $\tilde{\lambda}_{i} \in {\cal O}_{\tilde{X}}$ and, as above, one can
modify the $\tilde{\lambda}_{i}$ to get
$$\tilde{\omega} = \sum^n_{i=1} (\tilde{\lambda}_{i} + p \cdot
\tilde{\mu}_{i}) d \tilde{\varphi}_{i}.$$
As $\tilde{\pi}^{*} \Omega^{1}_{\AF^{n}_{\tilde{S}}} \>>>
\Omega^{1}_{\tilde{X} / \tilde{S}}$ is injective as well,
$\tilde{\pi}$ is \'{e}tale.\\
\end{Beweis}
\begin{satz}{Lemma} Let $X$ be a smooth $S$-scheme and $$D = \sum_{j=1}^{r}
D_{j}$$ be a normal crossing divisor over $S$. Let $\tilde{X}$ be
a lifting of $X$
to $\tilde{S}$ and $\tilde{D}_{j} \subseteq \tilde{X}$ subschemes with
\[ \tilde{D}_{j} \otimes_{\tilde{S}} S = D_{j}  \]
for $j = 1, \ldots, r$. Then the following conditions are equivalent:\\
{\rm a)} $$\tilde{D} = \sum_{j=1}^{r} \tilde{D}_{j} \ \subset \ \tilde{X}$$
is a lifting of $D \subset X$ to $\tilde{S}$.\\
\ \\
{\rm b)} The components of $\tilde{D}$ are Cartier divisors in $\tilde{X}$.\\
\ \\
{\rm c)} If in (8.13,d) one knows that $D\mid_{U}$ is the zero-set of
$\varphi_{1} \cdot \ldots \cdot \varphi_{s}$, then one can
choose $\tilde{\pi} :
\tilde{U} \>>> \AF^{n}_{\tilde{S}}$, such that $\tilde{D}_{j}
\mid_{\tilde{U}}$ is the zero set of $\tilde{\pi}^{*} (t_{j})$.
\end{satz}
\begin{Beweis}{Proof} If $\tilde{D} \subset \tilde{X}$ is a lifting of
$D \subset X$, then the flatness of $\tilde{X}$ and $\tilde{D}_{j}$ over
$\tilde{S}$ implies that the ideal sheaf $J_{\tilde{D}_{j}}$
of $\tilde{D}_{j}$ is flat over $\tilde{S}$. We have again an exact sequence
\[ 0 \>>> p \cdot J_{\tilde{D}_{j}} \>>>
J_{\tilde{D}_{j}} \>>> J_{D_{j}} \>>> 0 \]
where $J_{D_{j}}$ is the ideal sheaf of $D_{j}$, and an isomorphism
$$p : J_{D_{j}} \>>> p \cdot J_{\tilde{D}_{j}}.$$
If $\tilde{\varphi}_{j}$
is a lifting of $\varphi_{j}$ to $J_{\tilde{D}_{j}}$, then for any
$\tilde{g} \in J_{\tilde{D}_{j}}$ one has $ g = \lambda \cdot \varphi_{j}$
and
$$ \tilde{g} - \tilde{\lambda} \cdot \tilde{\varphi}_{j} \in p \cdot
I_{\tilde{D}_{j}}$$
is of the form $p \cdot \tilde{\mu} \cdot
\tilde{\varphi}_{j} = p (\mu \cdot \varphi_{j})$ for some $\tilde{\mu} \in
{\cal O}_{\tilde{X}}$. Hence $\tilde{\varphi}_{j}$ is a defining equation
for $\tilde{D}_{j}$.\\
By (8.13,d) b) implies c) and obviously c) implies a).\\
\end{Beweis}
\begin{defn}{Definition} Using the notations from (8.14,c) and (8.13,d) we
define for a lifting $\tilde{D} \subset \tilde{X}$ of $D \subset X$ to
$\tilde{S}$ the sheaf $$\Omega^{1}_{\tilde{X}/\tilde{S}} (\log \
\tilde{D})$$ \index{$\Omega^{a}_{\tilde{X}/\tilde{S}} (\log \
\tilde{D})$} to be the ${\cal O}_{\tilde{X}}$-sheaf generated by
$$\frac{d \tilde{\varphi}_{j}}{\tilde{\varphi}_{j}} \mbox{ \ for \ }
j = 1, \ldots, s \mbox{ \ and \ } d \tilde{\varphi}_{j} \mbox{ \ for \ }
j = s + 1, \ldots, n.$$
\end{defn}
\begin{defn}{Properties} \\
a) For all $a$ the sheaves
$$ \Omega^{a}_{\tilde{X}/\tilde{S}} (\log \ \tilde{D}) = \bigwedge^{a}
\Omega^{1}_{\tilde{X} / \tilde{S}} (\log \ \tilde{D})$$
are locally free over ${\cal O}_{\tilde{X}}$.\\
b) One has an exact sequence of ${\cal O}_{\tilde{X}}$-modules
\[ 0 \>>> p \cdot \Omega^{a}_{\tilde{X}/\tilde{S}} (\log \ \tilde{D})
\>>> \Omega^{a}_{\tilde{X} / \tilde{S}} (\log \ \tilde{D}) \>>>
\Omega^{a}_{X/S} (\log \ D) \>>> 0 \]
and an ${\cal O}_{\tilde{X}}$-isomorphism
\[ p : \Omega^{a}_{X/S} (\log \ D) \>>> p \cdot
\Omega^{a}_{\tilde{X}/\tilde{S}} (\log \ \tilde{D}). \]
\end{defn}
\begin{satz}{Proposition} Let $X$ be a smooth $S$-scheme.\\
\ \\
{\rm a)} Locally in the Zariski topology $X$ has a lifting $\tilde{X}$ to
$\tilde{S}$.\\
\ \\
{\rm b)} If $\tilde{X}$ is a lifting of $X$ to $\tilde{S}$,
if $\tilde{X}$ is affine and $Y$ a complete intersection in $X$,
then there exists a lifting $\tilde{Y}$
of $Y$ to $\tilde{S}$ and an embedding $\tilde{Y} \subset \tilde{X}$.\\
\ \\
{\rm c)} In particular, if $D$ is a
$S$-normal crossing divisor on $X$ then locally
in the Zariski topology $D \subset X$ has a lifting $\tilde{D} \subset
\tilde{X}$ to $\tilde{S}$.
\end{satz}
\begin{Beweis}{Proof} Locally $X$ is a complete intersection in an affine
space over $S$. Hence a) follows from b). In b) we may assume that $Y$ is
a divisor, let us say the zero set of $\varphi \in {\cal O}_{X}$. We can
choose $\tilde{Y}$ to be the zero set of any lifting $\tilde{\varphi} \in
{\cal O}_{\tilde{X}}$ of $\varphi$. In fact, the flatness follows easily
from (8.13,c) or from the following argument.\\
Choose
$$\pi : X \>>> \AF^{n}_{S} = \Spec \ {\cal O}_{X} [t_{1},
\ldots, t_{n} ]$$ with $\varphi = \pi^{*} (t_{1})$. By (8.13,d) $\pi$
extends to an \'{e}tale map $\tilde{\pi} : \tilde{X} \>>>
\AF^{n}_{\tilde{S}}$ with $\tilde{\varphi} = \tilde{\pi}^{*} (t_{1})$.\\
\end{Beweis}
\Abschnitt{Isomorphisms between liftings}
Let in the sequel $X$ be a smooth $S$-scheme, $D \subset X$ be an
$S$-normal crossing divisor and let, for $i = 1,2$, \ $\tilde{D}^{(i)}
\subset \tilde{X}^{(i)}$ be two liftings of $D \subset X$ to $\tilde{S}$.

\begin{defn}{Notations} A morphism $u : \tilde{X}^{(1)} \rightarrow
\tilde{X}^{(2)}$ is called an {\it isomorphism of liftings}
\index{Isomorphism of liftings}
\[ u: (\tilde{X}^{(1)}, \tilde{D}^{(1)}) \>>> (\tilde{X}^{(2)},
\tilde{D}^{(2)}) \]
if $u \mid_{X} = id_{X}$ and if
\[ u^{*} ({\cal O}_{\tilde{X}^{(2)}} (- \tilde{D}^{(2)})) =
{\cal O}_{\tilde{X}^{(1)}} (- \tilde{D}^{(1)}). \]
\end{defn}
\begin{defn}{Remark} We have seen in (8.12) that $(\tilde{X}^{(i)})_{top}
= (X)_{top}$ and hence $u$ is the identity on the topological
spaces. Henceforth,
giving $u$ is the same as giving the morphism $$u^{*}:
{\cal O}_{\tilde{X}^{(2)}} \>>> {\cal O}_{\tilde{X}^{(1)}}$$ of
sheaves of rings on $(X)_{top}$. The assumption $u \vert_{X} = id_{X}$ forces
$u^{*}$ to be an isomorphism.
\end{defn}
\begin{satz}{Lemma} Locally in the Zariski topology there exists an isomorphism
of liftings
\[ u : (\tilde{X}^{(1)}, \tilde{D}^{(1)}) \>>> (\tilde{X}^{(2)},
\tilde{D}^{(2)}). \]
\end{satz}
\begin{Beweis}{Proof} Locally the diagonal $\Delta \subset X \times X$
is a complete intersection and we can lift it to
$$\tilde{\Delta}
\subset \tilde{X}^{(1)} \times \tilde{X}^{(2)}.$$
For example, if $\varphi_{1}, \ldots, \varphi_{n}$
are local parameters on $X$ and $\tilde{\varphi}_{1}^{(i)}, \ldots,
\tilde{\varphi}_{n}^{(i)}$ liftings in ${\cal O}_{\tilde{X}^{(i)}}$ such
that $D^{(i)}$ is the zero locus of $\varphi_{1}^{(i)} \cdot \ldots \cdot
\varphi_{s}^{(i)}$, then we can choose $\tilde{\Delta}$ to be defined
by
\[ \tilde{\varphi}^{(1)}_{j} \otimes 1 - 1 \otimes \tilde{\varphi}^{(2)}_{j}
\ \ \mbox{for \ \ $ j = 1, \ldots, n$.} \]
We have isomorphisms of liftings
\[ p_{1} : \tilde{\Delta} \>>> \tilde{X}^{(1)}, \ \ \ p_{2}:
\tilde{\Delta} \>>> \tilde{X}^{(2)} \]
and $u  = p_{2} \circ p_{1}^{-1}$ satisfies
\[ u^{*} ({\cal O}_{\tilde{X}^{(2)}} ( - \tilde{D}^{(2)})) =
{\cal O}_{\tilde{X}^{(1)}} ( - \tilde{D}^{(1)}). \]
\end{Beweis}
Let
$$u, v: (\tilde{X}^{(1)}, \tilde{D}^{(1)}) \>>> (\tilde{X}^{(2)},
\tilde{D}^{(2)})$$ be two isomorphisms of liftings. For $\tilde{x} \in
{\cal O}_{\tilde{X}^{(2)}}$ one has
\[ (u^{*} - v^{*}) (p \cdot \tilde{x}) = p ( u^{*} - v^{*}) (\tilde{x}) =
p (id - id) (x) = 0 \]
therefore $(u^{*} - v^{*}) \vert_{p \cdot {\cal O}_{\tilde{X}^{(2)}}} = 0$.
Of course, the map
\[ {\cal O}_{X} = {\cal O}_{\tilde{X}^{(2)}}
/ p \cdot {\cal O}_{\tilde{X}^{(2)}}
\>>> {\cal O}_{X} = {\cal O}_{\tilde{X}^{(1)}} / p \cdot
{\cal O}_{\tilde{X}^{(1)}} \]
induced by $(u^{*} - v^{*})$
is zero as well, and $(u^{*} - v^{*})$ factors through
\[ (u^{*} - v^{*}) : {\cal O}_{X} \>>> p \cdot
{\cal O}_{\tilde{X}^{(1)}} = p ({\cal O}_{X}). \]
For $x,y \in {\cal O}_{X}$ with liftings $\tilde{x}, \tilde{y} \in
{\cal O}_{\tilde{X}^{(2)}}$ one has
\[ (u^{*} - v^{*}) (\tilde{x} \cdot \tilde{y}) = u^{*}(\tilde{x})
\cdot u^{*} (\tilde{y}) - v^{*} (\tilde{x})
\cdot v^{*}(\tilde{y}) = x \cdot (u^{*} - v^{*}) (\tilde{y}) +
y \cdot (u^{*} - v^{*}) (\tilde{x}). \]
Hence $p^{-1} \circ (u^{*} - v^{*}) : {\cal O}_{X} \>>>
{\cal O}_{X}$ is a derivation and factors through
\[ {\cal O}_{X} \> d >> \Omega_{X/S}^{1} \>>> {\cal O}_{X}  \]
where we denote the second morphism by $(u^{*} - v^{*})$ again. If
$\tilde{t}^{(i)}_{j}$ is a local equation for $\tilde{D}_{j}^{(i)}$,
$i = 1,2$, with reduction $t_{j} \in {\cal O}_{X}$, then
\[ u^{*}(\tilde{t}_{j}^{(2)}) =
\tilde{t}^{(1)}_{j} \cdot (1 + p \cdot \lambda) \]
and
\[ v^{*} (\tilde{t}_{j}^{(2)}) = \tilde{t}^{(1)}_{j} (1 + p \cdot \mu). \]
Hence
$$(u^{*} - v^{*}) (dt_{j}) = t_{j} \cdot (\lambda - \mu ) \in t_{j} \cdot
{\cal O}_{X}$$ and $p^{-1} \circ (u^{*} - v^{*})$ even factors through
\[ {\cal O}_{X} \> d >> \Omega^{1}_{X/S} (\log \ D) \>>> {\cal O}_{X}. \]
\begin{satz}{Proposition} Keeping the notations from (8.19) let
$$u:(\tilde{X}^{(1)}, \tilde{D}^{(1)}) \>>>
(\tilde{X}^{(2)}, \tilde{D}^{(2)})$$ be an isomorphism
of liftings. Then
\[ \{ v:(\tilde{X}^{(1)}, \tilde{D}^{(1)}) \>>>
(\tilde{X}^{(2)}, \tilde{D}^{(2)}); \ v \mbox{ isomorphism of liftings  $\}$} \]
is described by the affine space
\[ u^{*} + \Hom_{{\cal O}_{X}} (\Omega^{1}_{X/S} (\log \ D), {\cal O}_{X}). \]
\end{satz}
\begin{Beweis}{Proof}
It just remains to show that for $$\varphi \in \Hom_{{\cal O}_{X}}
(\Omega^{1}_{X/S}
(\log \ D), {\cal O}_{X})$$ we can find $v$. Define
\[ v^{*} : {\cal O}_{\tilde{X}^{(2)}} \>>>
{\cal O}_{\tilde{X}^{(1)}}  \]
by
\[ v^{*}(\tilde{x}) = u^{*} (\tilde{x}) - p \cdot \varphi (dx). \]
If $\tilde{t}^{(i)}_{j}$ is as above an equation of $\tilde{D}^{(i)}_{j}$,
\[ v^{*} (\tilde{t}^{(2)}_{j}) = \tilde{t}^{(1)}_{j} \cdot (1 + p \cdot
\lambda) - p \cdot t \cdot \gamma \ \ \mbox{for some \ \ $\gamma \in
{\cal O}_{X}$}, \]
or
\[ v^{*} (\tilde{t}^{(2)}_{j}) = \tilde{t}^{(1)}_{j} (1 + p (\lambda -
\gamma)) \]
and $v^{*}$ satisfies the conditions posed in (8.19.).\\
\ \\
\end{Beweis}
\begin{satz}{Proposition} Let $X$ be an affine scheme, smooth over $S$ and
let $D$ be a normal crossing divisor over $S$. Let $\tilde{D}^{(i)}
\subset \tilde{X}^{(i)}$ be two liftings of $D \subset X$ to $\tilde{S}$.
Then there exists an isomorphism of liftings $$u : (\tilde{X}^{(1)},
\tilde{D}^{(1)}) \>>> (\tilde{X}^{(2)}, \tilde{D}^{(2)}).$$
\end{satz}
\begin{Beweis}{Proof} Of course, (8.22) just says that the isomorphisms
of liftings over a fixed open set form a ``torseur'' under the group
$$\Hom_{{\cal O}_{X}}(\Omega^{1}_{X/S} (\log \ D), {\cal O}_{X})$$
and, since $X$ is affine and
\[ H^{1} (X, {{\cal H}om}_{{\cal O}_{X}}
(\Omega^{1}_{X/S} (\log \ D), {\cal O}_{X})) = 0 \]
one obtains (8.23). However, to state this in the
elementary language used up to
now, let us avoid this terminology:\\
\ \\
From (8.21) we know that there is an affine open cover ${\cal U} =
\{ X_{\alpha} \}$ of $X$ and isomorphisms of liftings
\[ u_{\alpha} : (\tilde{X}^{(1)}_{\alpha}, \tilde{D}^{(1)}_{\alpha})
\>>> (\tilde{X}^{(2)}_{\alpha}, \tilde{D}_{\alpha}^{(2)}). \]
By (8.22) \ $p^{-1} \circ (u^{*}_{\alpha} - u^{*}_{\beta})$ defines a
$1$-cocycle with values in the sheaf
$$ {{\cal H}om}_{{\cal O}_{X}}
(\Omega^{1}_{X/S} (\log \ D), {\cal O}_{X}).$$
Since $$H^{1} (X, {{\cal H}om}_{{\cal O}_{X}}
(\Omega^{1}_{X/S} (\log \ D), {\cal O}_{X}) = 0$$
we find
$$\varphi_{\alpha} \in \Gamma (X_{\alpha},
{{\cal H}om}_{{\cal O}_{X}}
(\Omega^{1}_{X/S} (\log \ D), {\cal O}_{X}))$$
in some possibly finer cover $\{ X_{\alpha} \}$
such that
$$p^{-1} \circ (u^{*}_{\alpha} - u^{*}_{\beta}) = \varphi_{\alpha} -
\varphi_{\beta}.$$
Hence the isomorphisms of liftings
$u^{*}_{\alpha} - {p \cdot \varphi^{*}_{\alpha}}$
glue together to $u : \tilde{X}_{1} \rightarrow \tilde{X}_{2}$.\\
\end{Beweis}
\Paragraph{The Frobenius and its liftings}
\ \\
Everything in this lecture is either elementary or taken from \cite{DI87}.\\
\ \\
Let $S$ be a noetherian scheme defined over $\F_{p}$ and let $X$ be a
noetherian $S$-scheme.

\begin{defn}{Definition} The {\it absolute Frobenius of} $S$ is the
endomorphism\index{Absolute Frobenius}
\[ F_{S} : S \>>> S  \]
defined by the following conditions.\\
$F$ is the identity on the topological space and
$$F^{*}_{S}: {\cal O}_{S} \>>> {\cal O}_{S}$$ is given by $F^{*}_{S}
(a) = a^{p}$.
In particular for $x \in {\cal O}_{X}$ and $\lambda \in {\cal O}_{S}$ one
has $$F_{X}^{*} (\lambda x) = \lambda^{p} x^{p} = F_{S}^{*} (\lambda)
\cdot F^{*}_{X} (x),$$ and therefore one has a commutative diagram
\[
\begin{CD}
X \> F_{X} >> X\\
\V f V V  \V V f V\\
S \>> F_{S} > S
\end{CD}
\]
For $X' = X \times_{F_{S}} S$ this allows to factorize $F_{X}$:
\[
\begin{CDS}
X \> F >> X' \> pr_{1} >> X \\
\novarr \SE f EE \V f' V V \novarr \V V f V \\
\noharr S \>> F_{S} > S
\end{CDS}
\]
with $F_{X}= pr_{1} \circ F$ and $f' = pr_{2}$. By abuse of notations
we write
$$F_{S}= pr_{1}: X' \>>> X.$$\index{Relative Frobenius}
$F$ is called the {\it relative Frobenius} (relative to $S$). For
$$x \otimes \lambda \in {\cal O}_{X'} = {\cal O}_{X}
\otimes_{F_{S}} {\cal O}_{S} \mbox{ \ one has \ }
F^{*} (x \otimes \lambda)
= x^{p} \cdot \lambda$$
and for
$$x \in {\cal O}_{X} \mbox{ \ one has \ } F^{*}_{S}(x) =
x \otimes 1.$$
\end{defn}
\begin{defn}{Remark}
The absolute Frobenius $F_{S}$ is a morphism of schemes. In fact,
$F^{*}_{S} : {\cal O}_{S,s} \>>> {\cal O}_{S,s}$ satisfies
\[ F^{*^{-1}}_{S} (m_{S,s}) = \{ x \in {\cal O}_{S,s}; \ x^{p} \in
m_{S,s} \} = m_{S,s} \]
for any prime ideal $m_{S,s}$,
and hence it is a local homomorphism on the local rings. If $S =
\Spec  A$, then $F_{S}$ is induced by the $p$-th power map
$A \rightarrow A$.
\end{defn}
\begin{defn}{Properties}\\
a) Since $F_{S}: (S)_{top} \rightarrow (S)_{top}$ is the identity,
\[ F_{S}: (X')_{top} \rightarrow (X)_{top} \]
is an isomorphism of topological spaces, as well as
\[ F: (X)_{top} \rightarrow (X')_{top}. \]
b) If $ t_{1}, \ldots, t_{m}$ are locally on $X,$ generators of ${\cal O}_{X}$,
i.e.
\[ {\cal O}_{X} = {\cal O}_{S} [t_{1}, \ldots, t_{m}] /_{< f_{1}, \ldots,
f_{s}>} \]
and $f = \sum \lambda_{\underline{i}} \cdot
\underline{t}^{\underline{i}}$ ,
for $\underline{t}^{\underline{i}} = t_{1}^{i_{1}} \cdot
\ldots \cdot t^{i_{m}}_{m},$ and $\lambda_{\underline{i}} \in
{\cal O}_{S}$, then one has
$$F^{*}_{S} (f) = \sum \lambda^{p}_{\underline{i}}
\underline{t}^{\underline{i}}\ .$$ Hence
\[ {\cal O}_{X'} = {\cal O}_{X} \otimes_{F_{S}} {\cal O}_{S} = {\cal O}_{S}
[t_{1}, \ldots, t_{m}] /_{< F^{*}_{S} (f_{1}), \ldots F_{S}^{*} (f_{s}) >} \ .
\]
For $g = \sum \ \mu_{\underline{i}} \underline{t}^{\underline{i}} \in
{\cal O}_{X'}$ one has
$$F^{*} (g) = \sum \ \mu_{\underline{i}} t^{p \cdot
\underline{i}} \mbox{ \  where \ }
t^{p \cdot \underline{i}} = t_{1}^{p \cdot i_{1}}
\cdot \ldots \cdot t_{m}^{p \cdot i_{m}}. $$
c) If $X$ is smooth over $S$, one has locally \'{e}tale
morphisms $\pi : X \rightarrow \A^{n}_{S}$, hence a diagram, where
the right hand squares are by definition cartesian:
\[
\begin{CDS}
X \> F >> X' \> F_{S} >> X \\
\V \pi V V  \novarr \V V \pi' V  \novarr \V V \pi V \\
\AF^{n}_{S} \> F >> (\AF_{S}^{n})' \> F_{S} >> \AF^{n}_{S} \\
\novarr \SE E E  \V V V  \novarr \V V V\\
\noharr S \> F_{S} >> S
\end{CDS}
\]
For
$$\A^{n}_{S} = \Spec  {\cal O}_{S} [t_{1} \ldots t_{n}] $$ we have
$$(\A^{n}_{S})' = \Spec  {\cal O}_{S} [t_{1} \ldots t_{n}]$$ and
$F^{-1}{\cal O}_{(\A^{n}_{S})'}$ is the subsheaf of
${\cal O}_{\A^{n}_{S}}$
given by ${\cal O}_{S} [t_{1}^{p}, \ldots, t_{n}^{p}]$. Hence $F_{*}
{\cal O}_{\A^{n}_{S}}$ is freely generated over
${\cal O}_{(\A^{n}_{S})'}$
by $$F_*(t_{1}^{a_{1}} \cdot \ldots \cdot t_{n}^{a_{n}}) \mbox{ \ for \ }
0 \leq a_{i} < p.$$
We have an isomorphism $f: X \>>> X' \times_{(\A^{n}_{S})'} \A^{n}_{S}$
and the left upper square in the diagram is cartesian as well. \\
In fact,
for $x \in X$ we may assume that the maximal ideal
$m_{X,x} \subset {\cal O}_{X,x}$ is
generated by $t_{1}, \ldots t_{n}$ and,
if ${\cal O}_{x}$ is the local ring of $x$ in
$X' \times_{(\AF^{n}_{S})'} \AF^{n}_{S}$, then the maximal ideal $m_x$ of
${\cal O}_x$ has the same generators. Hence
\[ f^{*} : {\cal O}_{x} \>>> {\cal O}_{X,x} \]
is a local
homomorphism inducing a surjection
\[ m_{x} \>>> m_{X,x}/m^{2}_{X,x}. \]
d) The sheaf $F_{*}{\cal O}_{X}$ is again a locally free ${\cal O}_{X}$-module.
Using the notation from part c) it is generated by
$$F_* \pi^{*} (t_{1}^{a_{1}} \ldots t_{n}^{a_{n}}) \mbox{ \ for \ }
0 \leq a_{i} < p.$$
Therefore, for any locally free sheaf ${\cal F}$ on $X$, the sheaf
$F_{*} {\cal F}$ is locally free over ${\cal O}_{X'}$.
For example, if $D$ is a normal crossing
divisor on $X$, then $F_{*} \Omega^{a}_{X/S}(\log \ D)$ is locally free.
\end{defn}
\begin{defn}{Definition}
When $S$ has a lifting $\tilde{S}$ to $\Z /p^{2}$ (see (8.5)), a
{\it lifting} $\tilde{F}_{\tilde{S}}$ of $F_{S}$ is a finite morphism
$$\tilde{F}_{\tilde{S}}: \tilde{S} \>>> \tilde{S}$$ whose
restriction to $S$ is $F_{S}$.\\
\ \\
Similarly, if $X$ and $X'$ have liftings $\tilde{X}$ and $\tilde{X}'$
to $\tilde{S}$, a {\it lifting} $\tilde{F}$ {\it of the relative Frobenius}
$F$ is a finite morphism $$\tilde{F}: \tilde{X} \>>> \tilde{X}'$$
which restricts to $F$.
\end{defn}
In particular, (8.13,c) gives rise to an exact sequence of
${\cal O}_{\tilde{X}'}$-modules
\[ 0 \>>> \tilde{F}_{*}p \cdot {\cal O}_{\tilde{X}} \>>>
\tilde{F}_{*} {\cal O}_{\tilde{X}} \>>> F_{*} {\cal O}_{X}
\>>> 0 \]
together with an ${\cal O}_{\tilde{X}'}$-isomorphism
\[ p: F_{*} {\cal O}_{X} \>>> \tilde{F}_{*} p \cdot {\cal O}_{\tilde{X}}
= p \cdot \tilde{F}_{*} {\cal O}_{\tilde{X}}. \]
\begin{defn}{Assumptions} For the rest of this lecture we assume $S$
to be a scheme over $\F_{p}$ with a lifting $\tilde{S}$ to $\Z /p^{2}$
and a lifting
\[ \tilde{F}_{\tilde{S}} : \tilde{S} \>>> \tilde{S} \]
of the absolute Frobenius.\\
\ \\
Moreover we keep the assumptions made in (8.9). Hence $X$ is supposed
to be smooth over $S$ and $D \subset X$ is a normal crossing divisor over
$S$. We write $D' = F^{*}_{S}(D)$ for  $F_S : X' \rightarrow X$.
\end{defn}
\begin{defn}{Example} If $k$ is a perfect field, $S = \Spec \   k$ and
$\tilde{S} = \Spec \   W_{2}(k)$, then one takes
$$\tilde{F}_{\tilde{S}}^{*} (x + y \cdot p) = x^{p} + y^{p} \cdot p \ .$$
Furthermore, in this case $F_{S}$ is an isomorphism of fields
and $X'$ is isomorphic to $X$. In particular, $X$ has a lifting to
$\tilde{S} = \Spec \   W_{2}(k)$ if and only if $X'$ does.
\end{defn}
\begin{satz}{Proposition} Let $D \subset X$ be an $S$-normal crossing
divisor of the smooth $S$-scheme $X$. Let
$$\tilde{D}' \subset \tilde{X}' \mbox{ \ be a lifting of \ }
D' \subset X'$$
to $\tilde{S}$. Then locally in the Zarisky
topology $$D \subset X \mbox{ \  has a lifting \ }
\tilde{D} \subset \tilde{X}$$ to
$\tilde{S}$ such that $F$ lifts to
\[ \tilde{F}: \tilde{X} \>>> \tilde{X}' \mbox{ \ with \ }
\tilde{F}^{*} {\cal O}_{\tilde{X}'} (-\tilde{D}') =
{\cal O}_{\tilde{X}} (-p \cdot \tilde{D}). \]
\end{satz}
\begin{Beweis}{Proof}
By (8.17,c) we know that a lifting $\tilde{D} \subset \tilde{X}$
exists locally. Let
\[ \pi : X \>>> \A^{n}_{S} = {\bf Spec} \ {\cal O}_{S} [t_{1},
\ldots, t_{n}] \]
be \'{e}tale and $D_{j}$ be the zero set of $\varphi_{j} = \pi^{*} (t_{j})$.
By (8.14,c) we can choose liftings of $\varphi_{j}$ to $\tilde{\varphi}_{j}
\in {\cal O}_{\tilde{X}}$ and of
$$\varphi'_{j} = F_{S}^{*} (\varphi_{j})
= \varphi_{j} \otimes 1$$
to $\tilde{\varphi}'_{j}$ such that $\tilde{D}_{j}$
is defined by $\tilde{\varphi}_{j}$ and $\tilde{D}'_{j}$ by
$\tilde{\varphi}'_{j}$. We can define
$$\tilde{F}^{*} \mbox{ \  by \ } \tilde{F}^{*}
(\tilde{\varphi'_{j}}) = \tilde{\varphi}^{p}_{j}.$$
By the explicit description of $F$ in (9.3,c) $\tilde{F}$
restricts to $F$ and
$$\tilde{F}^{*} {\cal O}_{\tilde{X}'} ( - \tilde{D}') =
{\cal O}_{\tilde{X}} ( - p \cdot \tilde{D}).$$
\end{Beweis}
\begin{defn}{Remark} We have seen in (8.12) that $(\tilde{X})_{top}
= (X)_{top}$ and\\
$(\tilde{X}')_{top} = (X')_{top}$. By (9.3,a) we have
$(X)_{top} \simeq (X')_{top}$ and hence we can regard a lifting $\tilde{F}$
as a morphism
\[ \tilde{F}^{*}: {\cal O}_{\tilde{X}'} \>>> {\cal O}_{\tilde{X}} \]
of sheaves of rings over $(X')_{top}$.
\end{defn}
\ \\
Similarly to (8.22) we have
\begin{satz}{Proposition} Keeping notations and assumptions from (9.7) assume
that
$$D \subset X \mbox{ \  has a lifting \ }
\tilde{D} \subset \tilde{X}$$ to $\tilde{S}$
and let $\tilde{F}_{0}: \tilde{X} \rightarrow \tilde{X}'$ be one lifting
of $F$ with
$$\tilde{F}^{*}_{0} {\cal O}_{\tilde{X}'} ( - \tilde{D}') =
{\cal O}_{\tilde{X}} ( - p \cdot \tilde{D}).$$
Then
$$ \{ \tilde{F}: \tilde{X} \>>> \tilde{X}'; \ \ \tilde{F}^{*}
{\cal O}_{\tilde{X}'} ( - \tilde{D}') = {\cal O}_{\tilde{X}} ( - p
\cdot \tilde{D}),
\ \tilde{F} \mbox{ \ lifting of \ }F \}$$
is described by the affine space
\[ \tilde{F}^{*}_{0} + \Hom_{{\cal O}_{X'}} (\Omega^{1}_{X'/S} (\log \ D'),
F_{*} {\cal O}_{X}). \]
\end{satz}
\begin{Beweis}{Proof} As in (8.22) for isomorphisms of liftings one finds
that
$ \tilde{F}^{*} - \tilde{F}^{*}_{0}$  is zero on $p \cdot {\cal O}_{\tilde{X}}$
and induces the zero map from ${\cal O}_{X'}$ to ${\cal O}_{X}$.
Hence $\tilde{F}^{*} - \tilde{F}^{*}_{0}$ induces
\[ \tilde{F}^{*} - \tilde{F}^{*}_{0}: {\cal O}_{X'} \>>>
p \cdot {\cal O}_{X}. \]
For $\tilde{x}', \tilde{y}' \in {\cal O}_{\tilde{X}'}$ one has
\[ (\tilde{F}^{*} - \tilde{F}^{*}_{0}) (\tilde{x}' \cdot \tilde{y}') =
F(x') (\tilde{F}^{*} - \tilde{F}^{*}_{0}) (\tilde{y}') +
F(y') (\tilde{F}^{*} - \tilde{F}^{*}_{0}) (\tilde{x}') \]
and
\[ p^{-1} (\tilde{F}^{*} - \tilde{F}^{*}_{0}) : {\cal O}_{X'} \>>>
{\cal O}_{X} \]
factorizes through
\[ {\cal O}_{X'} \> d >> \Omega^{1}_{X'/S} \>>> {\cal O}_{X} \]
where the right hand side morphism is ${\cal O}_{X'}$-linear and is again
denoted by \\
$(\tilde{F}^{*} - \tilde{F}^{*}_{0})$. For $\tilde{\varphi}_{j}
\in {\cal O}_{\tilde{X}}$ and $\tilde{\varphi}'_{j} \in
{\cal O}_{\tilde{X}'}$ , local
parameters for $\tilde{D}_{j}$ and $\tilde{D}'_{j}$ respectively, which lift
$\varphi_{j}$ and $\varphi'_{j} = \varphi_{j} \otimes 1$, one has
\[ \tilde{F}^{*} (\tilde{\varphi}'_{j}) = \tilde{\varphi}_{j}^{p} \cdot
(1 + p \cdot \tilde{\lambda}) \]
and
\[ \tilde{F}^{*}_{0} (\tilde{\varphi}'_{j}) = \tilde{\varphi}^{p}_{j} \cdot
(1 + p \cdot \tilde{\lambda}_{0})
\]
for some  $\tilde{\lambda}, \tilde{\lambda}_{0} \in {\cal O}_{\tilde{X}}$.
Therefore
\[ (\tilde{F}^{*} - \tilde{F}^{*}_{0}) (\tilde{\varphi}'_{j}) =
\tilde{\varphi}^{p}_{j} (p (\tilde{\lambda} - \tilde{\lambda}_{0})) \]
and
\[ (\tilde{F}^{*} - \tilde{F}^{*}_{0}) (d \tilde{\varphi}'_{j}) = p^{-1}
\circ (\tilde{F}^{*} - \tilde{F}^{*}_{0}) (\tilde{\varphi}'_{j}) \in
{\cal O}_{X} ( - p \cdot D_{j}). \]
Hence $$(\tilde{F}^{*} - \tilde{F}_{0}^{*}): \Omega^{1}_{X'/S} \>>>
{\cal O}_{X}$$ extends to
\[ (\tilde{F}^{*} - \tilde{F}^{*}_{0}) : \Omega^{1}_{X'/S}  (\log \ D')
\>>> {\cal O}_{X}. \]
Conversely, for
$$\varphi \in \Hom_{{\cal O}_{X'}} (\Omega^{1}_{X'/S} (\log \ D'),
{\cal O}_{X})$$
we define
$$\tilde{F}^{*} = \tilde{F}^{*}_{0} + \varphi^{*}\mbox{ \ by \ }
\tilde{F}^{*} (\tilde{x}) = \tilde{F}^{*}_{0} (\tilde{x}) -
p (\varphi (d x)).$$
We have
\[ {\cal O}_{X'} (- D') \> d >> \Omega^{1}_{X'/S} (\log \ D')( - D')
\> \varphi >> {\cal O}_{X} ( -p \cdot D) \]
and
$$\tilde{F}^{*} ({\cal O}_{\tilde{X}'} ( - \tilde{D}')) = {\cal O}_{\tilde{X}}
( - p \cdot \tilde{D}).$$
\end{Beweis}
\begin{satz}{Corollary} Under the assumption of (9.7) assume that
$X$ is affine and that
$$D \subset X\mbox{ \  has a lifting \ }
\tilde{D} \subset \tilde{X}$$ to $\tilde{S}$. Then there is a lifting
$\tilde{F}: \tilde{X} \rightarrow \tilde{X}'$ of $F$ with
\[ \tilde{F}^{*} {\cal O}_{\tilde{X}'} (- \tilde{D}') = {\cal O}_{\tilde{X}}
( - p \cdot \tilde{D}). \]
\end{satz}
\begin{Beweis}{Proof} One repeats the argument used to prove (8.23),
replacing $\tilde{X}^{(1)}$ by $\tilde{X}$ and $\tilde{X}^{(2)}$
by $\tilde{X}'$ and using (9.7) and (9.9)
instead of (8.21) and (8.22).\\
\end{Beweis}
\begin{defn}{Remark} Let $X$ be a smooth $S$-scheme, $\tilde{X}'$ a
lifting of $X'$ and $\tilde{X}^{(i)}$, for $i = 1,2$, two liftings
of $X$ to $\tilde{S}$. Assume that we have a lifting
$$\tilde{F}_{2}:\tilde{X}^{(2)} \>>> \tilde{X}'$$
and isomorphisms
$$u: \tilde{X}^{(1)}\>>> \tilde{X}^{(2)}
\mbox{ \ and \ } v: \tilde{X}^{(1)} \>>>
\tilde{X}^{(2)},$$
both lifting the identity. Then, considering again
${\cal O}_{\tilde{X}'}, {\cal O}_{\tilde{X}^{(1)}}$ and
${\cal O}_{\tilde{X}^{(2)}}$ as sheaves of rings on $(X')_{top}$, one
has
\[ {\cal O}_{\tilde{X}'} \> \tilde{F}^{*}_{2} >> {\cal O}_{\tilde{X}^{(2)}}
\> (u^{*} - v^{*}) >> {\cal O}_{\tilde{X}^{(1)}} \]
and
\[ (u^{*} - v^{*}) \circ \tilde{F}^{*}_{2} (\tilde{x}') = (u^{*} -
v^{*}) (d (F^{*} (x'))) = 0 \]
since $F^{*} (x')$ is a $p$-th power. We find:
\[ (\tilde{F}_{2} \circ u)^{*} =
(\tilde{F}_{2} \circ v)^{*} : {\cal O}_{\tilde{X}'} \>>>
{\cal O}_{\tilde{X}^{(1)}}. \]
In other words, $(\tilde{F}_2 \circ u)^*$ does not depend on the
choice of $u$.
\end{defn}

In (9.10) we used the fact that, for $X'$ affine, the higher cohomology
groups of coherent sheaves are zero to obtain the existence of the lifting
$$\tilde{F} : \tilde{X} \>>> \tilde{X}'$$
of $F$. We have more generally:
\begin{satz}{Corollary} Let $X$ be a smooth scheme and $D \subset X$ be
a normal crossing divisor over $S$. Given liftings
$$\tilde{D} \subset \tilde{X} \mbox{ \  and \ } \tilde{D}' \subset
\tilde{X}' \mbox{ \ of \ } D \subset X \mbox{ \ and \ }
D' \subset X'$$ (respectively) to $\tilde{S}$, the exact obstruction
for lifting $F$ to
$$\tilde{F}: \tilde{X} \>>> \tilde{X}'
\mbox{ \ with \ } \tilde{F}^{*} {\cal O}_{\tilde{X}'} ( - \tilde{D}') =
{\cal O}_{\tilde{X}} ( - p \cdot \tilde{D})$$
is a class
\[ [F_{\tilde{X}', \tilde{D}'}] \in H^{1} (X', {\cal H}om_{{\cal O}_{X'}}
(\Omega^{1}_{X'/S} (\log \ D'), F_{*} {\cal O}_{X})) \]
which does not
depend on $(\tilde{X}, \tilde{D})$.
\end{satz}
\begin{Beweis}{Proof} By (9.7) or (9.10) one can cover $\tilde{X}$ by affine
$\tilde{X}_{\alpha}$ such that $F$ lifts to $\tilde{F}_{\alpha}$ on
$\tilde{X}_{\alpha}$ with the required property for $D$. Then by (9.9)
$(\tilde{F}^{*}_{\alpha} - \tilde{F}^{*}_{\beta})$ describes
a $1$-cocycle with values in
$${\cal H}om_{{\cal O}_{X'}} (\Omega^{1}_{X'/S}
(\log \ D'), F_{*} {\cal O}_{X}).$$
Changing the $\tilde{F}_{\alpha}$
corresponds to changing the cocycle by a coboundary.
We define $[F_{\tilde{X}', \tilde{D}'}]$ to be the cohomology class
of this cocycle. If $[F_{\tilde{X}', \tilde{D}'}] = 0$ one finds
for a possibly finer cover $\{ X'_{\alpha} \}$
$$\varphi_{\alpha} \in \Gamma (X'_{\alpha}, {\cal H}om_{{\cal O}_{X'}}
(\Omega^{1}_{X'/S} (\log \ D'), F_{*} {\cal O}_{X}))$$ such that the
$\tilde{F}_{\alpha}^{*} + \varphi_{\alpha}$ glue together to give $\tilde{F}:
\tilde{X} \rightarrow \tilde{X}'$.\\
\ \\
If $\tilde{X}^{(i)}$ are two liftings, $\tilde{X}^{(i)}_{\alpha}$ coverings
and $\tilde{F}^{(i)}_{\alpha}$ liftings of $F$, for $i=1,2$ we can apply
(8.21) or (8.23) to get isomorphisms of liftings
$$u_{\alpha}:\tilde{X}_{\alpha}^{(1)} \>>> \tilde{X}^{(2)}_{\alpha}.$$
By (9.11) we have on $\tilde{X}'_{\alpha} \cap \tilde{X}'_{\beta}$
\[ (\tilde{F}^{(2)}_{\alpha} \circ u_{\alpha})^{*} - (\tilde{F}^{(2)}_{\beta}
\circ u_{\beta})^{*} = (\tilde{F}^{(2)}_{\alpha} \circ u_{\alpha})^{*}
- (\tilde{F}^{(2)}_{\beta} \circ u_{\alpha})^{*} =
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \]
\[ u^{*}_{\alpha} \circ (\tilde{F}^{(2)*}_{\alpha} - \tilde{F}^{(2)*}_{\beta})
\in \Gamma (X'_{\alpha} \cap X'_{\beta}, {\cal H}om_{{\cal O}_{X'}}
(\Omega^{1}_{X'/S}(\log \ D'), F_{*} {\cal O}_{X})) \]
As $u^{*}_{\alpha}$ is the identity on $X'_{\alpha}$ the cocycle defined
by $\tilde{F}^{(2)}_{\alpha}$ and $\tilde{F}^{(2)}_{\alpha} \circ u_{\alpha}$
are the same and $\tilde{F}^{(2)}_{\alpha}$ and $\tilde{F}^{(1)}_{\alpha}$
define the same cohomology class.\\
\ \\
\end{Beweis}
\Abschnitt{The Cartier operator}\index{Cartier operator}
Let $X$ be a smooth scheme over $S$
and $F: X \rightarrow X'$ be the Frobenius relative to $S$. The key observation
is that the differential $d$ in the de Rham complex
$\Omega^{\bullet}_{X/S}$ is ${\cal O}_{X'}$-linear as, using the
notations of (9.1),
\[ dF^{*}(x \otimes 1) = dx^{p} = 0. \]
If $D$ is a normal crossing divisor over $S$, the homology sheaves
\[ {\cal H}^{a} = {\cal H}^{a} (F_{*} \Omega^{\bullet}_{X/S}
(\log \ D)) \]
are ${\cal O}_{X'}$-modules computed by the following
\begin{satz}{Theorem (Cartier, see \cite{Car57} \cite{Kat70})}\\
One has an isomorphism
of ${\cal O}_{X'}$-modules
\[ C^{-1} : \Omega^{1}_{X'/S} (\log \ D') \>>> {\cal H}^{1} (F_{*}
\Omega^{\bullet}_{X/S}(\log \ D)) \]\index{$C^{-1}$}
such that:\\
a) For $x \in {\cal O}_{X}$ one has
\[ C^{-1} (d(x \otimes 1)) = x^{p-1} dx \ \ \mbox{in \ \ ${\cal H}^{1}$}. \]
b) If $t$ is a local parameter defining a component of $D$, then
\[ C^{-1} \biggl( \frac{d(t \otimes 1)}{t \otimes 1} \biggr) = \frac{dt}{t}
\ \ \mbox{in \ \ ${\cal H}^{1}$} .\]
c) $C^{-1}$ is uniquely determined by a) and b).\\
\ \\
d) For all $a \geq 0$ one has an isomorphism
\[ \bigwedge^{a} C^{-1} : \Omega^{a}_{X'/S} (\log \ D') \>>>
{\cal H}^{a} (F_{*} \Omega^{\bullet}_{X/S} (\log \ D)) \]
obtained by wedge product from $C^{-1}$.
\end{satz}
\begin{Beweis}{Proof} c) is obvious since $\Omega^{1}_{X'/S} (\log \ D')$
is generated by elements of the form
$$d(x \otimes 1) \mbox{ \ and \ } \frac{d(t \otimes 1)}{t \otimes 1}.$$
For the existence of $C^{-1}$ let us first assume that $D = \emptyset$. Then
\[ (x + y)^{p-1} (dx + dy) - x^{p-1} dx - y^{p-1} dy = df \]
where for
$$\gamma_{i} \in \F_{p} \mbox{ \ with \ }\gamma_{i} \equiv \frac{1}{p} \left(
\begin{array}{c}
p \\
i
\end{array}
\right) \ mod \ p$$
we take
\[ f = \sum^{p-1}_{i=1} \gamma_{i} \cdot x^{i} \cdot y^{p-i} =
\frac{1}{p} [ (x + y)^{p} - x^{p} - y^{p}]. \]
Moreover, one has
\[ (y \cdot x)^{p-1} d(y \cdot x) = x^{p} \cdot y^{p-1} dy + y^{p} \cdot
x^{p-1} dx \]
and $d(x^{p-1} dx) = 0$.
Hence, the property a) defines $C^{-1}$.\\
\ \\
For $D \neq \emptyset$, b) is compatible with the definition of $C^{-1}$ on
$\Omega^{1}_{X'/S}$. In fact,
\[ C^{-1} \biggl( t \otimes 1 \cdot \frac{d (t \otimes 1)}{ t \otimes 1}
\biggr) = F^{*} (t \otimes 1) C^{-1} \biggl( \frac{d(t \otimes 1)}{t
\otimes 1} \biggr) = t^{p} \frac{dt}{t}. \]
Having defined $C^{-1}$, we can define $\bigwedge^{a} C^{-1}$ as well.
As in (9.3,c) we have locally a cartesian square
\[
\begin{CD}
X \> F >> X'\\
\V \pi V V  \V V  \pi' V \\
\AF^{n}_{S} \> F >> (\AF^{n}_{S})'
\end{CD}
\]
with $\pi$ and $\pi'$ \'{e}tale. Hence to show that $\bigwedge^{a} C^{-1}$
is an isomorphism it is enough to consider the case
$$X = \A^{n}_{S} = {\bf Spec} \ {\cal O}_{S} [t_{1}, \ldots t_{n}]$$
and $D$ to be the zero set of $t_{1} \cdot \ldots \cdot t_{r}$.\\
\ \\
If ${\cal B}^{a}$ is the $\F_{p}$-vector space freely generated by
\[ t_{1}^{i_{1}} \cdot \ldots \cdot t_{n}^{i_{n}} \cdot
\omega_{\alpha_{1}} \wedge \ldots \wedge \omega_{\alpha_{a}} \]
for
\[ 0 \leq i_{\nu} < p \ \ \mbox{for \ \ $\nu = 1, \ldots, n$} \]
\[ 1 \leq \alpha_{1} < \alpha_{2} \ldots < \alpha_{a} \leq n \]
\[ \omega_{\nu} =
\left\{
\begin{array}{ll}
\frac{dt_{\nu}}{t_{\nu}} & \nu = 1, \ldots, r\\
dt_{\nu} & \nu = r+1, \ldots, n\\
\end{array}
\right. \]
then ${\cal B}^{\bullet}$, with the usual differential is a subcomplex
of $F_{*}\Omega^{\bullet}_{X/S} (\log \ D)$.
One has $F_{*} \Omega^{\bullet}_{X/S} (\log \ D) = {\cal O}_{X'}
\otimes_{\F_{p}} {\cal B}^{\bullet}$ and (9.14)
follows from the following claim.\\
\end{Beweis}
\begin{defn}{Claim}
One has\\
i) $H^{0} ({\cal B}^{\bullet}) = \F_{p}$ .\\
\ \\
ii) $H^{1}({\cal B}^{\bullet})$ has the basis
$ \{ \omega_{1}, \ldots, \omega_{r}, t^{p-1}_{r+1} \omega_{r+1}, \ldots,
t_{n}^{p-1} \omega_{n} \} $ .\\
\ \\
iii) $H^{a} ({\cal B}^{\bullet}) = \bigwedge^{a} H^{1} ({\cal B}^{\bullet})$.
\end{defn}
\begin{Beweis}{Proof}
For $n=1$, this is shown easily:\\
Obviously $\ker \ (d: {\cal B}^{0}
\>>> {\cal B}^{1})
= \F_{p}$.
For $D = \emptyset$ let us write ${\cal K}^{\bullet} = {\cal B}^{\bullet}$.
One has
$${\cal K}^{1} = <t^{i} dt; \ i=0, \ldots , p-1>_{\F_{p}}$$ and
$$d {\cal K}^{0} = < dt^{i+1} = (i+1) \cdot t^{i} dt; \ i= 0, \ldots,
p-2>_{\F_{p}}.$$
For $D \neq \emptyset$ write ${\cal L}^{\bullet}
= {\cal B}^{\bullet}$.
One has
$${\cal L}^{1} = < t^{i} \frac{dt}{t}; \ i = 0, \ldots, p-1>_{\F_{p}}$$
and
$$d{\cal L}^{0} = < dt^{i} = i \cdot t^{i} \cdot \frac{dt}{t};
i=1, \ldots p-1>_{\F_{p}}.$$
In both cases (9.15) is obvious.
For $n > 1$ one can write
\[ {\cal B}^{\bullet} = \underbrace{{\cal L} \otimes_{\F_{p}}
{\cal L}^{\bullet}
\otimes \ldots \otimes_{\F_{p}} {\cal L}^{\bullet}}_{r \ \mbox{times}}
\otimes_{\F_{p}}
\underbrace{{\cal K}^{\bullet} \otimes \ldots \otimes_{\F_{p}}
{\cal K}^{\bullet}}_{n-r \ \mbox{times.}}. \]
By the K\"unneth formula (A.8)
\[ H^{a} ({\cal B}^{\bullet}) = \sum_{\sum^{n}_{i=1}
{\varepsilon}_{i} = a} = H^{\varepsilon_{1}}
({\cal L}^{\bullet}) \otimes \ldots \otimes H^{\varepsilon_{r}}
({\cal L}^{\bullet}) \otimes
\ldots \otimes H^{\varepsilon_{n}} ({\cal K}^{\bullet}). \]
which implies a), b) and c).\\
\end{Beweis}
\begin{defn}{Notation} Following Deligne-Illusie, we define
\[ \Omega^{\bullet}_{X/S} (A,B) = \Omega^{\bullet}_{X/S}
(\log \ (A+B)) (-A) \]
where $A+B$ is a normal crossing divisor over $S$.
\end{defn}
\begin{satz}{Corollary}
The Cartier operator induces an isomorphism
\[ \Omega^{a}_{X'/S}(A',B') \>>> {\cal H}^{a} (F_{*}
\Omega^{\bullet}_{X/S} (A,B))  \]
\end{satz}
\begin{Beweis}{Proof}
By (2.7) the residues of
$$d: {\cal O}_{X} (-A) \>>> \Omega^{1}_{X/S}
(\log \ (A+B)) (-A) = \Omega^1_{X/S}(A,B)$$
along the components of $A$ are all $1$ and by (2.10)
\[ \Omega^{\bullet}_{X/S} (\log \ (A+B))(-p \cdot A) \>>>
\Omega^{\bullet}_{X/S} (A,B) \]
is a quasi isomorphism. Since
\[ F_{*} \Omega^{\bullet}_{X/S} (\log \ (A+B)) (-p \cdot A) =
F_{*} \Omega^{\bullet}_{X/S} (\log \ (A+B)) \otimes_{{\cal O}_{X'}}
{\cal O}_{X'} (- A') \]
we can apply (9.14).\\
\end{Beweis}
\Abschnitt{Duality} Let us keep the notation from (9.16).
The wedge product
\[ \Omega^{n-i}_{X/S} (\log \ D) \otimes \Omega^{i}_{X/S} (\log \ D)
\> \wedge >> \Omega^{n}_{X/S} \otimes {\cal O}_{X} (D) \]
is a perfect duality of locally free sheaves.
Hence one obtains:
\begin{satz}{Lemma}
$\Omega^{n-i}_{X/S} (A,B) \otimes \Omega^{i}_{X/S} (B,A) \> \wedge >>
\Omega^{n}_{X/S}$ is a perfect duality.
\end{satz}
\begin{satz}{Lemma}
One has a perfect duality
\[ F_{*} \Omega^{n-i}_{X/S} (A,B) \otimes F_{*} \Omega^{i}_{X/S} (B,A)
\>>> \Omega^{n}_{X'/S} \]
given by
\[ F_{*} \Omega^{n-i}_{X/S} (A,B) \otimes F_{*} \Omega^{i}_{X/S} (B,A)
\>> \wedge > F_{*} \Omega^{n}_{X/S} \>>> {\cal H}^{n} \>> C >
\Omega^{n}_{X'/S}\] where $C$ is the Cartier operator.
\end{satz}
\begin{Beweis}{Proof}
In fact, this is nothing but duality for finite flat morphisms
(\cite{Har77}, p 239). One has
\[
\begin{array}{lcl}
F_{*} \Omega^{n-i}_{X/S} (A,B) & = & F_{*} {\cal H}om_{{\cal O}_{X}}
(\Omega^{i}_{X/S} (B,A), \Omega^{n}_{X/S})\\
& \simeq & {\cal H}om_{{\cal O}_{X'}} (F_{*} \Omega^{i}_{X/S} (B,A),
\Omega^{n}_{X'/S})
\end{array}
\]
and (9.20) is just saying that $F_{*} \Omega^{n}_{X/S} \>>>
\Omega^{n}_{X'/S}$ is given by the Cartier operator.
One can do the calculations by hand.\\
\ \\
As in the proof of (9.14) it is enough to consider $X = \A^{n}_{S}$,
$A$ the zero set of $t_{1} \cdot \ldots \cdot t_{s}$ and $B$ the zero set
of $t_{s+1} \ldots t_{r}$.
Define, for $a > 0$, ${\cal B}^{a} (A,B)$ to be the $\F_{p}$-vector
space generated by all
\[ \varphi = t_{1}^{i_{1}} \ldots \cdot t_{n}^{i_{n}} \cdot \omega_{\alpha_{1}}
\wedge \ldots \wedge \omega_{\alpha_{a}} \]
with
\[
\omega_{\nu} =
\left\{
\begin{array}{ll}
\frac{dt_{\nu}}{t_{\nu}} & \mbox{for \ \ $\nu = 1, \ldots s, \ldots, r$}\\
dt_{\nu} & \mbox{for \ \ $\nu = r+1, \ldots, n$}\\
\end{array}
\right. \,
\]
where the indices are given by
\[
\begin{array}{ll}
0 < i_{\nu} \leq p & \mbox{for \ \ $\nu = 1, \ldots, s$}\\
0 \leq i_{\nu} < p & \mbox{for \ \ $\nu = s+1, \ldots, r, \ldots, n$}\\
\mbox{and by } & 1 \leq \alpha_{1} < \alpha_{2} < \ldots < \alpha_{a} \leq n.
\end{array}
\]
Similarly we have ${\cal B}^{a} (B,A)$ by taking as index set
\[
\begin{array}{ll}
0 \leq i_{\nu} < p & \mbox{for \ \ $\nu = 1, \ldots, s $ and
$\nu = r+1, \ldots, n$}\\
0 < i_{\nu} \leq p & \mbox{for \ \ $\nu = s+1, \ldots, r$}\\
\mbox{and } & 1 \leq \alpha_{1} < \alpha_{2} < \ldots < \alpha_{a} \leq n.
\end{array}
\]
For $a=n-i$ the only generator $\delta$ of ${\cal B}^{i} (B,A)$ with
$C(\varphi \wedge \delta) \neq 0$ is
\[ \delta = t^{j_{1}}_{1} \cdot \ldots \cdot t^{j_{n}}_{n} \omega_{\beta_{1}}
\wedge \ldots \wedge \omega_{\beta_{i}} \]
with
\[ \{ \beta_{1}, \ldots, \beta_{i} \} \cup \{ \alpha_{1}, \ldots,
\alpha_{n-i} \} = \{ 1, \ldots, n \} \]
and
\[ i_{\nu} + j_{\nu} =
\left\{
\begin{array}{cl}
p & \mbox{for \ \ $\nu =1, \ldots, r$}\\
p-1 & \mbox{for \ \ $\nu = r+1, \ldots,n$.}\\
\end{array}
\right.
\]
\end{Beweis}
\begin{defn}{Remark} For $\varphi \in {\cal B}^{n-i-1} (A,B)$ and
$\delta \in {\cal B}^{i} (B,A)$ the explicit description of the
duality in the proof of (9.20) shows that (up to sign)
\[ C(d \varphi \wedge \delta) = C (\varphi \wedge d \delta). \]
\end{defn}
Hence we obtain as well:
\begin{satz}{Corollary} Under the duality in (9.20) the transposed
of the differential $d$ is again $d$ (up to sign).
\end{satz}
\ \\
\Paragraph{The proof of Deligne and Illusie $[$12$]$}
\ \\
We keep the assumptions from Lectures 8 and 9. Hence $X$ is supposed
to be a smooth noetherian $S$-scheme, $D \subset X$ a $S$-normal crossing divisor,
and $S$ is a noetherian scheme over $\Z /p$ which admits a lifting $\tilde{S}$ to
$\Z /p^{2}$ as well as a lifting $ \tilde{F}_{\tilde{S}} : \tilde{S}
\rightarrow \tilde{S}$ of the absolute Frobenius $F_{S}$.

\Abschnitt {\bf The two term de Rham complex}\index{Two term
de Rham complex} \index{$\tau_{\leq 1} \ F_{*}
\Omega^{\bullet}_{X/S} \ (\log \ D).$}
is defined as $$\tau_{\leq 1} \ F_{*} \Omega^{\bullet}_{X/S} \ (\log \ D).$$
Hence, as explained in (A.26), it is the complex
\[ F_{*} {\cal O}_{X} \>>> Z^{1} \]
where $$Z^{1} = Ker \ (F_{*} \Omega^{1}_{X/S} \ (\log \ D) \>>>
F_{*} \Omega^{2}_{X/S} \ (\log \ D)).$$
One has a short exact sequence of complexes
\[ 0 \>>> {\cal H}^0 \>>>
\tau_{\leq 1} \ F_{*} \Omega^{\bullet}_{X/S} \ (\log \ D) \>>>
{\cal H}^1[-1] \>>> 0 \]
given by
\[
\begin{CD}
0 \>>> {\cal H}^{0} \>>> F_{*} {\cal O}_{X}\\
 \noarr  \noarr \V VV\\
 \noarr \noarr Z^{1} \>>> {\cal H}^{1} \>>> 0
\end{CD}
\]
where ${\cal H}^{0} = {\cal O}_{X'}$ and ${\cal H}^{1}$ is ${\cal O}_{X'}$-
isomorphic to $\Omega^{1}_{X'/S} (\log \ D)$ via the Cartier operator (9.14).

\begin{defn}{Definition} A {\it splitting} of $\tau_{\leq 1} F_{*}
\Omega^{\bullet}_{X/S} (\log \ D)$ is a diagram
\index{Splitting of $\tau_{\leq 1} F_{*}
\Omega^{\bullet}_{X/S} (\log \ D)$}
\[
\begin{CD}
\tau_{\leq 1} F_{*} \Omega^{\bullet}_{X/S} (\log \ D) \> \sigma >>
{\cal K}^{\bullet} \\
\noarr \A A \theta A\\
\noarr {\cal H}^{0} \oplus {\cal H}^{1} [ -1 ]
\end{CD}
\]
where ${\cal K}^{\bullet}$ is the \v{C}ech complex
$${\cal C}^{\bullet} ({\cal U}, \tau_{\leq 1} F_{*} \Omega^{\bullet}_{X/S}
(\log \ D))$$ associated to some affine open cover ${\cal U}$ of $X'$,
where $\sigma$ is the induced morphism, hence a quasi-isomorphism (see
(A.6)), and where $\theta$ is a quasi-isomorphism. We may
assume, of course, that $${\cal H}^{i} \> \sigma >> {\cal H}^{i}
\> \theta^{-1} >> {\cal H}^{i}$$ is the identity for $i = 0, 1$.
\end{defn}
\begin{defn}{Example} Assume that $D \subset X$ and $D' \subset X'$ both
lift to
$$\tilde{D} \subset \tilde{X} \mbox{ \ and \ } \tilde{D}' \subset \tilde{X}'$$
on $\tilde{S}$ and that $F$ lifts to $\tilde{F} : \tilde{X} \rightarrow
\tilde{X}'$ in such a way that $$\tilde{F}^{*} {\cal O}_{\tilde{X}'} ( -
\tilde{D}') = {\cal O}_{\tilde{X}} (- p \cdot \tilde{D}).$$ For example,
if $S = \Spec \ k$ for a perfect field $k$ and if $D \subset X$ has
a lifting $\tilde{D} \subset \tilde{X}$ then as we have seen in (9.6)
$D' \subset X'$ has a lifting as well. By (9.12) the existence
of $\tilde{F}$ is equivalent to $$[F_{\tilde{X}', \tilde{D}'}] = 0
\mbox{ \ in \ }
H^{1} (X', {\cal H}om_{{\cal O}_{X'}}
(\Omega^{1}_{X'/S}(\log \ D'), F_{*} {\cal O}_{X})).$$
For example it automatically exists if this group vanishes.\\
\ \\
Anyway, if the liftings $\tilde{D}, \tilde{X}, \tilde{D}', \tilde{X}'$ and
$\tilde{F}$ exist, the morphism
\[ \tilde{F}^{*} : \Omega^{1}_{\tilde{X}'/\tilde{S}} (\log \ \tilde{D}')
\>>> \Omega^{1}_{\tilde{X}/\tilde{S}} (\log \ \tilde{D}) \]
verifies $$\tilde{F}^{*} |_{p \cdot \Omega^{1}_{\tilde{X}'/
\tilde{S}} (\log \ \tilde{D}')} = 0 .$$ In fact,
\[ F^{*} : \Omega^{1}_{X'/S} (\log \ D') \>>> \Omega^{1}_{X/S}
(\log \ D) \]
is given by $F^{*} (d (t \otimes 1)) = d(t^{p})$ and hence it is the zero map.
We have a commutative diagram
\[
\minCDarrowwidth=1.0cm
\begin{CD}
\Omega^{1}_{X'/S} (\log \ D') \> p > \simeq > p \cdot \Omega^{1}_{\tilde{X}'
/ \tilde{S}} (\log \ \tilde{D}')\\
\V V F^{*} V  \V V \tilde{F}^{*} V \\
\Omega^{1}_{X/S} (\log \ D) \> p > \simeq > p \cdot \Omega^{1}_{\tilde{X}/
\tilde{S}} (\log \ \tilde{D})
\end{CD}
\]
and hence the vertical morphisms are both zero.
The same argument shows that the factorization
\[ \tilde{F}^{*} : \Omega^{1}_{X'/S} (\log \ D') \>>>
\Omega^{1}_{\tilde{X}/ \tilde{S}} (\log \ \tilde{D}) \]
takes values in $$p \cdot \Omega^{1}_{\tilde{X}/\tilde{S}} (\log \ \tilde{D})
= p \cdot \Omega^{1}_{X/S} (\log \ D).$$
The induced map
\[ p^{-1} \circ \tilde{F}^{*}: \Omega^{1}_{X'/S} (\log \ D') \>>>
\Omega^{1}_{X/S} (\log \ D) \]
can be written in coordinates as follows. For
$ x \in {\cal O}_{X}$ let $ \tilde{x} \in {\cal O}_{\tilde{X}} $
be a lifting of $ x $ and let $\tilde{x}' \in {\cal O}_{\tilde{X}'} $
be a lifting of $x' = x \otimes 1$.  One writes
\[ \tilde{F}^{*} (\tilde{x}') = \tilde{x}^{p} + p (u (\tilde{x},
\tilde{x}')) \]
for some $u(\tilde{x}, \tilde{x}') \in {\cal O}_{X}$. Then
\[ p^{-1} \circ \tilde{F}^{*} (d \tilde{x}') = x^{p-1} dx + du (\tilde{x},
\tilde{x}'). \]
In particular the image of $p^{-1} \circ \tilde{F}^{*}$ lies in
$$Z^{1} \subset F_{*} \Omega^{1}_{X/S} (\log \ D)$$
and the composition with
$ Z^{1} \>>> {\cal H}^{1} $
gives back the Cartier operator.\\
\ \\
In this example, i.e. if the liftings
$\tilde{D}, \tilde{X}, \tilde{D}', \tilde{X}'$ and
$\tilde{F}$ all exist, we can take ${\cal U} = \{ X' \}$ and define
\[ \theta : {\cal O}_{X'} \oplus \Omega^{1}_{X'/S} (\log \ D') [-1] \>>>
\tau_{\leq 1} F_{*} \Omega^{\bullet}_{X/S} (\log \ D) \]
by
\[
\begin{CD}
{\cal O}_{X'} \>>> F_{*} {\cal O}_{X} \\
\V V 0 V   \V V d V\\
\Omega^{1}_{X'/S} \>> p^{-1} \circ \tilde{F}^{*} >  Z^{1}
\end{CD}
\]
and, by (9.14), $\theta$ is a quasi-isomorphism.
\end{defn}
\begin{defn}{Notation} We call a cohomology class
\[ \varphi \in \h^{1} (X', {\cal H}om_{{\cal O}_{X'}} ({\cal H}^{1},
F_{*} {\cal O}_{X}) \rightarrow {\cal H}om_{{\cal O}_{X'}} ({\cal H}^{1},
Z^{1})) \]\index{Splitting cohomology class}
a {\it splitting cohomology class}, if $\varphi$ maps to the identity in
\[ H^{0} (X', {\cal H}om_{{\cal O}_{X'}} ({\cal H}^{1}, {\cal H}^{1})) =
\h^{1} (X', {\cal H}om_{{\cal O}_{X'}} ({\cal H}^{1}, {\cal H}^{1}) [ -1 ]).\]
\end{defn}
\begin{satz}{Proposition} The splittings of $$\tau_{\leq 1} F_{*}
\Omega^{\bullet}_{X/S} (\log \ D)$$ are in one to one correspondence with
the splitting cohomology classes
\[ \varphi \in \h^{1} (X', {\cal H}om_{{\cal O}_{X'}} ({\cal H}^{1}, F_{*}
{\cal O}_{X}) \>>> {\cal H}om_{{\cal O}_{X'}} ({\cal H}^{1}, Z^{1})). \]
\end{satz}
\begin{Beweis}{Proof} Let $\varphi$ be a splitting cohomology class,
realized as cocycle
\[ \varphi_{\alpha \beta} \in \Gamma (X'_{\alpha \beta},
{\cal H}om_{{\cal O}_{X'}} ({\cal H}^{1}, F_{*} {\cal O}_{X})) \]
and $$\psi_{\alpha} \in \Gamma (X'_{\alpha}, {\cal H}om_{{\cal O}_{X'}}
({\cal H}^{1}, Z^{1}))$$ for some affine open cover ${\cal U} = \{ X'_{\alpha}
\}$ of $X'$. Hence, using the notations from (A.6) for the differential in
the \v{C}ech complex, $\delta \varphi = 0$ and $d \varphi - \delta \psi = 0$.
By assumption $\psi_{\alpha}$ induces the identity in $$\Gamma (X'_{\alpha},
{\cal H}om_{{\cal O}_{X'}} ({\cal H}^{1}, {\cal H}^{1})).$$ Then $\theta =
(id, (\varphi_{\alpha \beta}, \psi_{\alpha}))$ is the map wanted, i.e.\\
\[
\minCDarrowwidth=1.0cm
\begin{CD}
{\cal O}_{X'} \> id >> \bigoplus_{\cal U} \ \varrho_{\alpha *} F_{*}
{\cal O}_{X}|_{X'_{\alpha}} \\
\V 0 V V    \V V \delta \oplus d V \\
{\cal H}^{1} \> (\varphi_{\alpha \beta}, \psi_{\alpha}) >> \bigoplus_{\cal U}
\ \varrho_{\alpha \beta *} F_{*} {\cal O}_{X}|_{X'_{\alpha \beta}} \oplus
\bigoplus_{\cal U} \ \varrho_{\alpha *} \ Z^{1}|_{X'_{\alpha}} \\
\V V V      \V V (- \delta \oplus d, \ 0 \oplus - \delta) V \\
0 \>>> \bigoplus_{\cal U} \varrho_{\alpha \beta \gamma *} F_{*} {\cal O}_{X}
|_{X'_{\alpha \beta \gamma}} \oplus \bigoplus_{\cal U} \ \varrho_{\alpha \beta
*} \ Z^{1}|_{X'_{\alpha \beta}}.
\end{CD}
\]
where $$\varrho_{\alpha_1, \cdots \alpha_r}:X'_{\alpha_1, \cdots \alpha_r}
\>>> X'$$ denotes the embedding, and where
$$(- \delta \oplus d, \ 0 \oplus - \delta)(x_{\alpha \beta}, z_{\alpha}) =
(-\delta(x), d(x) - \delta(z)).$$
Conversely, let for some ${\cal U}$
\[ \tau_{\leq 1} (F_{*} \Omega^{\bullet}_{X/S} (\log \ D)) \> \sigma >>
{\cal K}^{\bullet} \< \theta << {\cal H}^{0} \oplus {\cal H}^{1} [ -1] \]
be a splitting. As ${\cal H}^{1}$ is ${\cal O}_{X'}$-locally free,
$\sigma \otimes id$ and $\theta \otimes id$ are quasi-isomorphisms of the
corresponding complexes tensored with ${\cal H}om_{{\cal O}_{X'}} ({\cal H}^{1},
{\cal O}_{X'})$. We obtain therefore maps
\[
\minCDarrowwidth=0.5cm
\begin{CD}
H^{0} \Bigl( X', {\cal H}om_{{\cal O}_{X'}}({\cal H}^{1}, {\cal H}^{1})
\Bigr) = \h^{1} \Bigl( X', {\cal H}^{1}
\otimes {\cal H}om_{{\cal O}_{X'}}
({\cal H}^{1}, {\cal O}_{X'}) [ -1 ] \Bigr) \\
\V V V \\
\h^{1} \Bigl( X', \Bigl[ {\cal H}^{0} \oplus {\cal H}^{1} [ -1] \Bigr]
\otimes {\cal H}om_{{\cal O}_{X'}} ({\cal H}^{1}, {\cal O}_{X'}) \Bigr) \\
\V V \simeq V \\
\h^{1} \Bigl( X', \tau_{\leq 1} F_{*} \Omega^{\bullet}_{X/S} (\log \ D)
\otimes {\cal H}om_{{\cal O}_{X'}} ({\cal H}^{1}, {\cal O}_{X'}) \Bigr)\\
\V V = V \\
{\h^{1} (X', {\cal H}om_{{\cal O}_{X'}} ({\cal H}^{1},
F_{*} {\cal O}_{X}) \rightarrow {\cal H}om_{{\cal O}_{X'}} ({\cal H}^{1},
Z^{1}))}\\
\V V V \\
H^{0} \Bigl( X', {\cal H}om_{{\cal O}_{X'}}({\cal H}^{1}, {\cal H}^{1}) \Bigr)
\end{CD}
\]
where the last map comes from the short exact sequence in (10.1). By definition
of a splitting, the composed map is the identity.
Therefore, the image of
$$id_{{\cal H}^{1}} \in H^{0} (X',
{\cal H}om_{{\cal O}_{X'}} ({\cal H}^{1}, {\cal H}^{1}))$$
is a splitting cohomology class $\varphi$. Obviously, the both
constructions are inverse to each other.\\
\ \\
\end{Beweis}
\begin{defn}{Remark} In (10.2) one could have replaced in the definition of
a splitting the \v{C}ech complex by any complex ${\cal K}^{\bullet}$ bounded
below and quasi-isomorphic to $\tau_{\leq 1} F_{*} \Omega^{\bullet}_{X/S}
(log \ D)$. Then we would have proven as in (10.5),
that a splitting cohomology
class defines a splitting. However, to get the converse, we need that
$\sigma$ and $\theta$ define a map from
\[ \h^{0}(X', {\cal H}om_{{\cal O}_{X'}} ({\cal H}^{1}, {\cal O}_{X'}) \otimes
{\cal K}^{\bullet}) \]
to
\[ \h^{1} (X', {\cal H}om_{{\cal O}_{X'}} ({\cal H}^{1}, {\cal O}_{X'})
\otimes \tau_{\leq 1} F_{*} \Omega^{1}_{X/S} (\log \ D)). \]
This is of course the case when ${\cal K}^{\bullet}$ is a complex
of ${\cal O}_{X'}$-modules, but not in general. One needs a bit more knowledge
on the derived category; in particular one needs the global
Hom (  \ ,  \ ) in this
category.
\end{defn}
\begin{satz}{Main theorem} Let $X$ be a smooth scheme over $S$ and $D
\subset X$ be a $S$-normal crossing divisor. Then\\
\ \\
{\rm a)} A lifting $\tilde{D}' \subset \tilde{X}'$ of $D' \subset X'$ to
$\tilde{S}$ defines a splitting cohomology class $$\varphi =
\varphi_{(\tilde{X}', \tilde{D}')}.$$
{\rm b)} Every splitting cohomology class $\varphi$ is of the shape
$\varphi = \varphi_{(\tilde{X}', \tilde{D}')}$ for some lifting $\tilde{D}'
\subset \tilde{X}'$ of $D' \subset X'$ to $\tilde{S}$.
\end{satz}
\ \\

We will only need part a) in the proof of Theorem (8.3). Even if it might
be more elegant to use more formal arguments we will give the necessary
calculations in an explicit way for cycles in the \v{C}ech-cohomology.

\begin{Beweis}{Proof}
a) Let ${\cal U} = \{ X_{\alpha} \}$ be an affine
cover of $X$, such that the $X_{\alpha \beta}$
are affine, and such that (8.17) and (9.10) give liftings
$$\tilde{D}_{\alpha} \subset \tilde{X}_{\alpha} \mbox{ \  of \ }
D_{\alpha} \subset X_{\alpha}$$ to $\tilde{S}$
and liftings
$$\tilde{F}_{\alpha} : \tilde{X}_{\alpha} \>>>
\tilde{X}' \mbox{ \ of \ } F: X \>>> X'$$ satisfying
$$\tilde{F}_{\alpha}^{*}
{\cal O}_{\tilde{X}'} ( - \tilde{D}') = {\cal O}_{\tilde{X}_{\alpha}}
(- p \cdot \tilde{D}_{\alpha}).$$
For $\tilde{X}_{\alpha \beta} = \tilde{X}_{\alpha}|_{X_{\alpha \beta}}$
(8.23) implies the existence of isomorphisms of liftings
\[ u_{\alpha \beta}: \tilde{X}_{\alpha \beta} \>>>
\tilde{X}_{\beta \alpha} \]
As in the proof of (9.12) one uses (9.9) to define
\[ \varphi_{\alpha \beta} = p^{-1} \circ (\tilde{F}^{*}_{\alpha} -
(\tilde{F}_{\beta} \circ u_{\alpha \beta})^{*}) \in \Gamma (X'_{\alpha \beta},
{\cal H}om_{{\cal O}_{X'}} (\Omega^{1}_{X'/S} (\log \ D'), F_{*}
{\cal O}_{X})) \]and by (9.11) $\varphi_{\alpha \beta}$ is a cocycle.
On $\tilde{X}_{\alpha}$ we obtained in (10.3) a map
\[ \psi_{\alpha} = p^{-1} \circ \tilde{F}^{*}_{\alpha} \in \Gamma (X'_{\alpha},
{\cal H}om_{{\cal O}_{X'}} (\Omega^{1}_{X'/S} (\log \ D'), Z^{1})) \]
lifting the Cartier operator $C^{-1}$.\\
\ \\
\end{Beweis}
\begin{defn}{Claim} One has $$p^{-1} \circ \tilde{F}^{*}_{\beta}|_{X'_{\beta \alpha}}
= p^{-1} \circ (\tilde{F}_{\beta} \circ u_{\alpha \beta})^{*}$$ in
\[ \Gamma
(X'_{\beta \alpha}, {\cal H}om_{{\cal O}_{X'}} (\Omega^{1}_{X'/S} (\log \ D'),
Z')). \]
\end{defn}
\begin{Beweis}{Proof} For $\tilde{x}' \in {\cal O}_{\tilde{X}'}$ and
$\tilde{x}_{\beta} \in {\cal O}_{\tilde{X}_{\beta}}$ we can write
\[ \tilde{F}^{*}_{\beta}(\tilde{x}') = \tilde{x}^{p}_{\beta} + p \cdot u
(\tilde{x}_{\beta}, \tilde{x}').\]
Since $u^{*}_{\alpha \beta}|_{p \cdot {\cal O}_{X_{\alpha \beta}}}$ is the
identity, one obtains
\[ u^{*}_{\alpha \beta} \tilde{F}^{*}_{\beta}(\tilde{x}') = u^{*}_{\alpha \beta}
(\tilde{x}_{\beta})^{p} + p \cdot u(\tilde{x}_{\beta}, \tilde{x}').\]
By (10.3) we have
\[ p^{-1} \circ \tilde{F}^{*}_{\beta} (d \tilde{x}') = x^{p-1} dx +
du (\tilde{x}_{\beta}, \tilde{x}') = p^{-1} \circ (\tilde{F}_{\beta}
\circ u_{\alpha \beta})^{*}. \]
\end{Beweis}
Now (10.8) is just saying that $\delta \psi = d \varphi$
and therefore $(\varphi_{\alpha \beta}, \psi_{\beta})$ defines
a cohomology class $\varphi_{(\tilde{X}', \tilde{D}')}$ in
\[ \h^{1} (X', {\cal H}om_{{\cal O}_{X'}} (\Omega^{1}_{X'/S} (\log \ D'),
F_{*} {\cal O}_{X}) \>>> {\cal H}om_{{\cal O}_{X'}} (\Omega^{1}_{X'/S}
(\log \ D'), Z^{1})). \]
By construction its image in
\[ H^{0} (X', {\cal H}om_{{\cal O}_{X'}} (\Omega^{1}_{X'/S} (\log \ D'),
{\cal H}^{1})) \]
is given by $(\psi_{\beta})$ and hence it is the Cartier operator.\\
\ \\
{\rm b)} Conversely, let $(\varphi_{\alpha \beta}, \psi_{\alpha})$
be the cocycle giving the splitting cohomology class $\varphi$ for
some affine covering
${\cal U}' = \{ X'_{\alpha} \}$. First we want to add some coboundary
to get a new representative of $\varphi$. \\
\ \\
By (8.17), (9.10) and (8.21)
we can assume that we have:\\
\ \\
i) {\it Liftings to} $\tilde{S}:$
$$ \tilde{D}_{\alpha} \subset \tilde{X}_{\alpha}
 \mbox{ \ of \ } D_\alpha = D|_{X_{\alpha}} \subset X_{\alpha}
 = X|_{X'_{\alpha}},$$
$$\tilde{D}'_{\alpha} \subset \tilde{X}'_{\alpha} \mbox{ \ of \ } D'_{\alpha} =
D'|_{X'_{\alpha}} \subset X'_{\alpha}$$
and
$$\tilde{F}_{\alpha}: \tilde{X}_{\alpha} \>>> \tilde{X}'_{\alpha}
\mbox{ \ of \ } F_\alpha = F|_{X_\alpha}:X_\alpha \>>> X'_\alpha$$
with $$\tilde{F}^{*}_{\alpha}: {\cal O}_{\tilde{X}'_{\alpha}}
(- \tilde{D}'_{\alpha}) = {\cal O}_{\tilde{X}_{\alpha}} ( - p \cdot
\tilde{D}_{\alpha}).$$
ii) {\it Isomorphisms of liftings}:
\[
u'_{\alpha \beta} : \tilde{X}'_{\alpha \beta} \>>>
\tilde{X}'_{\beta \alpha }  \mbox{ \ and \ }
u_{\alpha \beta}: \tilde{X}_{\alpha \beta} \>>>
\tilde{X}_{\beta \alpha}
\]
were we keep the notation $\tilde{X}'_{\alpha \beta} = \tilde{X}'_{\alpha}
|_{X_{\alpha \beta}}$.\\
\ \\
For $\tilde{x}' \in {\cal O}_{\tilde{X}'_{\alpha \beta}}$ we can write
$u_{\alpha \beta}^{'*} (\tilde{x}') = \tilde{x}' + p \cdot \lambda_{\alpha
\beta} (\tilde{x}')$. Then
\[ \tilde{F}^{*}_{\alpha} u'^{*}_{\alpha \beta} (\tilde{x}') =
\tilde{F}_{\alpha}^{*} (\tilde{x}') + p \cdot F^{*} \lambda_{\alpha \beta}
(\tilde{x}') = \tilde{F}^{*}_{\alpha} (\tilde{x}') + p \cdot
\lambda_{\alpha \beta} (\tilde{x}')^{p}. \]
Since $d(\lambda_{\alpha \beta} (\tilde{x}')^{p}) = 0$ the explicit
description of $p^{-1} \circ \tilde{F}^{*}_{\alpha}$ in (10.3) gives
\begin{defn}{Claim} One has $p^{-1} \circ \tilde{F}^{*}_{\alpha}|_{X'_{\alpha \beta}}
= p^{-1} \circ (u'_{\alpha \beta} \circ \tilde{F}_{\alpha})^{*}$ in
\[ \Gamma (X'_{\alpha \beta}, {\cal H}om_{{\cal O}_{X'}} (\Omega^{1}_{X'/S}
(\log \ D'), \ Z^{1})). \]
\end{defn}
Define
\[ \theta_{\alpha} = p^{-1} \circ \tilde{F}^{*}_{\alpha} \in
\Gamma (X'_{\alpha}, {\cal H}om_{{\cal O}_{X'}} (\Omega^{1}_{X'/S} (\log \ D'),
\ F_{*} {\cal O}_{X})). \]
Replacing ${\cal U}'$ by some finer cover if necessary we find
$$f_{\alpha} \in \Gamma (X'_{\alpha}, {\cal H}om_{{\cal O}_{X'}}
(\Omega^{1}_{X'/S} (\log \ D'), \ F_{*} {\cal O}_{X}))$$ such that $df_{\alpha}
= \theta_{\alpha} - \psi_{\alpha}$.
\begin{defn}{Claim}For $\sigma'_{\alpha \beta} = \varphi_{\alpha \beta}
+ \delta f_{\alpha}$ the cohomology class $\varphi$
is represented by the cocycle $(\sigma'_{\alpha \beta}, \theta_{\alpha}).$
\end{defn}
\begin{Beweis}{Proof} This is obvious since
\[ (\sigma'_{\alpha \beta}, \theta_{\alpha}) = (\varphi_{\alpha \beta}
+ \delta f_{\alpha}, \psi_{\alpha} + d f_{\alpha}). \]
\end{Beweis}
The main advantage of $\sigma'_{\alpha \beta}$ is that it comes in
geometric terms:\\
Write, using (9.9)
\[ \sigma_{\alpha \beta} = p^{-1} \circ \Bigl( (u'_{\alpha \beta} \circ
\tilde{F}_{\alpha})^{*} - (\tilde{F}_{\beta} \circ u_{\alpha \beta})^{*}
\Bigr) \]
in
$$\Gamma \Bigl( X'_{\alpha \beta}, {\cal H}om_{{\cal O}_{X'}}
(\Omega^{1}_{X'/S} (\log \ D') \ , F_{*} {\cal O}_{X}) \Bigr).$$
Applying (10.9) to the first and (10.8) to the second summand one gets
\[ d \sigma_{\alpha \beta} = \theta_{\alpha} - \theta_{\beta} = df_{\alpha}
- df_{\beta} + \psi_{\alpha} - \psi_{\beta} = d (f_{\alpha} - f_{\beta})
+ d \varphi_{\alpha \beta}. \]
Hence $g_{\alpha \beta} = \sigma_{\alpha \beta} - \varphi_{\alpha \beta} -
(f_{\alpha} - f_{\beta})$ is closed and lives in
\[ \Gamma \Bigl(
X'_{\alpha \beta}, {\cal H}om_{{\cal O}_{X'}}
(\Omega^{1}_{X'/S} (\log \ D'), {\cal O}_{X'}) \Bigr). \]
By (8.22)
$u'^{*}_{\alpha \beta} - g_{\alpha \beta}$ defines a new isomorphism
of liftings
\[ v'_{\alpha \beta} : \tilde{X}'_{\alpha \beta} \>>>
\tilde{X}'_{\beta \alpha}. \]
As $\tilde{F}^{*}_{\alpha} = p F^{*}$ on $p \cdot {\cal O}_{X'_{\alpha}}$,
one has $p^{-1} \circ \tilde{F}^{*}_{\alpha} \circ g_{\alpha \beta} =
g_{\alpha \beta}$ and
\[
\begin{array}{ll}
\sigma'_{\alpha \beta} = & \varphi_{\alpha \beta} + (f_{\alpha} - f_{\beta})
= \sigma_{\alpha \beta} - g_{\alpha \beta} = \\
\ \\
 \ & p^{-1} \circ \Bigl[ (u'_{\alpha \beta} \circ \tilde{F}_{\alpha})^{*} -
(\tilde{F}_{\beta} \circ u_{\alpha \beta})^{*} - \tilde{F}^{*}_{\alpha}
\cdot g_{\alpha \beta} \Bigr] \ .
\end{array}
\]
One obtains
\begin{defn}{Claim} $\sigma'_{\alpha \beta} = p^{-1} \circ
\Bigl[ (v'_{\alpha \beta} \circ \tilde{F}_{\alpha})^{*} -
(\tilde{F}_{\beta} \circ u_{\alpha \beta})^{*} \Bigr].$ \\
\ \\
The proof of (10.7) ends with
\end{defn}
\begin{defn}{Claim} The cocycle condition for $\sigma'_{\alpha \beta}$
allows the glueing of $\tilde{X}'_{\alpha}$ to $\tilde{X}'$ using
$v'_{\alpha \beta}$.
\end{defn}
\begin{Beweis}{Proof}  One has to show that
\[ v'_{\alpha \gamma} = v'_{\beta \gamma} \circ v'_{\alpha \beta} \]
or, by (8.22), that the homomorphism defined there verifies
\[ v'^{*}_{\alpha \beta} \circ v'^{*}_{\beta \gamma} - v'^{*}_{\alpha
\gamma} = 0. \]
Since $\tilde{F}^{*}_{\alpha}$ is injective, it is enough to show
that
\[ p^{-1} \circ \tilde{F}^{*}_{\alpha} \circ [ v'^{*}_{\alpha \beta}
\circ v'^{*}_{\beta \gamma} - v'^{*}_{\alpha \gamma} ] = 0 \]
as homomorphism in
\[ {\cal H}om_{{\cal O}_{X'}} (\Omega^{1}_{X'/S} (\log \ D'), F_{*}
{\cal O}_{X}). \]
The cocycle condition for $\sigma'$ is
\[ p^{-1} \circ [ \tilde{F}^{*}_{\alpha} \circ v'^{*}_{\alpha \beta} -
u^{*}_{\alpha \beta} \circ \tilde{F}^{*}_{\beta} - \tilde{F}^{*}_{\alpha}
\circ v'^{*}_{\alpha \gamma} + u^{*}_{\alpha \gamma} \circ
\tilde{F}^{*}_{\gamma} + \tilde{F}^{*}_{\beta} \circ v'^{*}_{\beta \gamma}
- u^{*}_{\beta \gamma} \circ \tilde{F}^{*}_{\gamma} ] = 0. \]
Since
\[ \tilde{F}^{*}_{\alpha} \circ v'^{*}_{\alpha \beta} - u^{*}_{\alpha
\beta} \circ \tilde{F}^{*}_{\beta} \]
is a homomorphism from ${\cal O}_{X'}$ to $p \cdot {\cal O}_{X}$,
we can replace it by
\[ [ \tilde{F}^{*}_{\alpha} \circ v'^{*}_{\alpha \beta} - u^{*}_{\alpha \beta}
\circ \tilde{F}^{*}_{\beta} ] \circ v^{*}_{\beta \gamma} \]
as $v^{*}_{\beta \gamma}$ is the identity on ${\cal O}_{X'_{\beta \gamma}}$.
Similarly, we can add some $u^{*}_{\alpha \beta}$ at the right hand
side (see (9.11)) and get
\[
\begin{array}{l}
0 = p^{-1} \circ [ \tilde{F}^{*}_{\alpha} \circ v'^{*}_{\alpha \beta} \circ
v'^{*}_{\beta \gamma} - u^{*}_{\alpha \beta} \circ \tilde{F}^{*}_{\beta}
\circ v'^{*}_{\beta \gamma} - \tilde{F}^{*}_{\alpha} \circ
v'^{*}_{\alpha \gamma} + \\
\ \\
+ u^{*}_{\alpha \gamma} \circ \tilde{F}^{*}_{\gamma} - u^{*}_{\alpha \beta}
\circ \tilde{F}^{*}_{\beta} \circ v'^{*}_{\beta \gamma} - u^{*}_{\alpha \beta}
u^{*}_{\beta \gamma} \circ \tilde{F}^{*}_{\gamma} ]
\end{array}
\]
where all the summands are morphisms
from ${\cal O}_{\tilde{X}'_{\! \gamma}}
\>>> {\cal O}_{\tilde{X}_{\alpha}}$.
This is the same as
\[ 0 = p^{-1} \circ \tilde{F}^{*}_{\alpha} \circ (v'^{*}_{\alpha \beta} \circ
v'^{*}_{\beta \gamma} - v'^{*}_{\alpha \gamma}) +
p^{-1} \circ (u^{*}_{\alpha \gamma} - u^{*}_{\alpha \beta}
u^{*}_{\beta \gamma}) \tilde{F}_{\gamma}^{*}. \]
By (9.11) the term on the right is zero and
\[ 0 = p^{-1} \circ \tilde{F}^{*}_{\alpha} \circ (v'^{*}_{\alpha \beta}
\circ v'^{*}_{\beta \gamma} - v'^{*}_{\alpha \gamma}). \]
\end{Beweis}
\Abschnitt{Splittings of the de Rham complex.}
Let us generalize (10.2) to $\tau_{\leq i} F_{*} \Omega^{\bullet}_{X/S}
(\log \ D)$ for $i > 1$. As remarked in (10.6) one can, using the derived
category, replace the complex ${\cal K}^{\bullet}$ in the following
definition by any complex ${\cal K}^{\bullet}$ bounded below and
quasi-isomorphic to $\tau_{\leq i} F_{*} \Omega^{\bullet}_{X/S} (\log \ D)$.
\begin{defn}{Definition} A {\it splitting} of $\tau_{\leq i} F_{*}
\Omega^{\bullet}_{X/S} (\log \ D)$ is a diagram
\index{Splitting of $\tau_{\leq 1} F_{*}
\Omega^{\bullet}_{X/S} (\log \ D)$}
\[
\begin{CD}
\tau_{\leq i} F_{*} \Omega^{\bullet}_{X/S} (\log \ D) \> \sigma >>
{\cal K}^{\bullet} \\
\noarr   \A A \theta A \\
\noarr \bigoplus_{j \leq i} {\cal H}^{j} [-j]
\end{CD}
\]
where ${\cal K}^{\bullet}$ is the \v{C}ech complex
$${\cal C}^{\bullet}
({\cal U}, \tau_{\leq i} F_{*} \Omega^{\bullet}_{X/S} (\log \ D))$$
associated to some affine cover ${\cal U}$ of $X$ (and hence $\sigma$
a quasi-isomorphism) and where $\theta$ is a quasi-isormorphism.
Here again, $$\bigoplus_{j \leq i} {\cal H}^{j} [ - j]$$ is the complex with
zero differential and with ${\cal H}^{j}$ in degree $j$
and $\tau_{\leq i}$ is the filtration explained in (A.26).
\end{defn}
\begin{defn}{Example} Let us return to the assumptions made in (10.3), i.e.
that the liftings $\tilde{D}, \tilde{X}, \tilde{D}', \tilde{X}'$ and
especially $\tilde{F}$ exist. We had defined there a morphism
\[ \psi = p^{-1} \circ \tilde{F}^{*} : \Omega^{1}_{X'/S} (\log \ D')
\>>> Z^{1} \]
which was a lifting of $C^{-1}$.\\
\ \\
We define
\[ \psi^{j} (\omega_{1} \wedge \ldots \wedge \omega_{j}) =
\psi (\omega_{1}) \wedge \psi (\omega_{2}) \wedge \ldots \wedge \psi
(\omega_{j}) \]
where $\omega_{l} \in \Omega^{1}_{X'/S} (\log \ D')$. Since $\psi (\omega_{l})$
is closed, the image of $\psi^{j}$ lies in $Z^{j}$ and, since the Cartier
operator was defined as $\bigwedge^{j} C^{-1}$ the map $\psi^{j}$ induces
the Cartier operator on in
\[ {\cal H}om_{{\cal O}_{X'}} (\Omega^{j}_{X'/S} (\log \ D'), {\cal H}^{j}). \]
\end{defn}
\begin{satz}{Theorem} Let $X$ be a smooth $S$-scheme, $D \subset X$ be a normal
crossing divisor over $S$ and let
$$\tilde{D}' \subset \tilde{X}' \mbox{ \  be a lifting of \ }
D' \subset X'$$ to $\tilde{S}$. Then the splitting cohomology class
$\varphi_{(\tilde{X}', \tilde{D}')}$ of (10.7,a) induces a splitting of
$$\tau_{\leq i} F_{*} \Omega^{\bullet}_{X/S} (\log \ D) \mbox{ \ for \ }
i < p = \Char \ (S).$$ In particular, if $p > \dim_{S} X$, it induces a
splitting of the whole de Rham complex
\[ F_{*} \Omega^{\bullet}_{X/S} (\log \ D). \]
\end{satz}
\begin{Beweis}{Proof} Let $(\varphi_{\alpha \beta}, \psi_{\alpha})$ be a
\v{C}ech cocycle for $(\varphi_{\tilde{X}', \tilde{D}'})$ where we
regard \\
$(\varphi_{\alpha \beta}, \psi_{\alpha})$ as an
${\cal O}_{X'}$-homomorphism:
\[ (\varphi, \psi) :\Omega^{1}_{X'/S} (\log \ D') \>>> {\cal C}^{1}
(F_{*} {\cal O}_{X}) \oplus {\cal C}^{0} (Z^{1}). \]
We define
\[ (\varphi, \psi)^{\otimes j} (\omega_{1}, \otimes \cdots \otimes
\omega_{j}) \in {\cal C}^{j} (\tau_{\leq i} F_{*} \Omega^{\bullet}_{X/S}
(\log \ D)) \]
for all $ 0 < j \leq i$ and all
$$\omega_{1} \otimes \cdots \otimes \omega_{j} \in \bigotimes^{j}_{1}
\Omega^{1}_{X'/S} (\log \ D')$$ by the following inductive formula:\\
For any cocycle
\[
\underline{b} : = (b_{j}, \ldots, b_{0}), \ b_{l} \in {\cal C}^{l}
(F_{*} \Omega^{j - l}_{X/S} (\log \ D)), \]
with
\[ d b_{j-s} + (-1)^{j - s} \delta
b_{j-s - 1} = 0 \mbox{ \ for all \ } 0 \leq s \leq j, \]
we define
\[\underline{b} \otimes (\varphi, \psi) (\omega_{j+1}) =
(a_{j+1}, \ldots, a_{0}) = : \underline{a}, \]
with
\[a_{l} \in {\cal C}^{l}
(F_{*} \Omega^{j+1-l}_{X/s} (\log \ D)), \]
by the rule
\[a_{j+1-s} : = (-1)^{s} b_{j-s} \cup \varphi + b_{j+1-s} \cup \psi \]
where
\[ (b_{j-s} \cup \varphi)_{\alpha_{0} \ldots \alpha_{j+1-s}}
: = (b_{j-s})_{\alpha_{0} \ldots \alpha_{j-s}} \cdot \varphi_{\alpha_{j-s},
\alpha_{j+1-s}} (\omega_{j+1}) \]
and
\[(b_{j+1-s} \cup \psi)_{\alpha_{0} \ldots \alpha_{j+1-s}}: =
(b_{j+1-s})_{\alpha_{0} \ldots \alpha_{j+1- s}} \cdot \psi_{\alpha_{j+1-s}}
(\omega_{j+1}).
\]
One has
\[ d a_{j+1 - s} + (-1)^{j+1 - s} \delta a_{j-s} = 0 \]
and therefore $\underline{a}$ is a \v{C}ech cocycle.\\
\ \\
We have, for $j \leq i$, a diagram
\[
\minCDarrowwidth{1.7cm}
\begin{CD}
(\Omega^{1}_{X'/S} (\log \ D'))^{\otimes \ j} \> (\varphi, \psi)^{\otimes \ j}
>> {\cal C}^{j} (\tau_{\leq i} F_{*} \Omega^{1}_{X/S} (\log \
D ))_{\mbox{closed}}\\
\V V \pi^{j} V   \V V V \\
\Omega^{j}_{X'/S} (\log \ D') \> \bigwedge^{j} C^{-1} >> {\cal H}^{j}
\end{CD}
\]
and any section $\delta^{j}$ of $\pi^{j}$ allows to define
$$\theta_{j} = (\bigwedge^{j} C^{-1})^{-1} \circ \delta^{j} \circ (\varphi,
\psi)^{\otimes j}.$$
The splitting $$\theta : \bigoplus_{j \leq i} {\cal H}^{j} [-j] \>>>
{\cal K}^{\bullet} \mbox{ \ is \ } \theta = \bigoplus_{j \leq i} \theta_{j} [-j].$$
Such sections $\delta^{j}$ exist for $j \leq i < \Char \ (S)$:
\[ \delta^{j} (\omega_{1} \otimes \ldots \otimes \omega_{j}) =
\frac{1}{j!} \sum_{s \in \Sigma_{j}} \ sign \ (s) \cdot \omega_{s(1)} \wedge
\ldots \wedge \omega_{s(j)}, \]
where $\Sigma_j$ denotes the symmetric group.\\
\end{Beweis}
\begin{satz}{Corollary} Let $D = A + B$ in (10.16). Then for $i < p$
the splitting cohomology class $\varphi_{(\tilde{X}, \tilde{D})}$ induces
a splitting of $$\tau_{\leq i} F_{*} \Omega^{\bullet}_{X/S} (A, B)$$ as well,
i.e. a quasi-isomorphism
\[ \theta : \bigoplus_{j \leq i} \Omega^{j}_{X'/S} (A', B') [-j] \>>>
{\cal K}^{\bullet} (A, B) \]
where ${\cal K}^{\bullet} (A, B)$ is the \v{C}ech complex of
\[ \tau_{\leq i} F_{*} \Omega^{\bullet}_{X/S} (A, B). \]
\end{satz}
\begin{Beweis}{Proof} As in (9.17) one obtains from (2.7) and
(2.10) a quasi-isomorphism
\[
\begin{CD}
F_{*} \Omega^{1}_{X/S} (\log \ (A + B)) \otimes_{{\cal O}_{X'}}
{\cal O}_{X'} (- A') \\
\| \\
F_{*} (\Omega^{\bullet}_{X/S} (\log \ (A+B)) ( - p
\cdot A))\\
\V V V \\
F_{*} \Omega^{\bullet}_{X/S} (A, B) \ .\\
\end{CD}
\]
For ${\cal K}^{\bullet}$, the \v{C}ech complex of $\tau_{\leq i} F_{*}
\Omega^{\bullet}_{X/S} (\log \ D)$, we have a quasi-isomorphism
\[ {\cal K}^{\bullet} \otimes_{{\cal O}_{X'}} {\cal O}_{X'} (-A) \>>>
{\cal K}^{\bullet} (A, B) \]
and the existence of $\theta$ follows from (10.16).\\
\ \\
\end{Beweis}
\begin{defn}{Remark} In (10.3) and (10.15)
we have seen that if both $D' \subset
X'$ and $D \subset X$ lift to $\tilde{D}' \subset \tilde{X}'$ and
$\tilde{D} \subset \tilde{X}$, and if $F$ lifts to: $\tilde{F} :
\tilde{X} \rightarrow \tilde{X}'$ with
\[ \tilde{F}^{*} {\cal O}_{\tilde{X}'}
(- \tilde{D}') = {\cal O}_{\tilde{X}} (- p \cdot \tilde{D}), \]
then
\[ \psi = p^{-1} \circ \tilde{F}^{*} : \Omega^{1}_{X'/S} (\log \ D')
\>>> Z^{1} \subset \Omega^{1}_{X/S} (\log \ D) \]
lifts the Cartier operator, and gives an especially nice splitting
of
\[ \tau_{\leq 1} F_{*} \Omega^{\bullet}_{X/S} (\log \ D). \]
This defines
\[ \bigwedge^{l} \psi : \Omega^{l}_{X'/S} (\log \ D') \>>>
Z^{l} \subset \Omega^{l}_{X/S} (\log \ D) \]
lifting the Cartier operator, and therefore one obtains a quasi-isomorphism:
\[ \bigoplus_{j} \Omega^{j}_{X'/S} (\log \ D') [-j] \> \bigwedge^{\bullet}
\psi >> F_{*} \Omega^{\bullet}_{X/S} (\log \ D), \]
which gives via (10.17) a quasi-isomorphism
\[ \bigoplus_{j} \Omega^{j}_{X'/S} (A', B') [-j] \> \bigwedge^{\bullet}
\psi >> F_{*} \Omega^{\bullet}_{X/S} (A, B) \]
if $D = A + B$.
In particular, there is here no restriction on $\dim_{S} X$ in this case.
\end{defn}
In general one has
\begin{satz}{Proposition} Let $X$, $A$ and $B$ be as in (10.17). Then
the splitting cohomology class $\varphi_{(\tilde{X}', \tilde{D}')}$
induces a splitting of $$F_{*} \Omega^{\bullet}_{X/S} (A,B)$$
when $\dim_{S} X \leq p$ and $S$
is affine.
\end{satz}
\begin{Beweis}{Proof} Of course this is nothing but (10.17) if
$\dim_{S} X < p$.\\
For $\dim_{S} X \geq p$, we observe first that whenever
$j: U \rightarrow X$ is the embedding
of an open set such that $(X-U)$ is a divisor, then for coherent
sheaves ${\cal F}$ on $U$ and ${\cal G}$ on $X$ one has
\[
{\cal H}om_{{\cal O}_X} (j_{*} {\cal F}, {\cal G})_{x} =
\left\{
\begin{array}{ll}
{\cal H}om_{{\cal O}_X} ({\cal F}, {\cal G}
|_{U})_{x} & \mbox{for \ $x \in U$}\\
\ & \\
0 & \mbox{for \ $x \in (X - U),$}\\
\end{array}
\right.
\]
that is
$${\cal H}om_{{\cal O}_X} (j_{*} {\cal F}, {\cal G})
=  j_{!} {\cal H}om_{{\cal O}_X} ({\cal F}, {\cal G}|_{U}).$$
\ \\
Let us consider the ${\cal O}_{X'}$-maps defined in (10.17) for
$0 \leq l \leq p-1$:
\[
\Omega^{l}_{X'/S} (B', A')  \>>>  {\cal C}^{l} (F_{*}
{\cal O}_{X}) + \cdots  + {\cal C}^{0} (Z^{l})
\]
\[ \hspace{4cm}
(\varphi_{\alpha_{0}  \ldots \alpha_{l}}, \ \ldots \ ,\varphi_{\alpha_{0}})
\]
with the cocycle condition
\[ d \varphi_{\alpha_{0} \ldots \alpha_{k}} + (-1)^{l}
\delta \varphi_{\alpha_{0} \ldots \alpha_{k-1}} = 0.\]
The composite map
\[ \Omega^{l}_{X'/S} (B', A') \>>> {\cal C}^{0} (Z^{l}) \>>>
{\cal C}^{0} ({\cal H}^{l})\]
is just $\bigwedge^{l} C^{-1}$.\\
Applying for $1 \leq k \leq l \leq p-1$ the functor
${\cal H}om_{{\cal O}_{X'}} ( -, \Omega^{n}_{X'/S})$, where\\
$n = \dim_{S} X$,
one obtains ${\cal O}_{X'}$-linear maps (see (9.19) and (9.20))
\[(j_{\alpha_{0} \ldots \alpha_{k}})_! F_{*} \Omega^{n-(l-k)}_{X/S} (A,B)
\> \varphi_{\alpha_{0} \ldots \alpha_{k}}^{\vee} >>
\Omega^{n-l}_{X'/S} (A', B') \]
and therefore ${\cal O}_{X'}$-maps
\[ F_{*} \Omega^{n- (l-k)}_{X/S} (A,B) \> \varphi_{\alpha_{0}
\ldots \alpha_{k}}^{\vee} >> {\cal C}^{k} (\Omega^{n-l}_{X'/S}
(A',B')). \]
For $k = 0$, one has an exact sequence
\[ 0 \>>> Z^{l} \>>> F_{*} \Omega^{l}_{X/S} (B,A) \>>>
F_{*} \Omega^{l+1}_{X/S} (B,A), \]
and applying (9.20) again, one obtains that
\[ {\cal H}om_{{\cal O}_{X'}} (Z^{l}, \Omega^{n}_{X'/S}) =
\frac{F_{*} \Omega^{n-l}_{X/S} (A,B)}{d F_{*} \Omega^{n-l-1}_{X/S} (A,B)} \]
which gives similarly a ${\cal O}_{X'}$-linear map
\[ \frac{F_{*} \Omega^{n-l}_{X/S} (A,B)}{d F_{*} \Omega^{n-l-1}_{X/S}
(A,B)} \> \varphi_{\alpha_{0}}^{\vee} >>
{\cal C}^{0} (\Omega^{n-l}_{X'/S}
(A', B')). \]
The cocycle condition tells us that
\[ \varphi^{\vee}_{\alpha_{0} \ldots \alpha_{k}} \circ d + (-1)^{l}
\delta \varphi^{\vee}_{\alpha_{0} \ldots \alpha_{k-1}} = 0. \]
This means that $\varphi^{\vee}$ defines a map of complexes
$$\tau_{\geq n - l} F_{*} \Omega^{\bullet}_{X/S} (A,B) [(n-l)]
\>>> \tau_{\leq l} {\cal C}^{\bullet}
(\Omega^{n-l}_{X'/S} (A', B')),$$ where
$$\tau_{\geq n - l} F_{*} \Omega^{\bullet}_{X/S} (A,B) [(n-l)] :=
\hspace{4.7cm}$$
$$\frac{F_{*} \Omega^{n-l}_{X/S} (A,B)}{d F_{*} \Omega^{n-l}_{X/S}
(A,B)} \>>> F_{*} \Omega^{n-l+1}_{X/S} (A,B) \>>> \cdots
\>>> F_{*} \Omega^{n}_{X/S} (A,B).$$
The composite map
\[ {\cal H}^{n-l} \>>> \tau_{\geq n - l} F_{*} \Omega^{\bullet}_{X/S} (A,B)
[(n-l)] \>>> {\cal C}^{\bullet} (\Omega^{n-l}_{X'/S}
(A',B')) \]
is given by $\bigwedge^{n-l} C$.
As \ $\tau_{\geq n - l}$ \ maps to \ $\tau_{\geq n - l+1}$\ , we find
in this way a ${\cal O}_{X'}$-map
\[ \tau_{\geq n - p + 1} F_{*} \Omega^{\bullet}_{X/S} (A,B) \> \varphi^{\vee}
>> \bigoplus^{n}_{i = n - p + 1} \ \ {\cal C}^{\bullet} (\Omega^{i}_{X'/S}
(A',B')) [-i] \]
which is a quasi-isomorphism.
In particular, for any open set $U' \subset X'$ and any ${\cal O}_{X'}$-sheaf
${\cal F'}$ one has
\[ \h^{l} (U', \tau_{\geq n - p + 1} F_{*} \Omega^{\bullet}_{X/S} (A,B)
\otimes {\cal F'}) = \bigoplus^{n}_{i=n-p+1} \ H^{l-i}
(U', \Omega^{i}_{X'/S} (A', B') \otimes {\cal F'}) . \]
Consider now $n = p$ as needed to finish the proof of (10.19).
From the exact sequence
\[ 0 \>>> {\cal H}^{0} \>>> F_{*} \Omega^{\bullet}_{X/S}
(A,B) \>>> \tau_{\geq 1} F_{*} \Omega^{\bullet}_{X/S}
(A,B) \>>> 0 \]
we obtain
an exact sequence of hypercohomology groups
\[
\h^{p} (F_{*} \Omega^{\bullet}_{X/S} (A,B) \otimes {\cal H}^{p^{\vee}})
\>>> \bigoplus^{p}_{a=1} H^{p-a} (\Omega^{a}_{X'/S} (A', B') \otimes
{\cal H}^{p^{\vee}}) \>>> \]
\[ \hspace{6cm} \>>> H^{p+1} ({\cal H}^{0} \otimes {\cal H}^{p^{\vee}})
\]
where ${\cal H}^{p^{\vee}} : = {\cal H}om_{{\cal O}_{X'}} ({\cal H}^{p},
{\cal O}_{X'})$.\\
\ \\
As $\dim_{S} X = p$ and $S$ is affine, one has $H^{p+1} ({\cal H}^{0}
\otimes {\cal H}^{p^{\vee}}) = 0$, and therefore
\[ \bigwedge^{p} C \in H^{0} (\Omega^{p}_{X'/S} (A',B') \otimes
{\cal H}^{p^{\vee}}) \]
lifts to some
\[ \Gamma \in \h^{p} (F_{*} \Omega^{\bullet}_{X/S} (A,B) \otimes
{\cal H}^{p^{\vee}}). \]
Representing $\Gamma$ by a \v{C}ech cocycle
\[ {\cal H}^{p} \> [ \Gamma ] >> {\cal C}^{p} (F_{*} {\cal O}_{X}) +
\cdots + {\cal C}^{0} (F_{*} \Omega^{p}_{X/S} (A,B)), \]
and taking a common refinement ${\cal U}$ of the \v{C}ech covers defining
$\varphi$ and $\Gamma$, one obtains altogether a quasi-isomorphism
\[ (\varphi, [\Gamma ] \circ \bigwedge^{p} C^{-1}) : \bigoplus_{j}
\Omega^{j}_{X'/S} (A', B') [-j] \>>> {\cal C}^{\bullet} (F_{*}
\Omega^{\bullet}_{X/S} (A,B)). \]
\end{Beweis}
\begin{defn}{Remark}
If $\dim_{S} X = n > p$, and $S$ is affine, one considers the
exact sequence
\[ 0 \>>> \tau_{\leq n - p} F_{*} \Omega^{\bullet}_{X/S}
(A,B) \>>> F_{*} \Omega^{\bullet}_{X/S} (A,B) \>>>
\tau_{\geq n - p + 1} \Omega^{\bullet}_{X/S} (A,B) \>>> 0 \]
giving the short exact sequences
\[
\h^{q} (F_{*} \Omega^{\bullet}_{X/S} (A,B) \otimes {\cal H}^{q^{\vee}})
\>>> \bigoplus^{n}_{a=n - p + 1} \ H^{q - a} (\Omega^{a}_{X'/S}
(A', B') \otimes {\cal H}^{q^{\vee}}) \>>>  \]
\[ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\>>> H^{q+1} (\tau_{\leq n - p} F_{*} \Omega^{\bullet}_{X/S} (A,B)
\otimes {\cal H}^{q^{\vee}}) \]
for all $ p \leq q \leq n$.
If $n - p \leq p -1 $, then
\[
H^{q+1} (\tau_{\leq n -p} F_{*} \Omega^{\bullet}_{X/S} (A,B) \otimes
{\cal H}^{q^{\vee}}) = \bigoplus^{n-p}_{a = 0} H^{q+1 - a}
(\Omega^{a}_{X'/S} (A',B') \otimes {\cal H}^{q^{\vee}}). \]
If those groups are vanishing, then the same argument as above shows
that one obtains a splitting of $F_{*} \Omega^{\bullet}_{X/S}
(A,B)$. \\
\ \\
For example, take $D = \emptyset$.
For $n = p + 1$, one requires the vanishing of
\[ H^{p+1} ({\cal H}^{p^{\vee}}), \ H^{p} (\Omega^{1}_{X'/S} \otimes
{\cal H}^{p^{\vee}})
\mbox{ \ and of \ }
H^{p+1} (\Omega^{1}_{X'/S} \otimes {\cal H}^{p +1^{\vee}}), \]
that is, via duality, the vanishing of
\[ H^{0} (\Omega^{p}_{X'/S} \otimes \Omega^{p+1}_{X'/S}) \mbox{ \ and \ }
\ H^{1} (\Omega^{p \otimes 2}_{X'/S}). \]
\end{defn}

Using (10.19) it is now quite easy to prove theorem (8.3) and some
generalizations.
\begin{satz}{Theorem} Let $f: X \rightarrow S$ be a smooth proper
$S$-scheme, $\dim_{S} X \leq p$, and let $D \subset X$ be a $S$-normal
crossing divisor. Assume that there exists a lifting
$$\tilde{D}' \subset \tilde{X}' \mbox{ \  of \ } D' \subset X'$$
to $\tilde{S}$. Let $D = A + B$. Then one
has:\\
\ \\
{\rm a)} The ${\cal O}_{S}$-sheaves $$E^{ab}_{1} = R^{b} f_{*} \Omega^{a}_{X/S}
(A,B)$$ are locally free and compatible with arbitrary base change.\\
\ \\
{\rm b)} The Hodge to de Rham spectral sequence
\[ E^{ab}_{1} \Longrightarrow \R^{a+b} f_{*} \Omega^{\bullet}_{X/S} (A,B) \]
degenerates in $E_{1}$.\index{Degeneration!of the Hodge to de
Rham spectral sequence}
\end{satz}
\begin{Beweis}{Proof} Assuming a), part b) follows if one knows
that $\R^{l} f_{*} \Omega^{\bullet}_{X/S} (A,B)$ is a locally free
${\cal O}_{S}$-module of rank $$\sum_{a+b = l} \ \mbox{rank}_{{\cal O}_{S}}
(E^{ab}_{1}).$$ Hence for a) and b) we can assume $S$ to be affine.\\
\ \\
(10.17), for $ i = \dim_{S} X < p$, or (10.19) for
$\dim_{S} X = p$ imply that
\[ \R^{l} f_{*} \Omega^{\bullet}_{X/S} (A,B) = \R^{l} f'_{*} (F_{*}
\Omega^{\bullet}_{X/S} (A,B)) = \bigoplus_{a} R^{l-a} f'_{*} \Omega^{a}_{X'/S}
(A',B'). \]
Hence, if a) holds true for $f: X \rightarrow S$, then
\[ \R^{l} f_{*} \Omega^{\bullet}_{X/S} (A,B) = \bigoplus_{a} F_{S}^{*}
R^{l-a} f_{*} \Omega^{a}_{X/S} (A,B) \]
is locally free of the right rank and one obtains b). \\
\ \\
By ``cohomology and base change'' (\cite{Mum70}, I$\!$I \S 5, for example), there
exist for $a = 0, \ldots, l$ bounded complexes ${\cal E}^\bullet_{a}$ of vector bundles
on $S$, such that
\[  {\cal H}^{l} ({\cal E}^{\bullet}_{a}) =
R^{l-a} f_{*} \Omega^{a}_{X/S} (A,B)  \]
and, for any affine map $\varphi : T \>>> S$,
\[ {\cal H}^{l} (\varphi^{*} {\cal E}_{a}^{\bullet}) = R^{l-a} f_{T^{*}}
\Omega^{a}_{X_{T}/T} (A_{T}, B_{T}) \]
where $X_{T}, \ A_{T}, \ B_{T}, \ X'_T, \ f_{T} : X_{T} \rightarrow T$ and
$f'_T : X'_T \rightarrow T$
are obtained by pullback from $X, \ A, \ B, \ X', \ f$ and
$f': X' \rightarrow S$.\\
\ \\
For example, $R^{l-a} f'_{*} \Omega^{a}_{X'/S} (A',B')$ is given by
${\cal H}^{l} (F^{*}_{S} {\cal E}^{\bullet}_{a})$ and hence\\
$\R^{l} f_{*} \Omega^{\bullet}_{X/S} (A,B)$ by the $l$-th homology
of the complex
$$F_{S}^{*} {\cal E}^{\bullet} \mbox{ \ for \ } {\cal E}^{\bullet} =
\bigoplus_{a} {\cal E}^{\bullet}_{a}.$$
To prove a), we have to show that for all $l$, the sheaf
$${\cal H}^{l} ({\cal E}^{\bullet}) =
\bigoplus_{a} \ {\cal H}^{l} ({\cal E}^{\bullet}_{a})$$
is locally free. If this is wrong, then we take $l_{0}$ to be the maximal $l$
with ${\cal H}^{l} ({\cal E^\bullet})$ not locally free. Hence, if
$\partial_{\bullet}$ denotes the differential in ${\cal E}^{\bullet}$,
$\ker \ \partial_{l_{0}}$ is a vector bundle, let us say of rank $r$,
but the image of $$\partial_{l_{0} - 1} : {\cal E}^{l_{0} - 1} \>>>
\ker \ \partial_{l_{0}}$$ is not a subbundle. \\
\ \\
For some closed point $s \in S$ one finds an infinitesimal neighbourhood
$\hat{S}$, for example one of the form
$$ \hat{S} = \Spec({\cal O}_{S,s} / m_{S,s}^\mu) \mbox{ \ for \ } \mu >> 0 ,$$
such that $\im ( \partial_{l_0 - 1}|_{\hat{S}} )$ is not a subbundle of
$\ker ( \partial_{l_0}|_{\hat{S}} )$. Let us write $\hat{S} = \Spec R$
where $R$ is an Artin ring. For any $R$-module $M$ let $\mbox{lg}(M)$ denote
the lenght.\\
\ \\
$\partial_{l_{0} - 1}$ is represented by a matrix $\Delta_{l_{0} - 1}$.
For the point $s \in \hat{S}$ one has
\[ h = \mbox{rank}_{k(s)} {\cal H}^{l_{0}} ({\cal E}^{\bullet} \otimes k(s))
> \frac{\mbox{lg}({\cal H}^{l_{0}} ({\cal E}^{\bullet}
\otimes R))}{\mbox{lg}(R)}\ . \]
Let $n_0 \subset R$ be the ideal generated by the $(r - h + 1)$
minors of $\Delta_{l_{0} - 1}$. Then
$${\cal H}^{l_{0}}({\cal E}^{\bullet} \otimes
R/n_0)$$ is free of rank $h$ as an $R/n_0$-module.
If $n'_0 \subset n_0 \subset R$ is another ideal with $n'_0 \neq n_0$, then
\[ \mbox{lg}({\cal H}^{l_{0}} ({\cal E}^{\bullet} \otimes R/n'_0)) <
h \cdot \mbox{lg}(R/n'_0). \]
Repeating this construction, starting with $R/n_0$
instead of $R$ we find after finitly many steps some ideal $n$ such that
the $R/n$ modules $${\cal H}^{l}({\cal E}^{\bullet}
\otimes R/n)$$ are free of rank $h(l)$ over $R/n$ for all $l$,
but for some $l'$ and for all ideals $n' \subset n$
with $n' \neq n$ one has
$$\mbox{lg}({\cal H}^{l'}({\cal E}^{\bullet} \otimes R/n')) < h(l') \cdot
\mbox{lg}(R/n') \ .$$
In particular, this holds true for the ideal
$n'$ of $R$ generated by $F_{\hat{S}}^{*}n$. Let us write $T'
= \Spec (R/ n')$ and $T = \Spec (R/n)$. We have affine morphisms
\[
\begin{CD}
T' \> \delta >> T \> j >> T'\\
\V V V \V V V \V V V \\
S \> F_S >> S \> = >> S\ .
\end{CD}
\]
${\cal H}^{l} ({\cal E}^{\bullet}|_{T})$ is free of rank $h(l)$ over $R / n$
for all $l$ and hence
\[ {\cal H}^{l} (\delta^{*} j^{*} {\cal E}^{\bullet}|_{T'}) = {\cal H}^{l}
((F_{S}^{*} {\cal E}^{\bullet})|_{T'}) \]
is free of rank $h(l)$ over $R/n'$.  The Hodge to de Rham spectral sequence
implies that
\[
\mbox{lg}({\cal H}^{l} ({\cal E}^{\bullet}|_{T'})) =
\sum_{a} \mbox{lg}(R^{l - a} f_{T'*} \Omega^{a}_{X_{T'}/T'}
(A_{T'},B_{T'})) \geq
\]
\[ \hspace{1.8cm} \geq \mbox{lg}(\R^{l} f_{T'*} \Omega^{\bullet}_{X_{T'}
/T'} (A_{T'},B_{T'})) .
\]
On the other hand,
\[
\R^{l} f_{T'*} \Omega^{\bullet}_{X_{T'}/T'} (A_{T'},B_{T'}) =
\R^{l} f'_{*} (F_* \Omega^{\bullet}_{X/S} (A,B) \otimes f^{\prime *}
(R/n')) =
\]
\[\hspace{3cm}
= \bigoplus_{a} R^{l-a} f'_{*} (\Omega^{a}_{X'/S} (A',B') \otimes
f^{\prime *} (R/n')) .
\]
We can apply base change and find the latter to be
\[
\bigoplus_a {\cal H}^l (\delta^* {\cal E}^\bullet_a |_T)
= {\cal H}^l (\delta^* {\cal E}^\bullet |_T) \ .
\]
Since the sheaves ${\cal H}^{l} ({\cal E}^{\bullet}|_{T})$ are locally free,
we have altogether
\[
\mbox{lg}({\cal H}^l ({\cal E}^\bullet |_{T'})) \geq
\mbox{lg}({\cal H}^l (\delta^* {\cal E}^\bullet |_{T})) =
\]
\[
= \mbox{lg}( \delta^* {\cal H}^l ({\cal E}^\bullet |_{T})) =
h(l) \cdot \mbox{lg}(R / n') \ .
\]
For $l=l'$ this contradicts the choice of $n$ and $n'$.
Hence ${\cal H}^{l} ({\cal E}^{\bullet})$ is locally free.\\
\end{Beweis}
\begin{defn}{Remark} As shown in \cite{DI87}, 4.1.2. it is enough
to assume that for all $a$ and $b$ the dimension of $H^{b} (X_{s},
\Omega^{a}_{X_{s}} (A_{s}, B_{s}))$ is finite for closed points $s \in S$
and that the conjugate spectral sequence
\[ {}_{c}E^{ij}_{2} = R^{i} f'_{T*} {\cal H}^{j} (F_{*}
\Omega^{\bullet}_{X_T/T}
(A_T,B_T)) \Longrightarrow \R^{i+j} f_{T*} \Omega^{\bullet}_{X_T/T} (A_T,B_T)
\]
satisfies $${}_{c}E^{ij}_{2} = {}_{c}E^{ij}_{\infty} \mbox{ \ for \ } i + j = l$$
and for all T, in order to obtain (10.21).
\end{defn}
\begin{satz}{Corollary} Let $K$ be a field of characteristic zero, $X$ be
a smooth proper scheme over $K$ and $D = A + B$ be a reduced normal crossing
divisor defined over $K$. Then the Hodge to de Rham spectral sequence
\[ E^{ab}_{1} = H^{b} (X, \Omega^{a}_{X} (A,B)) \Longrightarrow \h^{a+b}
(X, \Omega^{\bullet}_{X} (A,B)) \]
degenerates in $E_{1}$.
\end{satz}
\begin{Beweis}{Proof} By flat base change we may assume that $K$ is
of finite type over $\Q$. Hence we find a ring $R$, of finite type over $\Z$,
such that $K$ is the quotient field over $R$. Let $f: {\cal X} \rightarrow
\mbox{Spec} R$ be a proper morphism with  $X = {\cal X} \times_{R} K$,
and ${\cal A}$ and ${\cal B}$ divisors on ${\cal X}$ with $A =
{\cal A}|_{X}$ and $B = {\cal B}|_{X}$.\\
Replacing $\Spec R$ by some open affine subscheme, we may assume that
$f$ is smooth, that ${\cal D} = {\cal A} + {\cal B}$ \ is a normal crossing
divisor over $\Spec R$ and that
$$\R^{l} f_{*} \Omega^{\bullet}_{{\cal X}/\mbox{Spec} R}
({\cal A}, {\cal B}) \mbox{ \ and \ }
R^{b} f_{*} \Omega^{a}_{{\cal X}/\mbox{Spec}\ R} ({\cal A}, {\cal B})$$
are locally free for all $l, a$ and $b$.
Take a closed point $s \in \Spec R$ with
$$\Char \ k(s) = p > \dim_{S} X.$$ Hence
$$X_{s} = {\cal X} \times_{R} k(s) \>>> S = k(s)\ , \ \ A_{s} =
{\cal A}|_{X_{s}} \mbox{ \ and \ } B_{s}= {\cal B}|_{X_{s}}$$
satisfy the assumptions made in (10.21) and
\[ \sum_{a+b=l} \ \mbox{rank} \ R^{b} f_{*}
\Omega^{a}_{{\cal X}/\mbox{Spec} \ R} ({\cal A}, {\cal B}) =
\mbox{rank} \ \R^{l} f_{*} \Omega^{\bullet}_{{\cal X}/\mbox{Spec} \ R}
({\cal A}, {\cal B}). \]
\end{Beweis}

As mentioned in (3.18) the corollary (10.23) finally ends the proof of (3.2) in
characteristic zero, and hence of the different vanishing theorems and
applications discussed in Lectures 5 - 7. A slightly different
argument, avoiding the use of (3.19) or (3.22)
can be found at the end of this lecture.\\
\ \\
As promised we are now able
to prove (3.2,b and c) for fields of characteristic $p \neq 0$ as well.
\begin{Beweis}{Proof of (3.2,b) and c) in characteristic $p \neq 0$}
Recall, that on the projective manifold $X$ we considered the invertible sheaves
$${\cal L}^{(i)} = {\cal L}^{(i, D)} = {\cal L}^{i} (- [\frac{i \cdot D}{N}])
 \mbox{ \ where }$$
$$D = \sum^{r}_{j = 1} \ \alpha_{i} D_{j}$$ is a normal crossing divisor and
${\cal L}$ an invertible sheaf with ${\cal L}^{N} = {\cal O}_{X} (D)$.
For $N$ prime to $\Char \ k$, we
constructed in \S 3, for $i = 0, \ldots, N-1$, integrable logarithmic
connections
\[ \nabla_{(i)} : {\cal L}^{(i)^{-1}} \>>> \Omega^{1}_{X}
(\log \ D^{(i)}) \otimes {\cal L}^{(i)^{-1}} \]
with poles along
\[ D^{(i)} = \sum^{r}_{\scriptstyle j=1 \atop \scriptstyle
\frac{i \cdot \alpha_{j}}{N} \not\in \Z} \ D_{j}.\]
The residue of $\nabla_{(i)}$ along $D_{j} \subset D^{(i)}$ is given
by multiplication with
\[(i \cdot \alpha_{j} - N \cdot [ \frac{i \cdot
\alpha_{j}}{N} ] ) \cdot N^{-1}.\]
If $A$ and $B$ are reduced divisors
such that $A,B$ and $D^{(i)}$ have pairwise no common component then we
want to prove:\\
\ \\
\end{Beweis}
\begin{defn}{Claim} Let $k$ be a perfect field, $N$ prime to $\Char  \ k = p$
and assume that $p \geq \dim \ X$. If $X,D,A$ and $B$ admit a lifting to
$W_{2} (k)$, then the spectral sequence
\[ E_{1}^{ab} = H^{b} (X, \Omega^{a}_{X} (\log \ (A + B + D^{(i)})) (-B)
\otimes {\cal L}^{(i)^{-1}}) \Longrightarrow \hspace{2cm} \]
\[ \hspace{2cm} \h^{a+b} (X, \Omega^{\bullet}_{X} (\log \ (A + B + D^{(i)})) ( - B) \otimes
{\cal L}^{(i)^{-1}}) \]
degenerates in $E_{1}$.
\end{defn}
\begin{Beweis}{Proof} Let $F: X \rightarrow X'$ be the relative Frobenius
morphism and ${\cal L}', D', A'$ and $B'$ be the sheaf and the divisors on $X'$
obtained by field extensions. Then
\[ F^{*} {\cal L}^{\prime (i, D')} = F^{*} {\cal L}^{\prime (i)} = {\cal L}^{p
\cdot i} (-p \cdot [ \frac{i \cdot D}{N} ] ) \]
contains
$${\cal L}^{(p \cdot i)} = {\cal L}^{p \cdot i} ( - [ \frac{p \cdot i \cdot
D}{N} ] ).$$
Since $p$ is prime to $N$ one has $D^{(i)} = D^{(p \cdot i)}$. The
connection
\[ \nabla_{(p \cdot i)^{-1}} : {\cal O}_X (-B) \otimes
{\cal L}^{(p \cdot i)^{-1}}  \>>> \Omega^{1}_{X} (\log \ (A + B + D^{(i)}))(-B) \otimes
{\cal L}^{(p \cdot i)^{-1}} \]
induces a connection with the same poles on
$${\cal O}_X(-B) \otimes F^{*} {\cal L}^{\prime (i)^{-1}} $$ whose residues
along $D_{j} \subset D^{(i)}$ are given by multiplication with
\[ N^{-1} \cdot p \cdot i \cdot \alpha_{j} -
[ \frac{p \cdot i \cdot \alpha_{j}}{N} ] + ( [ \frac{p \cdot i \cdot
\alpha_{j}}{N} ] - p [ \frac{i \cdot \alpha_{j}}{N} ] ) \ . \]
Obviously this number is zero modulo $p$. Since
\[ [ \frac{p \cdot i \cdot \alpha_{j}}{N} ] \leq \frac{p \cdot i \cdot
\alpha_{j}}{N} < p \cdot ( [ \frac{i \cdot \alpha_{j}}{N} ] + 1) =
p \cdot [ \frac{i \cdot \alpha_{j}}{N} ] + p \]
(2.10) implies that the complexes
\[ \Omega^{\bullet}_{X} \ (\log \ (A + B + D^{(i)}))( - B) \otimes
F^{*} {\cal L}^{\prime (i)^{-1}} \]
and
\[ \Omega^{\bullet}_{X} (\log \ (A + B + D^{(i)})) (-B) \otimes
{\cal L}^{(p \cdot i)^{-1}} \]
are quasi-isomorphic.
\begin{satz}{Claim} The complex
\[ F_{*} (\Omega^{\bullet}_{X} (\log \ (A + B + D^{(i)})) (-B) \otimes
F^{*} {\cal L}^{\prime (i)^{-1}}) \]
is isomorphic to the complex
$$F_{*} (\Omega^{\bullet}_{X} (\log \ (A + B + D^{(i)})) (-B))$$ tensorized with
${\cal L}^{\prime (i)^{-1}}$.
\end{satz}
\begin{Beweis}{Proof} Let $\pi : Y \rightarrow X$ be the cyclic cover
obtained by taking the $N$-th root out of $D$, let $F: Y \rightarrow Y'$
be the relative Frobenius of $Y$. We have the induced diagram
\[
\begin{CD}
Y \> F >> Y' \\
\V \pi V V  \V V \pi ' V \\
X \> F >> X'
\end{CD}
\]
and, on $X' - \Sing (D_{red})$,\\
\ \\
$\pi '_{*} {\cal O}_{Y'} = Ker (d: \pi '_{*} F_{*} {\cal O}_{Y} \>>>
\pi ' _{*} F_{*} \Omega^{1}_{Y}) = $
\[Ker (d: F_{*} ( \bigoplus^{N-1}_{j=0} {\cal L}^{(j)^{-1}}) \>>>
F_{*} (\bigoplus^{N-1}_{j=0} \Omega^{1}_{X} (\log \ D^{(j)}) \otimes
{\cal L}^{(j)^{-1}})).
\]
Since $F^{*}$ of the $i$-th eigenspace ${\cal L}'^{ (i)}$ of $\pi '_{*}
{\cal O}_{Y'}$ lies in the $p \cdot i$-th eigenspace ${\cal L}^{(i \cdot p)}$
of $\pi_{*} {\cal O}_{Y}$ one has
\[
\begin{array}{ll}
{\cal L}^{\prime (i)^{-1}} & = Ker (\nabla_{(p \cdot i)} : F_{*} {\cal L}^{(p
\cdot i)^{-1}} \>>> F_{*} (\Omega^{1}_{X} (\log \ D^{(i)}) \otimes
{\cal L}^{(p \cdot i)^{-1}}))\\
\ \\
& = Ker (\nabla_{(p \cdot i)} : (F_{*} {\cal O}_{X}) \otimes {\cal L}^{\prime
(i)^{-1}} \>>> (F_{*} \Omega^{1}_{X} (\log \ D^{(i)})) \otimes
{\cal L}^{\prime (i)^{-1}})
\end{array}
\]
By the Leibniz rule $\nabla_{(p \cdot i)}$ restricted to $(F_{*} {\cal O}_{X})
\otimes {\cal L}^{\prime (i)^{-1}}$ is nothing but $d \otimes id$ as claimed.\\
\ \\
\end{Beweis}
From (10.19) and by base change
\[
\dim \h^{l} (X, \Omega^{\bullet}_{X} (\log \ (A + B + D^{(i)})) (- B)
\otimes {\cal L}^{(p \cdot i)^{-1}}) =
\]
\[
\dim \h^{l} (X', F_* (\Omega^{\bullet}_{X} (\log \ (A + B + D^{(i)})) (- B)
\otimes {\cal L}^{(p \cdot i)^{-1}})) =
\]
\[
\sum_{a + b = l} \ \dim H^{b} (X', \omega^{a}_{X'} (\log \ (A' + B' + D'^{(i)}))
(-B') \otimes {\cal L'}^{(i)^{-1}} =
\]
\[
\sum_{a + b = l} \ \dim H^{b} (X, \omega^{a}_{X} (\log \ (A + B + D^{(i)}))
(-B) \otimes {\cal L}^{(i)^{-1}} \geq
\]
\[
\dim \h^{l} (X, \Omega^{\bullet}_{X} (\log \ (A + B + D^{(i)})) (-B)
\otimes {\cal L}^{(i)^{-1}}).
\]
For some $\nu > 0$ one has $p^{\nu} \equiv 1 \ mod \ N$. Repeating the argument
$\nu - 1$ times one finds
\[
\dim \h^{l} (X, \Omega^{\bullet}_{X} (\log \ (A + B + D^{(i)})) ( - B)
\otimes {\cal L}^{(i)^{-1}}) \geq
\]
\[
\hspace{1.2cm}
\dim \h^{l} (X, \Omega^{\bullet}_{X} (\log \ (A + B + D^{(i)})) ( - B)
\otimes {\cal L}^{(p^{\nu - 1} \cdot i)^{-1}}) \geq \cdots
\]
\[
\hspace{1cm}
\cdots \geq \sum_{a + b = l} \ \dim H^{b} (X, \Omega^{a}_{X} (\log \ (A + B + D^{(i)}))
(-B) \otimes {\cal L}^{(i)^{-1}}) \geq
\]
\[
\dim \h^{l} (X, \Omega_{X}^{\bullet} (\log \ (A + B + D^{(i)})) (-B)
\otimes {\cal L}^{(i)^{-1}}) \ . \hspace{0.4cm}
\]
Hence all the inequalities must be equalities and one obtains (10.24).\\
\ \\
\end{Beweis}
\begin{Beweis}{2. Proof of (3.2,b) and c) in characteristic $0$} \\
The arguments used in (10.23) to reduce (10.23) to (10.21) show as well
that 3.2,b and c in characteristic $0$ follow from (10.24).\\
\end{Beweis}
\Paragraph{Vanishing theorems in characteristic $p$.}
\ \\
In this lecture we start with the elegant proof of the Akizuki-Kodaira-Nakano
vanishing theorem, due to Deligne, Illusie and Raynaud \cite{DI87}.
Then we will discuss some generalizations. However they only seem to be of
interest if one assumes that one has
embedded resolutions of singularities in characteristic $p$.
\ \\
\begin{satz}{Lemma} Let $k$ be a perfect field, let $X$ be a proper smooth
$k$-scheme and $D$ a normal crossing divisor, both
admitting a lifting $\tilde{D} \subset \tilde{X}$ to $W_{2} (k)$. Let ${\cal M}$ be a locally
free ${\cal O}_{X}$-module. Then, for
$l <$  \Char $(k)$ one has
\[ \sum_{a+b=l} \ \dim \ H^{b} (X, \Omega^{a}_{X} (\log \ D)
\otimes {\cal M}) \leq
\sum_{a + b = l} \ \dim \ H^{b} (X, \Omega^{a}_{X} (\log \ D)
\otimes F_{X}^{*} {\cal M}). \]
\end{satz}
\begin{Beweis}{Proof} By (10.16) we have
\[ \dim \ \h^{l} (X, \Omega^{\bullet}_{X} (\log \ D)
\otimes F^{*}_{X} {\cal M})=
\sum_{a+b=l} \dim \ H^{b} (X', \Omega^{a}_{X'} (\log \ D') \otimes {\cal M}') \]
for the sheaf ${\cal M}' = pr_{1}{}^{*} {\cal M}$ on $X' = X \times_{F_S} S$
and $D' = D \times_{F_S} S$.
By base change the right hand side is
$$\sum_{a+b=l} \dim \ H^{b} (X, \Omega^{a}_{X} (\log \ D) \otimes {\cal M}).$$
The Hodge to de Rham spectral sequence implies that the left hand side
is smaller than or equal to
\[ \sum_{a+b=l} \ \dim \ H^{b} (X, \Omega^{a}_{X} (\log \ D)
\otimes F^{*}_{X} {\cal M}). \]
\end{Beweis}
\begin{satz}{Corollary} Under the assumptions of (11.1) assume that ${\cal M}$
is invertible. Then
\[ \sum_{a+b=l} \ \dim \ H^{b} (X, \Omega^{a}_{X} (\log \ D)
\otimes {\cal M}) \leq
\sum_{a+b=l} \ \dim \ H^{b} (X, \Omega^{a}_{X} (\log \ D)
\otimes {\cal M}^{p}). \]
\end{satz}
\begin{satz}{Corollary (Deligne, Illusie, Raynaud, see
\cite{DI87})} \index{Vanishing theorem!Deligne Illusie Raynaud}\\
For $a+b < \mbox{{\rm Min}} \ \ \{ \Char (k), \ \dim \ X \}$
and ${\cal L}$ ample and invertible, one has under the assumptions of (11.1.)
\[ H^{b} (X, \Omega^{a}_{X} (\log \ D) \otimes {\cal L}^{-1}) = 0. \]
\end{satz}
\begin{Beweis}{Proof} For $\nu$ large enough, and ${\cal L}^{-1} = {\cal M}$ one
has
\[ H^{b} (X, \Omega^{a}_{X} (\log \ D) \otimes {\cal M}^{p^{\nu}}) = 0 \]
for $b < \dim  \ X$.
By (11.2) one has
$$H^{b}(X, \Omega^{a}_{X} (\log \ D)
\otimes {\cal M}^{p^{\nu-1}}) = 0$$ and
after finitely many steps one obtains (11.3)\\
\end{Beweis}
The following corollary is, as well known in characteristic zero,
a direct application of (11.3) for $a=0$. It will be needed
in our discussion of possible generalizations of (11.3).
\begin{satz}{Corollary} Let $k$ be a perfect field, let $X$ be
a proper smooth $k$-scheme with $\dim X \leq \Char \ k$ and let ${\cal L}$
be a numerically effective sheaf (see (5.5)). Assume that ${\cal A}$ is a very
ample sheaf, such that $X$ and ${\cal A}$ lift to $\tilde{X}$ and
$\tilde{\cal A}$ over $W_{2} (k)$, with $\tilde{\cal A}$ very ample over
$\tilde{S}$. Then one has:
\begin{quote}
{\rm a)} ${\cal A}^{\dim X+1} \otimes {\cal L} \otimes \omega_{X}$ is
generated by global sections.\\
\ \\
{\rm b)} ${\cal A}^{\dim X+2} \otimes {\cal L} \otimes \omega_{X}$ is
very ample.
\end{quote}
\end{satz}
\begin{Beweis}{Proof} By (11.3) and Serre duality $H^{1} (X, {\cal A}^{\dim
X} \otimes {\cal L} \otimes \omega_{X}) = 0$. Hence one has a surjection
\[ H^{0} (X, {\cal A}^{\dim X + 1} \otimes {\cal L} \otimes \omega_{X} )
\>>> H^{0} (H, {\cal A}^{\dim X} \otimes {\cal L} \otimes
\omega_{H}) \]
where $H$ is a smooth zero divisor of a general section of ${\cal A}$.
Since $\cal A$ lifts to $W_2 (k)$, we can choose $H$ such that it lifts to
$W_2 (k)$ as well. By
induction on $\dim  \ X$ we can assume that
$${\cal A}^{\dim X} \otimes {\cal L} \otimes \omega_{H}$$
is generated by global sections and, moving $H$, we obtain a).\\
Part b) follows directly from a) (see \cite{Har77}, II. Ex.7.5).\\
\end{Beweis}
\begin{satz}{Proposition} Let $k$ be a perfect field of characteristic
$p > 0$, let $X$ be a proper smooth $k$-variety, let $D$ be an effective
normal crossing divisor and ${\cal L}$ be an invertible sheaf on $X$.
Assume that $(X,D)$ and ${\cal L}$ admit liftings to $W_{2} (k)$ and that
one has:\\
\ \\
$(\ast)$ For some $\nu_{0} \in \N$ and all $\nu \geq 0$ the sheaf
${\cal L}^{\nu_{0} + \nu} \otimes {\cal O}_{X} (-D)$ is ample.\\
\ \\
Then, for $a + b < \dim \ X \leq \Char \ k $ one has
\[ H^{b} (X, \Omega^{a}_{X} (\log \ D) \otimes {\cal L}^{-1} ) = 0. \]
\end{satz}
\begin{Beweis}{Proof} By (5.7), the assumption $(\ast)$ implies that ${\cal L}$
is numerically effective. Let us choose $\mu_{0}$ such that ${\cal L}^{\mu_{0}
\cdot \nu_{0}} (- \mu_{0} \cdot D)$ is very ample and
$${\cal L}^{\mu_{0} \cdot \nu_{0}} ( - \mu_{0} \cdot D) \otimes \omega_{X}^{-1}
\mbox{ \ and  \ } {\cal L}^{\mu_{0}
\cdot \nu_{0}} (- \mu_{0} \cdot D + D_{red}) \otimes \omega^{-1}_{X}$$
are both ample. From (11.4) we find for $n = \dim X$ that both,
\[
{\cal L}^{\mu_{0} \cdot \nu_{0} (n+3) + \nu} (- \mu_{0} (n+3) \cdot D)
\mbox{ \ and \ } {\cal L}^{\mu_{0} \cdot \nu_{0} (n+3) + \nu}
( - \mu_{0} (n+3) \cdot D + D_{red})
\]
are very ample for all $\nu \geq 0$.
Hence the assumption $(\ast)$ in (11.5) can be replaced by\\
\ \\
$(\ast \ast)$ {\it For some $\nu_{0} \in \N$ and all $\nu \geq 0$ the sheaves
${\cal L}^{\nu_{0} + \nu} \otimes {\cal O}_{X} (-D)$ and ${\cal L}^{\nu_{0} +
\nu} \otimes {\cal O}_{X} (- D + D_{red})$ are very ample.}\\
\ \\
Choose $\eta \in \N - \{ 0 \}$ such that $N = p^{\eta} + 1 > \nu_{0}$ and
$ [ \frac{D}{N} ] = 0$.
Let $H$ be the zero set of a general section of ${\cal L}^{N} \otimes
{\cal O}_{X} (-D)$. In \S 3 we constructed an integrable logarithmic
connection $\nabla_{(i)}$ on the sheaf
$${\cal L}^{(i)^{-1}} = {\cal L}^{-i} ( [
\frac{i \cdot (D+H)}{N} ] ) \ .$$
Let $F: X \rightarrow X'$ be the relative Frobenius
morphism and ${\cal L}', D', H'$ the sheaf and the divisors on $X'$, obtained
by field extension via $ F_{\Spec \ k}: k \rightarrow k$
from ${\cal L}, D$ and $H$.\\
\ \\
As we have seen in the proof of (10.24) and in (10.25) one has an inclusion
of complexes
\[ (F_{*} \Omega^{\bullet}_{X} (\log \ (D+H))) \otimes {\cal L}'^{(i)^{-1}}
\>>> F_{*} (\Omega^{\bullet}_{X} (\log \ (D +H)) \otimes {\cal L}^{(p
\cdot i)^{-1}}). \]
This inclusion is a quasi-isomorphism and as in (10.24) one obtains from
(10.19) and base change the inequalities
\[
\dim \h^{l} (X, \Omega^{\bullet}_{X} (\log \ (D +H)) \otimes
{\cal L}^{(1)^{-1}}) \leq
\]
\[
\sum_{a+b=l} \dim H^{b} (X, \Omega^{a}_{X} (\log \ (D+H)) \otimes
{\cal L}^{(1)^{-1}}) =
\]
\[
\sum_{a+b=l} \dim H^{b} (X', \Omega^{a}_{X'} (\log \ (D'+H')) \otimes
{\cal L'}^{(1)^{-1}}) =
\]
\[
\dim \h^{l} (X, \Omega^{\bullet}_{X} (\log \ (D+H)) \otimes
{\cal L}^{(p)^{-1}}) \leq \cdots
\]
\[
\dim \h^{l} (X, \Omega^{\bullet}_{X} (\log \ (D+H)) \otimes
{\cal L}^{(p^{\gamma})^{-1}}) \leq
\]
\[
\sum_{a+b=l} H^{b} (X, \Omega^{a}_{X} (\log \ (D+H)) \otimes
{\cal L}^{(p^{\gamma})^{-1}})
\]
for all $\gamma > 0$.
For $\gamma = \eta$ we have $p^{\eta} = N - 1$ and
\[
\begin{array}{l}
{\cal L}^{(p^{\eta})} = {\cal L}^{(N - 1)} = {\cal L}^{N-1}
( - [ \frac{(N-1) \cdot (D + H)}{N} ] ) = \\
\ \\
= {\cal L}^{N-1} ( - D - [ \frac{-D}{N} + \frac{(N-1) \cdot H}{N}
]  ) = {\cal L}^{N-1} ( - D + D_{red}).
\end{array}
\]
Hence ${\cal L}^{(p^{\eta})}$ is ample and from (11.3) we obtain, for
$$l < \dim  \ X \leq \Char \ k ,$$ that
\[ \sum_{a+b=l} \ H^{b} (X, \Omega_{X}^{a} (\log \ (D+H)) \otimes
{\cal L}^{(p^{\eta})^{-1}}) = 0 . \]
Since ${\cal L}^{(1)} = {\cal L}$ we obtain for
$a+b < \dim  \ X \leq \Char \ k $
\[ H^{b} (X, \Omega^{a}_{X} (\log \ (D+H)) \otimes {\cal L}^{-1} ) = 0. \]
Finally, since ${\cal L}^N \otimes {\cal O}_X (-D)$ lifts to
$W_2 (k)$, we can choose $H$ such that $H$ and $D|_H$ both lift to
$W_2 (k)$ and the exact sequence
\[ H^{b-1} (H, \Omega^{a-1}_{H} (\log \ D)|_{H}) \otimes {\cal L}^{-1})
\>>> H^{b} (X, \Omega^{a}_{X} (\log \ D) \otimes {\cal L}^{-1})
\>>> \]
\[ \hspace{4cm} \>>> H^{b} (X, \Omega^{a}_{X} (\log \ (D+H)) \otimes
{\cal L}^{-1}). \]
allows to prove (11.5) by induction.\\
\end{Beweis}
\begin{defn}{Remarks.} \\
a) By (5.7) and (5.4,d) the assumption $(\ast)$ in (11.5) implies that ${\cal L}$
is numerically effective and of maximal Iitaka dimension.\\
\ \\
b) If, on the other hand, ${\cal L}$ is numerically effective and of
maximal Iitaka dimension, then there exists some effective divisor $D$
such that the sheaf
$${\cal L}^{\nu_{0} + \nu} \otimes {\cal O}_{X} (-D)$$
is ample for some
$\nu_{0} \in \N$ and all $\nu \geq 0$. However, in general, this divisor
is not a normal crossing divisor and henceforth (11.5) is of no use.\\
\ \\
If one assumes that the embedded resolution of singularities holds true
over $k$ and even over $W_{2} (k)$, (11.5) would give an affirmative
answer to
\end{defn}
\begin{satz}{Problem} Let $k$ be a perfect field of characteristic
$p > 0$, let $X$ be a proper smooth $k$-variety and ${\cal L}$ an
invertible sheaf. Assume that $X$ and ${\cal L}$ admit liftings to
$W_{2} (k)$, that ${\cal L}$ is numerically effective and that
$\kappa({\cal L}) = \dim X$.\\
\ \\
Does this imply that
\[ H^{b} (X, {\cal L}^{-1}) = 0  \ \  \mbox{for} \ \  b < \dim X
\leq \Char \ k  \ \mbox{?}  \]
\end{satz}
\begin{defn}{Remark} \\
a) If $\dim X = 2$ then (11.5) gives an affirmative answer to the problem
(11.7), since we have imbedded resolution of singularities for curves on
surfaces. In the surface case however, \cite{DI87}, Cor. 2.8, gives the
vanishing of $H^{b} (X, {\cal L}^{-1})$, for $b < 2$, without assuming
that $\cal L$ lifts to $W_2 (k)$.\\
b) As mentioned in Lecture 1 and 8, even if we restrict ourselves
to the case where $\cal L$ is semi-ample and of maximal Iitaka dimension,
we do not know the answer to problem (11.7) for higherdimensional $X$.
\end{defn}
\ \\
\Paragraph{Deformation theory for cohomology groups}
\ \\
In this lecture we will recall D. Mumford's description of higher
direct image sheaves, already used in (10.21),
and their base change properties and, following
Green and Lazarsfeld \cite{GL87}, deduce the deformation theory for
cohomology groups.\index{Deformation!of cohomology groups}\\
\ \\
\begin{satz}{Theorem (Mumford)} Let $g : Z\>>> Y$
be a projective flat morphism of
noetherian schemes, let $Y_0\subset Y$ be an affine open
subscheme, $$Z_0 = g^{-1}(Y_0) \ \ , \ \ g_0 = g|_{Z_0}$$
and let $\cal B$ be a locally free sheaf on $ Z$.
Then there exists a bounded complex $({\cal E}^\bullet, \delta_\bullet)$
of locally free ${\cal O}_{Y_0}$ modules of finite
rank such that $${\cal H}^{b} ({\cal E}^\bullet \otimes \ {\cal F}) =
R^{b}g_{0_*} ({\cal B}|_{Z_0}\otimes g_0^* {\cal F})$$ for all
coherent sheaves $\cal F$ on $Y_0$.\\
\end{satz}
In order to construct ${\cal E}^\bullet$, D. Mumford uses in \cite{Mum70},
II, \S 5, the description of higher direct images by \v{C}ech complexes. The
`` Coherence Theorem'' of Grauert-Grothendieck allows to realize
$({\cal E}^\bullet)$ as a complex of locally free sheaves
of finite rank.
The proof of (12.1) can be found as well in \cite{Har77}, III, \S 12. From
(12.1) one obtains easily the base change theorems of Grauert and Grothendieck,
as well as the ones used at the end of Lecture 10.
\begin{defn}{Example} Let $y \in Y_0$ be a point and ${\cal F} = k(y)$.
For $Z_y = g^{-1}(y)$ one
obtains
\[
\begin{CD}
{\cal H}^{b}({\cal E}^\bullet \otimes k(y)) \> = >> H^{b}(Z_y, B|_{Z_y})\\
\A \tau A A    \A \eta A A \\
{\cal H}^{b}({\cal E}^\bullet ) \otimes k(y) \> = >> R^{b}g_{*}({\cal B})
\otimes k(y),
\end{CD}
\]
\ \\
where $\eta$ is the base change morphism (\cite{Har77}, III, 9.3.1). In general,
due to the fact that the images of
$$\delta_{b-1} : {\cal E}^{b-1}
\>>> {\cal E}^{b} \mbox{ \ and \ } \delta_{b} : {\cal E}^{b} \>>> \
{\cal E}^{b+1}$$ are not subbundles of ${\cal E}^{b}$ and $ {\cal E}^{b+1},
\tau$ and hence $\eta$ will be neither injective nor surjective.\\
\end{defn}
\begin{defn}{Example} Let $X$ be a projective manifold, defined over an
algebraically closed field $k$ and let $Y \subset \Pic^{0}(X)$ be a closed
subscheme, $$Z = X \times Y \mbox{ \  and \ } g = pr_{2} :Z \>>> Y.$$
Recall that on Z we have a Poincar\'{e} bundle $\cal P$
(see for example \cite{Mum70}), i.e. an invertible
sheaf $\cal P$ such that
$${\cal P}|_{g^{-1}(y)} \simeq {\cal N}_{y},$$
if ${\cal N}_{y}$ is the
linebundle on $X$ corresponding to $$y\in Y\subset \Pic^{0}(X).$$
In more fancy terms, the functor
$$ T \longmapsto {\cal P}ic(X \times T / T)$$
is represented by a locally noetherian group-scheme \Pic(X), whose connected
component containing zero is $\Pic^{0}(X)$. The invertible sheaf $\cal P$
is the restriction of the universal bundle on $X \times \Pic(X)$ to
$$X\times Y \subset X \times \Pic^0(X)$$ (see \cite{Gro62}).
For $y \in Y$ let $T_{y,Y} = (m_{y,Y}/m^{2}_{y,Y})^{*}$ be the
Zariski tangent space. We have an exact sequence
$$0 \>>> \ T^{*}_{y,Y} \>>> {\cal O}_{y,Y}/m^{2}_{y,Y}
\>>> \ k(y) \>>> 0 .$$
Since
$$g^{*}(T^{*}_{y,Y})\otimes_{{\cal O}_Z}{\cal P} = g^{*}(T^{*}_{y,Y})
\otimes {\cal N}_y$$
one obtains the exact sequence
$$0 \>>> g^{*}(T^{*}_{y,Y})\otimes {\cal N}_y \>>> g^{*}
({\cal O}_{y,Y}/m^{2}_{y,Y}) \otimes {\cal P}\>>> {\cal N}_y
\>>> 0$$ on
$X \simeq \ g^{-1}(y)$. If, identifying $X$ with $g^{-1}(y)$,
$$\zeta \in H^{1}(X, g^{*}(T^{1}_{y,Y})) =
H^{1}(X, {\cal O}_X) \otimes T^{*}_{y,Y}$$ is the extension class of this sequence,
the induced edge morphism
\[
H^{b}(X, {\cal N}_{y}) \>>> H^{b+1}(X,g^{*}(T^{*}_{y,Y})
\otimes {\cal N}_{y}) =
H^{b+1}(X, {\cal N}_{y}) \otimes_{k(y)} T^{*}_{y,Y} \]
is the cup-product with $\zeta$.
\end{defn}
\Abschnitt{} Keeping the notations from (12.3), let ${\cal M}$ be a
locally free sheaf on $X$ and ${\cal B} = {\cal P} \otimes pr_{1}^{*}
{\cal M}$. Since the exact sequence
\[ 0 \>>> g^{*}(T^{*}_{y,Y}) \otimes {\cal N}_y \otimes
{\cal M} \>>> g^{*}({\cal O}_{y,Y}/m^{2}_{y,Y})
\otimes {\cal B} \>>> {\cal N}_y \otimes {\cal M} \>>> 0 \]
is obtained from
$$0 \>>> g^{*}(T^{*}_{y,Y})
\otimes {\cal N}_y \>>> g^{*}({\cal O}_{y,Y}/m^{2}_{y,Y})
\otimes {\cal P} \>>> {\cal N}_y \>>> 0$$
by tensorproduct with ${\cal M}$, the
induced edge morphism $$H^{b}(X, {\cal M} \otimes {\cal N}_y)
\>>> H^{b+1}
(X, {\cal M} \otimes {\cal N}_y)\otimes _{k(y)} T^{*}_{y,Y}$$
is again the cup-product
with $\zeta$. Let us finally  remark that $\zeta$ induces a morphism
\[
\begin{CD}
T_{y,Y} \> \zeta >> H^{1}(X, {\cal O}_X)
\end{CD}
\]
which, due to the
universal property of ${{\cal P}}$ is injective. In fact, if we represent\\
$\tau\in T_{y,Y}$ by a morphism
$$\zeta': D = \Spec \ \ k[\epsilon] \>>> Y \mbox{ \ with \ }
\zeta'(<\epsilon>) = y \ ,$$
where $k[\epsilon] = k[t]/t^{2}$
is the ring of dual numbers, then for $\tau \neq 0$ the pullback of
${\cal P}$ to ${X \times D}$ is non
trivial and hence the extension class $\zeta (\tau)$ of
\[
\begin{CD}
0 \> >> {\cal N}_y \> \cdot \epsilon >> {\cal P}|X \times D
\> >> {\cal N}_y \> >> 0
\end{CD}
\]
is non zero. If $Y = \Pic^0(X)$ one has
$\dim T_{y,Y} = \dim H^{1}(X,{\cal O}_X)$ and $\zeta$ is surjective as well.
\begin{defn}{Notations} For $X, {\cal M}$ as above let us write
$$S^{b}(X,{\cal M}) = \{y \in \Pic^0(X); H^{b}(X, {\cal N}_y\otimes
{\cal M})\neq 0\}.$$\index{$S^{b}(X,{\cal M})$}
\end{defn}
The first part of the following lemma is well known and an easy
consequence of the semicontinuity of the dimensions of cohomology
groups. To fix notations we will prove it nevertheless.
\begin{satz}{Lemma}\\
a) $S^{b}(X,{\cal M})$ is a closed subvariety of $\Pic^0(X)$.\\
\ \\
b) If $Y\subset S^{b}(X,{\cal M})$ is an ireducible component and
$$m =  \mbox{{\rm Min}}
\{\dim H^{b}(X,{\cal N}_y \otimes {\cal M}); \ y \in Y\},$$  then
the set $$U = \{y\in
Y; \ \dim H^{b}(X,{\cal N}_{y} \otimes {\cal M}) =m\}$$
is open and dense in $Y$.\\
\ \\
c) For $y \in U$ and $\zeta: T_{y,Y}\hookrightarrow H^{1}(X,{\cal O}_{X})$ as in
(12.4) the cup-products $$\zeta(T_{y,Y}) \otimes H^{b-1}(X,{\cal N}_{y}
\otimes {\cal M})
\>>> H^{b}(X,{\cal N}_{y}\otimes {\cal M})$$ and
$$\zeta(T _{y,Y})\otimes H^{b}(X, {\cal N}_{y}\otimes {\cal M})
\>>> H^{b+1}(X,{\cal N}_{y}\otimes {\cal M})$$
are both zero.
\end{satz}
\begin{Beweis}{Proof} For any open affine $P_{0} \subset \Pic^0(X)$ let ${\cal E}^
\bullet$ be the complex from (12.1) describing the higher direct images of
$${\cal B}|_{X\times P_o} = pr_1^*{\cal M}\otimes {\cal P}|_{X\times P_o}$$
and there base change.
If we write $${\cal W}_{b} = \mbox{Coker}(\delta_{b-1}:{\cal E}^{b-1}\>>>
{\cal E}^{b}),$$ then for ${\cal F}$ coherent on $P_0$ we have
$${\cal W}_{b}\otimes {\cal F} = \mbox{Coker}(\delta_{b-1}: {\cal E}^{b-1}
\otimes {\cal F}\>>> {\cal E}^{b}\otimes {\cal F})$$
and an exact sequence
$$0 \>>> {\cal H}^{b}({\cal E}^\bullet \otimes {\cal F})
\>>> {\cal W}_{b}\otimes {\cal F}\>>> {\cal E}^{b+1}
\otimes {\cal F}\>>> {\cal W}_{b+1} \otimes {\cal F}
\>>> 0.$$
If $S^{b}(X,{\cal M}) \neq \Pic^0(X)$, then $S^{b}(X,{\cal M})$
is just the locus where ${\cal W}_{b}\rightarrow {\cal E}^{b+1}$ is
not a subbundle. Obviously this condition defines a closed subscheme
of $P_0$. For $y \in Y\cap P_{0} = Y_{0}$ we have
$$\dim {\cal H}^{b}({\cal E}^
\bullet\otimes k(y)) = -\rank({\cal E}^{b+1}) +\dim ({\cal W}_{b}\otimes
k(y)) + \dim ({\cal W}_{b+1}\otimes k(y))$$
and the open set $U$ in b) is
nothing but the locus where both,
$${\cal W}_{b}\otimes {\cal O}_{Y_0} \mbox{ \ \  and \ \ }
{\cal W}_{b+1}\otimes {\cal O}_{Y_0}$$
are locally free ${\cal O}_{Y_0}$ modules.
On $U$ the sequence
$$0 \>>> {\cal H}^{b}({\cal E}^\bullet |_U)\>>>
{\cal W}_{b}|_{U}\>>> {\cal E}^{b+1}|_{U}\>>>
{\cal W}_{b+1}|_{U}\>>> 0$$
is an exact sequence of vector bundles
and $$ {\cal H}^{b}({\cal E}^\bullet \otimes {\cal F}) = {\cal H}^{b}
({\cal E}|_U) \otimes
{\cal F}$$ for all coherent ${\cal O}_{U}$ modules ${\cal F}$.
In particular for $y\in U$ the exact sequence
$$0 \>>> {\cal E}^
\bullet \otimes T^{*}_{y,Y}\>>> {\cal E}^\bullet\otimes
{\cal O}_{y,Y}/m^{2}_{y,Y}\>>> {\cal E}^\bullet\otimes k(y)\>>>
0$$ induces
\[
\begin{CD}
{{\cal H}^b}({{\cal E}^\bullet}\otimes T^{*}_{y,Y}) \> >> {{\cal H}^b}
({{\cal E}^\bullet} \otimes {\cal O}_{y,Y}/m^{2}_{y,Y})\> >> {\cal H}^b
({\cal E}^\bullet \otimes k(y))\\
 \V = VV  \V = VV  \V = VV\\
{{\cal H}^b}({\cal E}^\bullet) \otimes T^{*}_{y,Y}\> >> {{\cal H}^b}({\cal E}^{\bullet})
\otimes {\cal O}_{y,Y}/m^{2}_{y,Y} \> >> {{\cal H}^b}({\cal E}^{\bullet})
\otimes k(y)
\end{CD}
\]
and the edge morphisms
\[
{\cal H}^i( {\cal E}^{\bullet} \otimes k(y)) \>>> {\cal H}^{i+1}
({\cal E}^{\bullet}\otimes T^{*}_{y,Y})
\]
are zero for $i=b$ and $b-1$. Using (12.1) we have identified in
(12.4) this edge morphism with the cup-product
$$H^{i}(X,{\cal M}\otimes{\cal N}_y)
\>>> H^{i+1}(X,{\cal M}\otimes{\cal N}_y) \otimes_{k(y)} T^{*}_{y,Y}$$
with the extension class $\zeta \in H^{1}(X,{\cal O}_{X})\otimes
T^{*}_{y,Y}.$\\
\end{Beweis}
\begin{satz}{Corollary(Green, Lazarsfeld \cite{GL87})}
If $Y \subset S^{b}(X,{\cal M})$  is an irreducible component and
$y \in Y$  is a point in general position then
$$\codim_{\Pic^0(X)}(Y) \geq \codim (\Gamma \subset H^{1}(X,{\cal O}_{X}))$$
where $$\Gamma = \{\varphi \in H^{1}(X,{\cal O}_{X}) ; \ \alpha \cup \varphi =0
\mbox{\ \ and \ \ } \beta \cup \varphi  = 0 \mbox{\ \  for all \ \ }$$
$$\alpha \in
H^{b-1}(X,{\cal N}_y \otimes {\cal M}) \mbox{\ \  and \ \ }
\beta \in H^{b}(X,{\cal N}_y \otimes {\cal M})\}.$$
\end{satz}
\begin{defn}{Remark} Even if one seems to loose some information,
in the applications of (12.7) in Lecture 13
we will replace $\Gamma$ by the larger space
$$\{\varphi \in H^{1}(X,{\cal O}_{X}) ; \ \beta \cup \varphi  = 0
\mbox{\ \  for all \ \ }
\beta \in H^{b}(X,{\cal N}_y \otimes {\cal M})\}$$
in order to obtain lower bounds for $\codim_{\Pic^0(X)}(S^b(X,{\cal M}))$
for certain invertible sheaves $\cal M$.
\end{defn}
\Paragraph{Generic vanishing theorems $[$26$]$, $[$14$]$}
\ \\
In this section we want to use (12.7) and Hodge-duality to prove some bounds
for $$\codim_{\Pic^{0}(X)} (S^{b} (X, {\cal M}))$$ for the subschemes $S^{b} (X,
{\cal M})$ introduced in \S 12. In particular, we lose a little bit
the spirit of the previous lectures, where we tried to underline as much as
possible the {\it algebraic} aspects of vanishing theorems.
Everything contained in this lecture is either
due Green-Lazarsfeld \cite{GL87} or to H. Dunio \cite{Dun91}. The use of
Hodge duality will force us to assume that $X$ is a complex manifold. Without
mentioning it we will switch from the algebraic to the analytic language and use
the comparison theorem of \cite{Ser56} whenever needed.

\begin{defn}{Notations} Let $X$ be a projective complex manifold. The Picard
group $\Pic (X)$ is $H^{1} (X, {\cal O}^{*}_{X})$ and, using
the exponential sequence, $\Pic^{0}(X)$ is identified with $$H^{1} (X,
{\cal O}_{X}) / H^{1} (X, \Z ).$$\index{Poincar\'{e} bundle}
Let ${\cal P}$ be the Poincar\'{e} bundle on $X \times \Pic^{0} (X)$, and
$g=pr_2$. If $$\zeta : T_{y, \Pic^{0}(X)} \>>> H^{1} (X, {\cal O}_{X})$$
is the extension class of
\[ 0 \>>> g^{*} T_{y, \Pic^{0}(X)} \otimes {\cal N}_{y}
\>>> g^{*} ({\cal O}_{y, \Pic^{0}}/m^{2}_{y, \Pic^{0}})
\otimes {\cal P} \>>>
{\cal N}_{y} \>>> 0 \]
then $\zeta$ is the identity.
Let
$$\mbox{Alb} (X) = H^{0} (X, \Omega^{1}_{X} )^{*} / H_{1} (X, \Z ) $$
be the Albanese variety of $X$ and\index{Albanese variety}
\[ \alpha : X \>>> \mbox{Alb} (X)  \]
be the Albanese map. The morphism $\alpha$ induces an isomorphism
\[\alpha^{*} : H^{0} (\mbox{Alb} (X), \Omega^{1}_{\mbox{Alb}(X)}) \>>>
H^{0} (X, \Omega^{1}_{X}). \]
In particular,
$$\dim \alpha (X) = \rank _{{\cal O}_X} (\im (H^{0} (X, \Omega^{1}_{X})\otimes_{\C}
{\cal O}_X \>>> \Omega^1_X)) \ .$$
\end{defn}
\begin{satz}{Theorem (Green-Lazarsfeld \cite{GL87})}
\index{Generic vanishing!Green Lazarsfeld}
Let $X$ be a complex projective manifold and
$$S^{b} (X) = \{ y \in \Pic^{0} (X); \ H^{b} (X, {\cal N}_{y}) \neq 0 \}.$$
Then $$\codim_{\Pic^{0}(X)} (S^{b} (X)) \geq \dim (\alpha (X)) - b.$$
In particular, if ${\cal N} \in \Pic^{0} (X)$ is a generic line bundle, then
$H^{b} (X, {\cal N})= 0$ for $b < \dim \alpha (X)$.
\end{satz}
\begin{Beweis}{Proof} Using (12.7) or (12.8) one obtains (13.2) from\\
\end{Beweis}
\begin{satz}{Claim} Assume that $H^{b} (X, {\cal N}_{y}) \neq 0$
then for
\[ \Gamma = \{ \varphi \in H^{1} (X, {\cal O}_{X}); \ \beta \cup \varphi = 0
\ \ \mbox{for all} \ \ \beta \in H^{b} (X, {\cal N}_{y}) \} \]
one has
$$\codim (\Gamma \subset H^{1} (X, {\cal O}_{X})) \geq \dim (\alpha (X)) - b.$$
\end{satz}
\begin{Beweis}{Proof}\\
\noindent
{\it Step 1:} ${\cal N}_{y}$ is a flat unitary bundle on $X$, obtained from
a unitary representation of the fundamental group. In particular the
conjugation of harmonic forms with values in ${\cal N}_y$ gives a complex
antilinear isomorphism, the so called Hodge duality,
\[ \iota : H^{b} (X, \Omega^{a}_{X} \otimes {\cal N}_{y}) \>>>
H^{a} (X, \Omega^{b}_{X} \otimes {\cal N}^{-1}_{y}) \]
(see (13.5) and (13.6) for generalizations). Moreover, if $\varphi \in
H^{1} (X, {\cal O}_{X})$ and if $\omega = {\bar \varphi} \in
H^{0} (X, \Omega^{1}_{X})$ is the Hodge-dual of $\varphi$, then for $\beta \in
H^{b} (X, \Omega^{a}_{X} \otimes {\cal N}_{y})$ one has
\[ \iota (\beta \cup \varphi ) = \iota (\beta ) \wedge \omega \in H^{a} (X,
\Omega^{b+1}_{X} \otimes {\cal N}_{y}^{-1}). \]
Hence
$$\iota(\Gamma) = {\bar \Gamma} \subset H^{0} (X, \Omega^{1}_{X})$$
is the subspace of forms $\omega \in H^{0} (X, \Omega^{1}_{X})$ such that
\[ \beta \wedge \omega = 0 \ \ \mbox{for all} \ \ \beta \in H^{0} (X,
\Omega^{b}_{X} \otimes {\cal N}_{y}^{-1}). \]
\noindent
{\it Step 2:} Consider the natural map
\[ \gamma : H^{0} (X, \Omega^{1}_{X}) \otimes {\cal O}_{X} \>>>
\Omega^{1}_{X}. \]
Since all one-forms are pullback of one-forms on $\alpha (X) \subset
\mbox{Alb}(X)$, the subsheaf $\im (\gamma )$ of $\Omega^{1}_{X}$
is of rank \ $\dim \alpha (X)$ and
\[
r = \rank \gamma (\overline{\Gamma} \otimes {\cal O}_X)= \dim
\overline {\Gamma} - \rank (\ker ({\gamma}) \cap \overline {\Gamma}\otimes
{\cal O}_{X})  \geq
\]
\[
\dim \overline{\Gamma} - \dim \ker ({\gamma})
= \dim \overline{\Gamma} - (\dim H^0(X, \Omega^1_X)- \dim \alpha(X))
\]
\[
= \dim \alpha (X) - \codim (\Gamma \subset H^1(X, {\cal O}_X)) \ . \hspace{2cm}
\]
We assumed that $H^0 (X, \Omega^b_X \otimes {\cal N}_y^{-1})\neq 0$.
Hence we have at least one element
$$\beta \in H^0 (X, \Omega^b_X \otimes {\cal N}_y^{-1})
\mbox{ \ and \ } \beta \wedge \gamma
(\overline{\Gamma}\otimes {\cal O}_X) = 0.$$
Since $$(\wedge^b \Omega ^1_X) \otimes (\wedge^{n-b}\Omega ^1_X)
\>>> \Omega^n_X$$ is a nondegenerate pairing, for $n = \dim X$,
we find some meromorphic differential form
$$\delta \in \Omega^{n-b}_X \otimes \C (X) \mbox{ \ with \ }
\delta \wedge \beta \neq 0.$$
Hence $\delta$ lies in
$$\Omega^{n-b}_X \otimes \C (X) -\{ \gamma (\overline{\Gamma} \otimes
{\cal O}_X) \wedge \Omega^{n-b-1}_X \otimes \C (X)\}.$$
This however is only possible if $n - b \leq n - r$ or $ b \geq r$.
Altogether we find $$b \geq \dim \alpha(X) - \codim (\Gamma \subset H^1
(X, {\cal O}_X)).$$
\end{Beweis}
\Abschnitt{ }
If one tries to use the same methods for $S^b(X, {\cal M})$ one has to
make sure that $H^b(X,{\cal M} \otimes {\cal N}_y)$ is in Hodge duality with
$H^0(X, \Omega^b_X \otimes {\cal M'}\otimes {\cal N}_y^*)$
for some sheaf ${\cal M'}$.
As shown in (3.23) this holds true for the sheaves ${\cal L}^{(i)}$
arising from cyclic coverings, at least if one considers
$\Omega^b_X(\log \ D)$ instead of $\Omega^b_X$. More generally one has:
\begin{satz}{Theorem (K. Timmerscheidt \cite {Tim87})}
\index{Degeneration!of the Hodge to de Rham spectral sequence!
for unitary local systems}
Let $D$ be a normal crossing divisor on $X$, let ${\cal V}$ be a locally
free sheaf and $$\nabla: {\cal V} \>>> \Omega^1_X
(\log \ D) \otimes {\cal V}$$ an integrable logarithmic connection.
Assume that for all components $D_i$ of $D$, the real part of all eigenvalues
of ${\rm res}_{D_i}(\nabla)$ lies in $(0,1)$\\
(which implies that conditions
(*) and (!) of (2.8) are satisfied, and that ${\cal V}$ is the
canonical extension defined by Deligne \cite {Del70}).\\
Assume moreover that the local constant system
$$V = \ker (\nabla: {\cal V}|_U \>>> \Omega^1
_U \otimes {\cal V}|_U)$$ is unitary for $U=X-D$. Then one has:\\
a) The Hodge to de Rham spectral sequence $$E^{ab}_1 = H^b(X,\Omega^a_X
(\log \ D)\otimes {\cal V}) \>>> \h^{a+b}(X,
\Omega^\bullet_X(\log \ D)\otimes {\cal V})$$
degenerates at $E_1$.\\
b) There exists a $\C$ - antilinear isomorphism $$\iota:H^b(X,\Omega^a_X
(\log \ D)\otimes {\cal V}) \>>> H^a(X,\Omega^b_X(\log \ D)\otimes
{\cal V}^*(-D_{\rm red}))$$
such that for $\varphi \in H^1(X, {\cal O}_X)$ and $\omega = \overline{\varphi}
\in H^0(X, \Omega^1_X)$ the diagram
\[
\begin{CD}
H^b(X,\Omega^a_X(\log \ D)\otimes {\cal V}) \> \iota >> H^a(X,\Omega^b
(\log \ D)\otimes {\cal V}^*(-D_{\rm red}))\\
\V \cup \varphi V V   \V \wedge \omega V V\\
H^{b+1}(X,\Omega^a_X(\log \ D)\otimes {\cal V})\> \iota >> H^a
(X,\Omega^{b+1}_X(\log \ D)\otimes {\cal V}^*(-D_{\rm red}))
\end{CD}
\]
commutes.
\end{satz}
\begin{defn}{Examples}\\
a) If $D = 0$ and if ${\cal N}_y$ is the invertible sheaf corresponding to $y \in
\Pic^0(X)$, then ${\cal N}_y$ has an integrable connection $\nabla$, whose
kernel is a unitary rank one local constant system. In this case (13.5)
is  wellknown and proven by the usual arguments from classical Hodge-theory,
applied to ${\cal N}_y \otimes {\cal N}_y^{*}$ see \cite{Del72}.\\
b) If the sheaf ${\cal V}$ in (13.5) is of the form
${\cal V}={\cal L}^{(i)^{-1}}$
for $$D=\sum^r_{j=1} \alpha_j D_j \mbox{\ \ \ and \ \ \ }
{\cal L}^N={\cal O}_X(D),$$
then the assumptions made
in (13.5) are satisfied whenever
$$\frac{i \cdot \alpha_j}{N} \notin \Z
\mbox{ \ for \ } j=1,...,r.$$
Hence, using the notations from (3.2), one has $D^{(i)}=D^{(N-i)}=D_{red}$ and
$${\cal L}^{(i)}(-D_{red})={\cal L}^i(- [\frac{i\cdot D}{N} ]-D_{red})
={\cal L}^i(- \{\frac{i\cdot D}{N} \})=$$
$${\cal L}^i( [\frac{-i\cdot D}{N} ])={\cal L}^{i-N}( [
\frac{(N-i)\cdot D}{N} ])={\cal L}^{{(N-i)}^{-1}}$$
Hence (3.2) and (3.23) imply (13.5) for ${\cal M}={\cal L}^{(i)^{-1}}.$\\
c) Finally, for ${\cal M}={\cal L}^{(i)^{-1}}\otimes {\cal N}_y$ one can use
(13.6,a) on the finite covering $Y$ of $X$ obtained by taking the $N-$th root
out of $D$ and the arguments used to prove (3.23) imply (13.5)in that case.
\end{defn}
\begin{satz}{Corollary (H. Dunio \cite{Dun91})}
\index{Generic vanishing theorems!for nef $\Q$-divisors}
Keeping the assumptions made in (13.5) and the notations introduced in (13.1)
and (12.5) one has $$\codim_{\Pic_0(X)}(S^b(X,{\cal V})) \geq
\dim (\alpha (X))-b.$$
\end{satz}
\begin{Beweis}{Proof}
Again, it is sufficient to give a lower bound for
$$\codim (\Gamma \subset H^1
(X, {\cal O}_X))$$
where
$$\Gamma = \{\varphi \in H^1(X,{\cal O}_X); \ \beta
\cup \varphi = 0 \mbox{ \ for all \ } \beta\in H^b(X,{\cal N}_y
\otimes {\cal V})\},$$
or using (13.5,b), for
$$\codim (\bar{\Gamma} \subset H^0(X,\Omega^1_X)) $$
where
$$ \bar{\Gamma}= \{\omega \in H^0(X,\Omega^1_X); \beta \wedge \omega=0
\mbox{ \ for all \ } \hspace{1.5cm} $$
$$\beta\in H^0(X,\Omega^b_X(\log \ D)\otimes {\cal N}_y^*
\otimes {\cal V}^*(-D_{red}))\}.$$
As in (13.3), if $$\gamma:H^0(X,\Omega^1_X)\otimes {\cal O}_X \>>>
\Omega^1_X$$ is the natural map, one has $$r=\rank(\gamma(\bar{\Gamma}\otimes
{\cal O}_X)) \geq \dim \alpha (X)-\codim (\bar{\Gamma} \subset
H^0(X,\Omega^1_X)).$$
Assume that one has some $$0 \neq \beta \in H^0(X,\Omega^b_X(\log \ D)\otimes
{\cal N}_y^*\otimes {\cal V}^*(-D_{red})).$$
Let $v_1,....,v_s$ be a basis of
$${\cal N}_y^*\otimes {\cal V}^*(- D_{red})\otimes \C (X),$$
then one has $\beta=\sum_{i=1}^s \beta_i v_i$
for some $\beta_i \in \Omega^b_X \otimes \C (X)$
and $\beta_i\wedge\omega=0$ for all $i$ and all
$$\omega\in\gamma(\bar{\Gamma}\otimes{\cal O}_X)\otimes\C (X).$$
As in (13.3) this is only possible if $b\geq r$.\\
\end{Beweis}
\begin{defn}{Example}
If $\cal L$ is an invertible sheaf on $X$ and if $D$ is a normal crossing
divisor let $N\in\N$ be larger than the multiplicities of the components of $D$.
If ${\cal L}^N = {\cal O}_X(D)$ then ${\cal V}={\cal L}^{-1}$
satisfies the assumptions made in (13.7) and
$$H^b(X,{\cal L}^{-1}\otimes{\cal N}_y)=0$$
for $y\in \Pic^0(X)$
in general position and $b < \dim \alpha(X) $.\\
On the other hand, (5.12,e) tells us, that
$$H^b(X,{\cal L}^{-1}\otimes {\cal N}_y)=0$$
for all $y\in \Pic^0(X)$ and $b < \kappa({\cal L})$.
Hence (13.7) is only of interest if
$$\dim \alpha(X) - \kappa({\cal L}) > 0.$$
\end{defn}
In this situation the bounds given in (13.7) can be improved. The generic
vanishing theorem remains true for
$$b < \dim \alpha(X) + \kappa({\cal L}) - \dim \alpha'(Z)$$
where $Z$ is a desingularization of the image of the rational map
$$\Phi_\nu : X \>>> \P(H^0(X,{\cal L}^\nu))$$
for $\nu$ sufficiently large (see (5.3)) and where
$$\alpha' : Z \>>> \mbox{Alb}(Z)$$ is the Albanese map of $Z$.
To be more precise:
\begin{defn}{Assumptions and Notations}
Let $X$ be a complex projective manifold, let ${\cal L}$ be an invertible
sheaf on $X$, let $$D=\sum^r_{j=1} \alpha_j D_j$$ be a normal crossing
divisor and let $N$ be a natural number with $$0 < \alpha_j < N \mbox{ \ for \ }
j = 1,...,r.$$
Assume that either ${\cal L}^N(-D)$ is semi-ample or that
(more generally) ${\cal L}^N(-D)$ is numerically effective and
$$\kappa({\cal L}^N(-D)) = \nu({\cal L}^N(-D))$$
( see (5.9) and (5.11)). For some $\mu > 0$ the rational map ( see (5.3))
$$\Phi_\mu : X \>>> \Phi_\mu(X)\subset\P(H^0(X,{\cal L}^\mu))$$
has an irreducible general fibre and $\dim(\Phi_\mu(X)) = \kappa({\cal L}).$
For such a $\mu$ let $Z$ be a desingularization of $\Phi_\mu(X)$ and
$X'$ a blowing up of $X$ such that the induced rational map
$$\Phi' : X' \>>> Z$$
is a morphism of manifolds. $\Phi'^*$ defines a morphism
$$\Phi^* : \Pic^0(Z) \>>> \Pic^0(X') = \Pic^0(X)$$
and $\Phi^*(\Pic^0(Z))$ is an abelian subvariety of $\Pic^0(X)$
independent of the desingularization choosen. Let
$$\alpha : X \>>> \mbox{Alb}(X) \mbox{ \ and \ }
\alpha' : Z \>>> \mbox{Alb}(Z)$$ be the Albanese maps.
\end{defn}
\begin{satz}{Theorem (H. Dunio \cite{Dun91})}
Under the assumptions made in (13.9) one has:
\begin{quote}
a)  $S^b(X,{\cal L}^{-1}) = 0$ for $b < \kappa({\cal L})$.\\
\ \\
b)  $S^b(X,{\cal L}^{-1})$ lies in the subgroup of
$\Pic^0(X)$, which is generated by torsion elements and by
$ \Phi^*(\Pic^0(Z))$, for $b = \kappa({\cal L})$.\\
\ \\
c) $\codim_{\Pic^0(X)}(S^b(X,{\cal L}^{-1})) \geq
\dim \ \alpha(X) - \dim \ \alpha'(Z) + \kappa({\cal L}) - b$.
\end{quote}
\end{satz}
\begin{Beweis}{Proof}
a) is nothing but (5.12,e) and it has already be shown twice in these notes.
Nevertheless, when we prove (13.10,c) it will come out again.\\

First of all, since $S^b(X,{\cal L}^{-1})$ is compatible with blowing ups of $X$, we
may assume that the rational map $\Phi : X \rightarrow Z$ is a morphism.
Moreover, as in the proof of (5.12), we can assume that ${\cal L}^N(-D)$ is semi-ample
or  even, replacing $N$ and $D$ by some common multiple, that
$${\cal L}^N(-D) = {\cal O}_X(H)$$ where $H$ is a non singular
divisor and $D+H$ a normal crossing divisor. Since
$${\cal L}={\cal L}^{(1,D)}={\cal L}^{(1,D+H)}$$
we can as well assume that ${\cal L}^N = {\cal O}_X(D)$.
By (13.5,b) or (13.6,c) the space $$H^b(X,{\cal L}^{-1}\otimes{\cal N}_y) $$
is Hodge dual to
$$H^0(X,\Omega^b_X(\log \ D)\otimes{\cal L}^{(N-1)^{-1}}
\otimes{\cal N}_y^{-1}).$$
Let ${\cal G}^b_\Phi \hookrightarrow \Omega^b_X(\log \ D)$ be the largest subsheaf
which over some open non empty subvariety of $X$ coincides with
$$ \Phi^* \Omega_Z^\kappa \wedge \Omega_X^{b-\kappa}(\log \ D).$$
Of course ${\cal G}_\Phi^b=0$ for $b<\kappa =\kappa ({\cal L})$ and
$\delta \in {\cal G}_\Phi^b \otimes \C (X)$ if and only if $\delta$ is a
meromorphic $b$-form with $\delta\wedge\Phi^*\Omega^1_Z = 0$.\\
\ \\
Since ${\cal L}^{(N-1)} \subseteq {\cal L}^{N-1}$ and since
$$({\cal L}^{(N-1)})^N = {\cal O}_X(N\cdot D_{red}-D),$$
we have $\kappa({\cal L}^{(N-1)}) = \kappa
({\cal L})$ and $({\cal L}^{(N-1)})^\mu$ contains ${\cal L}^{N}$ for some
$\mu > 0$. \\
\begin{defn}{Claim}
If ${\cal M}$ is an invertible sheaf such that ${\cal M}^\mu$ contains
${\cal L}^N$ for some $\mu > 0$, then
$$H^0(X,{\cal G}^b_\Phi\otimes{\cal M}^{-1}\otimes{\cal N}_y^{-1}) =
H^0(X,\Omega_X^b(\log \ D)\otimes{\cal M}^{-1}\otimes{\cal N}_y^{-1}).$$
\end{defn}
\begin{Beweis}{Proof} The methods used to prove (13.11) are due to
F. Bogomolov \cite{Bog78}.\\
A section $\beta\in
H^0(X,\Omega_X^b(\log \ D)\otimes{\cal M}^{-1}\otimes{\cal N}_y^{-1})$ gives
an inclusion
$$\beta : {\cal M} \>>> \Omega_X^b(\log \ D)\otimes{\cal N}^{-1}_y$$
and we have to show that $\Phi^*\Omega_Z^1\wedge\beta({\cal M}) = 0.$\\
\ \\
If $\tau : X' \>>> X$ is generically finite and $D' = \tau^* D$
a normal crossing divisor, then $\tau^*\Omega^b_X(\log \ D)$ is a subsheaf of
$\Omega^b_{X'}(\log \ D').$
In fact, if (locally) $D'$ is the zero set of $x'_1,....,x'_r$ and if $x$
is a local parameter on $X$ defining one component of $D$, then
$$ \tau^* x = \prod^r_{j=1} {x'}_j^{\nu_j} \mbox{  \  and  \ }
\tau^*\frac{dx}{x} = \sum^r_{j=1}\nu_j\frac{dx'_j}{x'_j}\in
\Omega^b_{X'}(\log \ D').$$
Hence $\beta$ induces
$$ \beta ' : \tau^*{\cal M} \>>> \Omega^b_{X'}(\log \ D') \otimes
\tau^*({\cal N}_y^{-1}).$$
Since $$\beta '(\tau^*{\cal M})\wedge\tau^*\Phi^*\Omega^1_Z = 0$$ implies that
$$\beta ({\cal M})\wedge\Phi^*\Omega^1_Z = 0,$$
we can replace $X$ by $X'$ whenever we like.
For example, if
$$s_0,....,s_{\kappa} \in H^0(X,{\cal L}^\nu) \subset
H^0(X,{\cal M}^\mu)$$
are choosen such that the functions
$$\frac{s_1}{s_0},....,\frac{s_\kappa}{s_0}$$ are algebraic independent,
we can take $X'$ as a desingularization of the covering obtained by taking the
$\mu$-th root out of $s_0,s_1,....,s_\kappa$.
Hence to prove (13.11) we may assume that ${\cal M}$ itself has sections
$$s_0,....,s_\kappa \mbox{  \ with  \ } f_1=\frac{s_1}{s_0},....,f_\kappa=
\frac{s_\kappa}{s_0}$$
algebraic independent. From (13.5) we know that
$d(\beta(s_i)) = 0$, which by the Leibniz rule implies
$$ 0 = d(\beta(s_i)) = d(f_i \cdot \beta(s_0)) = d(f_i)\wedge\beta(s_0).$$
However, $d(f_1),....,d(f_\kappa)$ are generators of $\Phi^*\Omega^1_Z$
over some non empty open subset.\\
\end{Beweis}
Part a) of (13.10) follows from (13.11) since for $b<\kappa$ the sheaf
${\cal G}^b_\Phi = 0$.\\
\ \\
If $b=\kappa$ then
$${\cal G}^b_\Phi = \Phi^*\omega_Z \otimes {\cal O}_X(\Delta)$$
for some effective divisor $\Delta$ on $X$, not meeting the general fibre
$F$ of $\Phi$. Hence, for $y\in S^\kappa(X,{\cal L}^{-1})$
(13.11) implies that ${\cal L}^{(N-1)^{-1}} \otimes {\cal N}_y^{-1}|_F$ has a
non trivial section and therefore
${\cal N}_y^{-1}|_F = {\cal L}^{(N-1)}|_F$.
The divisor $D+H$ does not meet the general fibre $F$ and, as we claimed in
(13.10,b), $N\cdot y\in\Phi^*(\Pic^0(Z))$.
\begin{defn}{Remark}
If ${\cal L}$ is semi-ample and $b > \kappa$, then a similar argument shows
that ${\cal L}^{(N-1)^{-1}} \otimes {\cal N}_y^{-1}|_F \otimes
\Omega_F^{b-\kappa}$ has a non trivial section.
This implies, as we have seen in the proof of (13.2), that those
${\cal N}_y|_F$ are corresponding to points $y$ in a subvariety of $\Pic^0(F)$
of codimension larger than or equal to
$$\dim(\alpha(F))-b+\kappa=\dim(\alpha(X))-\dim(\alpha(Z))-b+\kappa,$$
which gives (13.10,c).
\end{defn}
\ \\
We instead generalise the argument used in step 2 of the proof of (13.3):\\
If
$$\Gamma = \{ \varphi\in H^1(X,{\cal O}_X); \ \beta\cup\varphi=0 \mbox{ \ for all \ }
\beta \in H^b(X,{\cal L}^{-1}\otimes{\cal N}_y) \}$$
then the Hodge dual of $\Gamma$ is
$$\bar{\Gamma} = \{ \omega\in H^0(X,\Omega^1_X); \ \beta\cup\omega=0 \mbox{ \ for all \ }
\beta \in H^0(X,{\cal G}^b_\Phi\otimes {\cal L}^{(N-1)^{-1}}
\otimes{\cal N}_y^{-1}) \}.$$
If $\gamma$ is the composed map
$$H^0(X,\Omega^1_X)\otimes {\cal O}_X \>>> \Omega^1_X \>>> \Omega^1_{X/Z}$$
then $$H^0(X,{\cal G}^b_\Phi\otimes {\cal L}^{(N-1)^{-1}}
\otimes{\cal N}_y^{-1}) \neq 0$$ implies that
$$\gamma(\bar{\Gamma}\otimes {\cal O}_X)\wedge\bar{\beta} = 0 \mbox{ \ for some \ }
\bar{\beta}\in\Omega^{b-\kappa}_{X/Z}\otimes\C (X)$$ or, in other terms, that
$$\Omega^{n-b}_{X/Z}\otimes\C (X) \neq \gamma(\bar{\Gamma}\otimes
{\cal O}_X)\wedge\Omega^{n-b-1}_{X/Z}\otimes\C (X).$$
Again this is only possible for
$$n-b \leq n-\kappa-\rank(\gamma(\bar{\Gamma}\otimes{\cal O}_X))$$
or
$$ b - \kappa \geq \rank(\gamma(\bar{\Gamma}\otimes{\cal O}_X)).$$
However,
$$\rank(\gamma(\bar{\Gamma}\otimes{\cal O}_X)) \geq \dim(\alpha(X)) -
\dim(\alpha'(Z)) - \codim(\bar{\Gamma}\subset H^0(X,\Omega^1_X))$$
and (13.10,c) follows from (12.7)\\
\end{Beweis}
\begin{defn}{Remarks}\\
a) If $Z(\varphi)$ denotes the zero locus of a global one-form $\varphi$
and $$ w(X) = \mbox{Max} \{ \codim_X Z(\varphi); \varphi \in H^0(X,\Omega^1_X)\},$$
then a second result of Green and Lazarsfeld \cite{GL87} says that, for a
generic line bundle $${\cal N} \in \Pic^0(X) \mbox{ \  and \ } a+b < w(X),$$ one has
$$H^b(X,\Omega^a_X\otimes{\cal N}) = 0.$$
b) In \cite{GL91} Green and Lazarsfeld obtain moreover
a more explicit description of the subvarieties $S^b(X)$ of $\Pic^0(X)$.
They show that the irreducible components of $S^b(X)$ are
translates of subtori of $\Pic^0(X)$.
This description generalizes results due to A. Beauville \cite{Bea88},
who studied $S^1(X)$  and showed the same result in this case.\\
\ \\
c) Finally, C. Simpson recently gave in \cite{Sim92} a complete description
of the $S^b(X)$  and similar ``degeneration loci''. In particular he showed
that the components of $S^b(X)$ are even translates of subtori of
$\Pic^0(X)$ by points of finite order, a result conjectured and proved for
$b=1$ by A. Beauville.\\
\ \\
d) Writing these notes we would have liked to prove the generic
vanishing theorems for invertible sheaves in the algebraic language
used in the first part. However, we were not able to replace the use of Hodge duality
by some algebraic argument.
\end{defn}
\newpage
\appendix{Hypercohomology and spectral sequences}
\ \\
\Abschnitt{} In this appendix, we list some formal properties
of cohomology of complexes that we are using throughout
these notes. However we do not pretend making a complete account
on this topic. In particular, we avoid the use of the derived category, which is
treated to a broad extend in the literature (see \cite{Ver77}, \cite{Har66},
\cite{Bor84}, \cite{Bor87}, \cite{Ive86}, \cite{Ill90}).
\Abschnitt{} Through this section $X$ is a variety over a commutative
ring $k$.
\Abschnitt{} We consider {\it complexes} ${\cal F}^{\bullet}$ {\it of sheaves}
of ${\cal O}$-modules, where ${\cal O}$ is a sheaf of commutative rings.
For example $${\cal O} = \Z \mbox{ , } {\cal O} = k \mbox{ \ or \ }
{\cal O} = \mbox{ structure sheaf of }X.$$
Any map of ${\cal O}$-modules
$$\sigma : {\cal F}^{\bullet} \>>> {\cal G}^{\bullet}$$
between two such complexes induces a map of
{\it cohomology sheaves}:
\[ {\cal H}^{i} (\sigma) : {\cal H}^{i} ({\cal F}^{\bullet}) \>>>
{\cal H}^{i} ({\cal G}^{\bullet}) \]
where ${\cal H}^{i}({\cal F}^{\bullet})$ is the sheaf associated to the
presheaf
\[ U \mapsto \frac{\ker \ \Gamma (U, {\cal F}^{i}) \rightarrow \Gamma (U,
{\cal F}^{i+1})}{\im \ \Gamma (U, {\cal F}^{i-1}) \rightarrow \Gamma
(U, {\cal F}^{i})} \]
in the given topology.\\
One says that $\sigma$ is a {\it quasi-isomorphism} if ${\cal H}^{i} (\sigma)$
is an isomorphism for all $i$.\index{Quasi-isomorphism}
\Abschnitt{} We will only consider complexes ${\cal F}^{\bullet}$ which are
{\it bounded below}, that means ${\cal F}^{i} = 0$ for $i$ sufficiently negative.
\\
\Abschnitt{Example: the analytic de Rham complex.}
$X$ is a complex manifold. Then the standard map
\index{Analytic de Rham complex}
$$\C \>>> \ \big( \ {\cal O}_X \rightarrow \Omega^{1}_{X}
\rightarrow \Omega^{2}_{X}
\rightarrow \Omega^{3}_{X} \rightarrow \cdots \ \big)$$
from the constant sheaf $\C$ to the analytic de Rham
complex $\Omega^{\bullet}_{X}$ is a quasi-isomorphism as by the so called
``Poincar\'{e} lemma'' $${\cal H}^{i} (\Omega^{\bullet}_{X}) = 0
\mbox{ \ for \ } i > 0 \mbox{, and } {\cal H}^{0} (\Omega^{\bullet}_{X}) = \C.
$$
\Abschnitt{Example: the \v{C}ech complex.}
\index{\v{C}ech complex}
Let ${\cal U} = \{ U_\alpha ;\alpha \in A \}$, for
$A \subset \N$, be some open
covering of the variety $X$ defined over $k$. To a bounded below complex
${\cal F}^{\bullet}$ one associates its \v{C}ech complex ${\cal G}^{\bullet}$
defined as follows.
\[ {\cal G}^{i} : = \bigoplus_{a \geq 0} {\cal C}^{a} ({\cal U},
{\cal F}^{i-a}) \]
where \[{\cal C}^{a} ({\cal U}, {\cal F}^{i-a}) =
\prod_{\alpha_{0} < \alpha_{1} < \ldots < \alpha_{a}} \varrho_*
{\cal F}^{i-a}|_{U_{\alpha_{0} \ldots \alpha_{a}}}. \]
Here, for any $$U_{\alpha_{0} \ldots \alpha_{a}} : = U_{\alpha_{0}} \cap \ldots
\cap U_{\alpha_a}$$
$\varrho$ denotes the open embedding
\[ \varrho = \varrho_{\alpha_{0} \ldots \alpha_{a}} :
U_{\alpha_{0} \ldots \alpha_{a}} \>>> X, \]
and for any sheaf ${\cal F}$, one writes  $\varrho_{*} {\cal F}$ for the sheaf
associated to the presheaf
$$U \mapsto \Gamma (U_{\alpha_{0} \ldots \alpha_{a}}
\cap U, {\cal F}).$$
As ${\cal F}^{i} = 0$ for $i << 0$, the direct sum in the definition of
${\cal G}$ has finitely many summands.\\
\ \\
The differential $\Delta$ of ${\cal G}^{\bullet}$ is defined by
\[ \Delta (s) = (-1)^{i} \delta s + d_{{\cal F}^{\bullet}} s \mbox{ \ for \ }
s \in {\cal C}^{a} ({\cal U}, {\cal F}^{i-a}),\]
where $\delta$ is the \v{C}ech differential defined by
\[ (\delta s)_{\alpha_{0} \ldots
\alpha_{a+1}} = \sum^{a+1}_{0} (-1)^{l} s_{\alpha_{0} \ldots
\hat{\alpha_{l}} \ldots \alpha_{a+1}}|_{U_{\alpha_{0} \ldots
\alpha_{a+1}}} \]
and $d_{{\cal F}^{\bullet}}$ is the differential of ${\cal F}^{\bullet}$.
Then the natural map
\[ \sigma : {\cal F}^{\bullet} \>>> {\cal G}^{\bullet} \]
defined by
\[ {\cal F}^{i} \> \varrho >> \prod_{\alpha \in A} \varrho_{*}
{\cal F}^{i}|_{U_{\alpha}} = {\cal C}^{0} ({\cal U}, {\cal F}^{i}) \]
is a quasi-isomorphism.

To show this one considers first a single sheaf ${\cal F}$
(see \cite{Har77}, III 4.2)
and then one computes that, whenever one has a double complex
\ \\
\[
\begin{CD}
\noarr \A A A  \A A A  \A A A \noarr\\
\>>> {\cal K}^{1, i - 1} \>>> {\cal K}^{1,i} \>>> {\cal K}^{1,i+1} \>>> \\
\noarr \A A A  \A A A  \A A A   \A A d_{vert} A \\
\>>> {\cal K}^{0,i-1} \>>> {\cal K}^{0,i} \>>> {\cal K}^{0, i+ 1} \>>> \\
\noarr \A A A  \A A A  \A A A \noarr\\
\>>> {\cal F}^{i-1} \>>> {\cal F}^{i} \>>> {\cal F}^{i+1} \>>> \ldots
\>> d_{hor} >
\end{CD}
\]
\ \\
such that ${\cal F}^{i} \>>> {\cal K}^{\bullet, i}$ is a
quasi-isomorphism for all $i$, then ${\cal F}^{\bullet} \>>>
{\cal D}^{\bullet}$ is a quasi-isomorphism as well, where ${\cal D}^{\bullet}$
is the associated {\it double complex}:
\[ {\cal D}^{i} : = \bigoplus_{a} {\cal K}^{a, i-a}
\>>> {\cal D}^{i+1}
: = \bigoplus_{a} {\cal K}^{a, i+1 -a} \]
with differential $(-1)^{i} d_{vert} + d_{hor}$.
\Abschnitt{} If ${\cal F}^{\bullet}$ and ${\cal G}^{\bullet}$ are
two complexes of ${\cal O}$-modules bounded below one defines the {\it tensor
product} ${\cal F}^{\bullet} \otimes {\cal G}^{\bullet}$ by
\index{Tensor product of complexes}
\[ ({\cal F}^{\bullet} \otimes {\cal G}^{\bullet})^{i}: = \bigoplus_{a}
{\cal F}^{a} \otimes {\cal G}^{i-a} \]
with differential from
\[ {\cal F}^a \otimes {\cal G}^{i-a} \mbox{ \ to \ }
{\cal F}^{a+1} \otimes {\cal G}^{i-a} \oplus {\cal F}^{a} \otimes
{\cal G}^{i+1-a}. \mbox{ \ given by \ }\]
\[ d( f_{a} \otimes g_{i-a}) = df_{a} \otimes g_{i-a} + (-1)^{a} f_{a} \otimes
dg_{i-a}. \]
As both ${\cal F}^{\bullet}$ and ${\cal G}^{\bullet}$ are bounded below, $a$ takes
finitely many values, and $({\cal F}^{\bullet} \otimes {\cal G}^{\bullet})$ is
a complex of ${\cal O}$-modules bounded below.
\Abschnitt{} If in 7 we assume moreover that {\it locally} the
${\cal O}$-modules ${\cal F}^{a}$ and ${\cal G}^{b}$ are {\it free}
and $d {\cal F}^{a-1}$ as well as $d {\cal G}^{b-1}$ are subbundles
for all $a$ and $b$, then locally one has some decomposition
\[
\begin{array}{lllllll}
{\cal F}^{a} & = & d {\cal F}^{a-1} & \oplus & {\cal H}^{a} ({\cal F}^{\bullet})
& \oplus & {\cal F}^{\prime a} \\
{\cal G}^{b} & = & d {\cal G}^{b-1} & \oplus & {\cal H}^{b} ({\cal G}^{\bullet})
& \oplus & {\cal G}^{\prime b}
\end{array}
\]
where
\[
\begin{array}{llll}
d: & {\cal F}^{\prime a} & \>>> & d {\cal F}^{a}\\
d: & {\cal G}^{\prime b} & \>>> & d {\cal G}^{b}
\end{array}
\]
are isomorphisms.
In particular,
\[
\begin{array}{lll}
{\cal F}^{a} \otimes {\cal G}^{b} & = & d({\cal F}^{a-1} \otimes d
{\cal G}^{b-1} + {\cal F}^{a-1} \otimes {\cal H}^{b} + {\cal H}^{a} \otimes
{\cal G}^{b-1})\\
& \oplus & {\cal H}^{a} ({\cal F}^{\bullet}) \otimes {\cal H}^{b}
({\cal G}^{\bullet}) \\
& \oplus & (d {\cal F}^{a-1} + {\cal H}^{a} ({\cal F}^{\bullet})) \otimes
{\cal G}^{\prime b} \\
& \oplus & {\cal F}^{\prime a} \otimes (d {\cal G}^{b-1} + {\cal H}^{b}
({\cal G}^{\bullet})) \\
& \oplus & {\cal F}^{\prime a} \otimes {\cal G}^{\prime b}
\end{array}
\]
and therefore one has the K\"unneth decomposition
\[ {\cal H}^{i} ({\cal F}^{\bullet} \otimes {\cal G}^{\bullet}) =
\bigoplus_{a} {\cal H}^{a} ({\cal F}^{\bullet}) \otimes
{\cal H}^{i-a} ({\cal G}^{\bullet}). \]
\Abschnitt{} The map $\sigma : {\cal F}^{\bullet} \>>>
{\cal I}^{\bullet}$ is\index{Injective resolution}
called an {\it injective resolution of} ${\cal F}^{\bullet}$ if
${\cal I}^{\bullet}$ is a complex of ${\cal O}$-modules bounded below,
$\sigma$ is a quasi-isomorphism, and the sheaves ${\cal I}^{i}$ are
injective for all $i$, i.e.:
$${\cal H}om_{\cal O} ({\cal B}, {\cal I}^{i})
\>>> {\cal H}om_{\cal O} ({\cal A}, {\cal I}^{i})$$
is surjective for any injective map ${\cal A}
\>>> {\cal B}$ of sheaves of ${\cal O}$-modules.
It is an easy fact that if ${\cal O}$ is a constant commutative ring, for
example ${\cal O} = \Z, \ {\cal O} = k$, then every complex of
${\cal O}$-modules which is bounded below admits an injective resolution
(see \cite{Ive86}, (6.1)).
\Abschnitt{} From now on we assume that ${\cal O}$ is a constant
commutative ring. Let ${\cal F}^{\bullet}$ be a complex of ${\cal O}$-modules,
bounded below. One defines the {\it hypercohomology group} $\h^{a} (X,
{\cal F}^{\bullet})$, \index{Hypercohomology group}\index{$\h^{a} (X, {\cal F}^{\bullet})$}
to be the ${\cal O}$-module
\[ \h^{a} (X, {\cal F}^{\bullet}) : = \frac{\ker \ \Gamma (X, {\cal I}^{a})
\rightarrow \Gamma (X, {\cal I}^{a+1})}{\im \ \Gamma
(X, {\cal I}^{a-1})
\rightarrow \Gamma (X, {\cal I}^{a})}. \]
One verifies that this definition does not depend on the injective
resolution choosen (see \cite{Har77} III, 1.0.8).\\
\ \\
In particular, if $\sigma : {\cal F}^{\bullet} \>>> {\cal G}^{\bullet}$
is a {\it quasi-isomorphism}, then $\sigma$ induces an {\it isomorphism
of the hypercohomology groups}:
\[ \h^{a} (X, {\cal F}^{\bullet}) \> \sim >> \h^{a} (X, {\cal G}^{\bullet}) \]
(by taking an injective resolution ${\cal I}^{\bullet}$ of ${\cal G}^{\bullet}$
which is also an injective resolution of ${\cal F}^{\bullet}$).
\Abschnitt{} By definition, $H^{a} (X, {\cal I}) = 0$ for $a > 0$ if
${\cal I}$ is an injective sheaf. We will verify in (A.28) that
if $\sigma : {\cal F}^{\bullet} \>>> {\cal G}^{\bullet}$ is a
quasi-isomorphism and if $H^{a} (X, {\cal G}^{i}) = 0$ for all $a > 0$
and all $i$, then
\[ \h^{a}(X, {\cal F}^{\bullet}) = \frac{\ker \ \Gamma (X, {\cal G}^{a})
\rightarrow \Gamma (X, {\cal G}^{a+1})}{\im \ \Gamma (X, {\cal G}^{a-1})
\rightarrow \Gamma (X, {\cal G}^{a})}. \]
We call $({\cal G}^{\bullet}, \sigma)$ an {\it acyclic resolution} of
${\cal F}^{\bullet}$ in this case.\index{Acyclic resolution}
\Abschnitt{} $\h$ transforms {\it short exact sequences}
\[ 0 \>>> {\cal A}^{\bullet} \>>> {\cal B}^{\bullet} \>>>
{\cal C}^{\bullet} \>>> 0 \]
of complexes of ${\cal O}$-modules which are bounded below into
{\it long exact sequences}
\[ \cdots \>>> \h^{i} ({\cal A}^{\bullet}) \>>> \h^{i}({\cal B}^{\bullet}) \>>>
\h^{i} ({\cal C}^{\bullet}) \>>> \h^{i+1} ({\cal A}^{\bullet})
\>>> \ldots \]
of ${\cal O}$-modules.
\Abschnitt{} We assume now that the complex ${\cal F}^{\bullet}$ of
${\cal O}$-modules, bounded below, has subcomplexes
\[
\begin{array}{ll}
& \ldots \subset Filt_{i-1} \ \subset Filt_{i} \subset \ldots \subset
{\cal F}^{\bullet} \\
\ \\
(\mbox{or} & \ldots \subset Filt^{i} \subset Filt^{i-1} \subset \ldots
\subset {\cal F}^{\bullet})
\end{array}
\]
such that
\[
\begin{array}{ll}
& \bigcup_{i} \ Filt_{i} = {\cal F}^{\bullet}\\
\ \\
(\mbox{or} & \bigcup_{i} \ Filt^{i} = {\cal F}^{\bullet})
\end{array}
\]
and such that
\[
\begin{array}{ll}
& Filt_{i} = 0 \ \ \mbox{for} \ \ i << 0\\
\ \\
(\mbox{or} & Filt^{i} = 0 \ \ \mbox{for} \ \ i \gg 0).
\end{array}
\]
One says that ${\cal F}^{\bullet}$ is {\it filtered} by the subcomplexes
$Filt_{i}$ (or $Filt^{i}$). Via (A.12), this filtration defines a
{\it filtration on the hypercohomology groups}:
\index{Filtrations!on hypercohomology groups}
\[
\begin{array}{rl}
Filt_{i} \h^{a} (X, {\cal F}^{\bullet}): = & \im ( \h^{a} (X, Filt_{i})
\>>> \h^{a} (X, {\cal F}^{\bullet})) \\
\ \\
(\mbox{or} \ \ Filt^{i} \h^{a} (X, {\cal F}^{\bullet}): = & \im
( \h^{a} (X, Filt^{i}) \>>> \h^{a} (X, {\cal F}^{\bullet}))).
\end{array}
\]
This just means that the group $\h^{a} (X, {\cal F}^{\bullet})$ has
subgroups
\[
\begin{array}{ll}
& \ldots \subset Filt_{i-1} \h^{a} (X, {\cal F}^{\bullet}) \subset Filt_{i}
\h^{a} (X, {\cal F}^{\bullet}) \subset \ldots \subset \h^{a} (X,
{\cal F}^{\bullet})\\
\ \\
(\mbox{or} & \ldots \subset Filt^{i} \h^{a} (X, {\cal F}^{\bullet}) \subset
Filt^{i-1} \h^{a} (X, {\cal F}^{\bullet}) \subset \ldots \subset
\h^{a} (X, {\cal F}^{\bullet})).
\end{array}
\]
We pass from {\it increasing} to {\it decreasing} filtrations by setting
\[ Filt_{i}: = Filt^{-i}. \]
\Abschnitt{} We define
\[ Gr_{i}: = Filt_{i}/Filt_{i-1}. \]
One has a diagram of exact sequences
\ \\
\[
\minCDarrowwidth=0.8cm
\begin{CD}
\noarr \noarr 0  \noarr 0 \noarr \noarr\\
\noarr \noarr \V V V   \V V V \\
\noarr 0 \>>> Filt_{i-1} \>>> Filt_{i} \>>> Gr_{i} \>>> 0\\
\noarr \noarr \V V V    \V V V \\
\noarr \noarr {\cal F}^{\bullet}  \> \mbox{identity} >>  {\cal F}^{\bullet}\\
\noarr \noarr \V V V   \V V V \\
0 \>>> Gr_{i} \>>> {\cal F}^{\bullet}/Filt_{i-1} \>>>
{\cal F}^{\bullet}/Filt_{i} \>>> 0\\
\noarr \noarr \V V V   \V V V \\
\noarr \noarr 0   \noarr 0
\end{CD}
\]
\ \\
which gives via (A.12):
\[
\begin{array}{rl}
Gr_{i} ( Filt_{\bullet} \h^{a} ({\cal F}^{\bullet})) \ := & Filt_{i}
\h^{a} ({\cal F}^{\bullet}) / Filt_{i-1} \h^{a} ({\cal F}^{\bullet})\\
\ \\
= & \mbox{\large{$\frac{\h^{a} (Filt_{i})}{\h^{a} (Filt_{i-1}) + \h^{a-1}
({\cal F}^{\bullet}/Filt_{i})}$}}\\
\ \\
= & \mbox{\large{$\frac{\ker ( \h^{a} (Gr_{i})
\rightarrow \h^{a+1} (Filt_{i-1}))}{\ker ( \h^{a}
(Gr_{i}) \rightarrow \h^{a} ({\cal F}^{\bullet}/Filt_{i-1}))}$}}.
\end{array}
\]
We define
\[
E_{\infty}^{a-i,i} \ :=  Gr_{i} ( Filt_{\bullet} \h^{a} ({\cal F}^{\bullet}))
\mbox{ \ and \ } E_{2}^{a-i,i} \ :=  \h^{a} (Gr_{i}).
\]
Obviously $E_{\infty}^{a-i,i}$ is a {\it subquotient} of $E_{2}^{a-i,i}$.
\Abschnitt{} The formations of {\it spectral sequences} has the aim to compute
$E_{\infty}^{a-i,i}$ only in terms of $E_{2}^{s,t}$, by filtering the terms $\h^{a+1} (Filt_{i-1})$ and $\h^{a}({\cal F}^{\bullet}/
Filt_{i-1})$ appearing in the description (A.14) by the induced filtrations:
\index{Spectral sequence}
\[ Filt_{l} \ \h^{a+1} (Filt_{i-1}): = \im ( \h^{a+1} (Filt_{l})
\>>> \h^{a+1}(Filt_{i-1}))\]
for $ \ l \leq i-1 $ and
\[ Filt_{l} \ \h^{a} ({\cal F}^{\bullet}/Filt_{i-1}):= \im ( \h^{a}
(Filt_{l}/Filt_{i-1}) \>>> \h^{a} ({\cal F}^{\bullet}/Filt_{i-1})) \]
for $ \ l \geq i-1,$ and by computing the corresponding graded quotients.
\Abschnitt{} If we assume that for some $a$, the filtration $Filt_{i}
\h^{a} ({\cal F}^{\bullet})$ is {\it exhausting}, that is:
\[ \bigcup_{i} Filt_{i} \h^{a} ({\cal F}^{\bullet}) = \h^{a}
({\cal F}^{\bullet}), \]
then
$$Gr \ \h^{a} ({\cal F}^{\bullet}) = \bigoplus_{i} E^{a-i,i}_{\infty}$$
is the corresponding graded group. In particular, assume that $\h^{a}
({\cal F}^{\bullet})$ and $E_{\infty}^{a-i,i}$ are {\it free ${\cal O}$-modules}
(where ${\cal O}$ is $\Z$ or $k$), and that $\h^{a} ({\cal F}^{\bullet})$
is of finite rank. Then
\[ \rank_{{\cal O}} \h^{a} ({\cal F}^{\bullet}) =
\sum_{i} \ \rank_{{\cal O}} E_{\infty}^{a-i,i}. \]
If $E_{2}^{a-i,i}$ is also free, then $\rank_{{\cal O}} \h^{a}
({\cal F}^{\bullet}) \leq \sum_{i} \ \rank_{{\cal O}} E_{2}^{a-i,i}$ and one
has equality if and only if
\[ E_{\infty}^{a-i,i} = E_{2}^{a-i,i} \ \ \mbox{for all} \ \ i. \]
\Abschnitt{Example: One step filtration:}
\index{One step filtration}
\[ 0 = Filt_{s-1} \subset Filt_{s} = {\cal F}^{\bullet}. \]
Then $\h^{a} ({\cal F}^{\bullet}) = \h^{a} (Gr_{s}) = E_{\infty}^{a-s,s}
= E_{2}^{a-s,s}$.\index{$E_{\infty}^{a-s,s}$}\index{$E_{2}^{a-s,s}$}
\Abschnitt{Example: Two steps filtration:}
\index{Two step filtration}
\[ 0 = Filt_{s-2} \subset Filt_{s-1} \subset Filt_{s} = {\cal F}^{\bullet} \]
Then one has the exact sequence
\[ 0 \>>> Gr_{s-1} \>>> {\cal F}^{\bullet} \>>>
Gr_{s} \>>> 0 \]
and
\[ 0 \>>> E_{\infty}^{a - (s-1), (s-1)} \>>> \h^{a}
({\cal F}^{\bullet}) \>>> E_{\infty}^{a-s,s} \>>> 0 \]
with
\[
\begin{array}{ll}
E_{\infty}^{a-(s-1), (s-1)} & = \h^{a} (Gr_{s-1})/\h^{a-1} (Gr_{s})\\
\ \\
E_{\infty}^{a-s,s} & = \ker \ \h^{a} (Gr_{s}) \>>> \h^{a+1} (Gr_{s-1})
\end{array}
\]
\Abschnitt{}
Define the differential $d_2$ as the connecting morphism of the
vertical exact sequence on the right hand side (or of the above
horizontal exact sequence) of the diagram
\[ \ \\
\minCDarrowwidth{0.9cm}
\begin{CD}
\noarr \noarr 0 \noarr 0 \noarr \noarr \\
\noarr \noarr \V V V  \V V V  \noarr \noarr \\
0 \>>> Gr_{i-1} \>>> Filt_{i}/Filt_{i-2} \>>> Gr_{i} \>>> 0\\
\noarr \|\noarr  \V V V   \V V V \\
0 \>>> Gr_{i-1} \>>> Filt_{i+1}/Filt_{i-2} \>>> Filt_{i+1}/Filt_{i-1}
\>>> 0\\
\noarr \noarr \V V V  \V V V  \noarr \\
\noarr \noarr Gr_{i+1}  \> \mbox{identity} >>  Gr_{i+1} \noarr\\
\noarr \noarr \V V V  \V V V  \noarr\\
\noarr \noarr 0 \noarr 0 \noarr
\end{CD}
\]
\ \\
and the ${\cal O}$-module $ E_{3}^{a-i,i}$ by
\[ E_{3}^{a-i,i} = \frac{\ker \ E_{2}^{a-i,i} \> d_{2} >> E_{2}^{a-i+2, i-1}}
{\im \ E_{2}^{a-i-2,i+1} \> d_{2} >> E_{2}^{a-i,i}}. \]
For $\varepsilon = 1,2$ and the $\varepsilon$-steps filtration one has
\[ E_{\infty}^{a-i,i} = E_{\varepsilon +1}^{a-i,i}, \]
as we saw in (A.17) and (A.18), and the filtration on $\h^{a}
({\cal F}^{\bullet})$ is exhausting.
\Abschnitt{}
The right vertical exact sequence of the diagram (A.19) gives a surjection
$$\h^a(Filt_{i+1}/Filt_{i-1}) \> \gamma >> \ker(\h^a(Gr_{i+1}) \rightarrow
\h^{a+1}(Gr_{i})) $$
and the middle horizontal sequence induces a morphism
$$\ker(\h^a(Gr_{i+1}) \rightarrow
\h^{a+1}(Gr_{i})) \> d'_3 >> \h^{a+1}(Gr_{i-1})/\im \ \h^a(Gr_{i})$$
Replacing $i$ by $i-1$, one obtains
maps
$$\h^a(Filt_i/Filt_{i-2}) \> \gamma >> \ker(\h^a(Gr_i) \rightarrow
\h^{a+1}(Gr_{i-1})) \> d'_3 >> \frac{\h^{a+1}(Gr_{i-2})}{\im \ \h^a(Gr_{i-1})}$$
where $\gamma$ is surjective. As the extension
$$0 \>>> Gr_{i-2} \>>> Filt_{i}/Filt_{i-3} \>>> Filt_i/Filt_{i-2} \>>>0$$
lifts to an extension
$$0 \>>> Filt_{i-2}/Filt_{i-4} \>>> Filt_{i}/Filt_{i-4} \>>> Filt_i/Filt_{i-2}
\>>> 0 $$
the image of $d'_3 \circ \gamma$ lies in fact in
\[ \frac{ \ker \ \h^{a+1}(Gr_{i-2}) \rightarrow \h^{a+2}(Gr_{i-3})}
{\im \ \h^a(Gr_{i-1}) \rightarrow \h^{a+1}(Gr_{i-2})} = E_3^{a-i+3,i-2} \ ,\]
as well as the image of $d'_3$. The map $d'_3$ factorizes through $E_3^{a-i,i}$.
The resulting map is written as
\[ d_3 : E_3^{a-i,i} \>>> E_3^{a-i+3,i-2}. \]
One defines
\[ E_4^{a-i,i} = \frac{ \ker \ E_3^{a-i,i} \> d_3 >> E_3^{a-i+3,i-2}}
{ \im \ E_3^{a-i-3,i+2} \> d_3 >> E_3^{a-i,i}} \ . \]
By construction a class in $E_3^{a-i,i}$ may be represented by a class
in
$$\h^a(Filt_i/Filt_{i-2})$$
and similarly a class in $E_4^{a-i,i}$ may be represented by a class in
$$\h^a(Filt_i/Filt_{i-3}).$$
More generally, one defines inductively in the same vein differentials
\[ E_{r}^{a-i,i} \> d_{r} >> E_{r}^{a-i+r, i-r+1}, \]
and ${\cal O}$-modules:
\[ E_{r+1}^{a-i,i} : = \frac{\ker \ E_{r}^{a-i,i} \> d_{r} >>
E_{r}^{a-i+r, i-r+1}}{\im \ E_{r}^{a-i-r, i+r-1} \> d_{r} >> E_{r}^{a-i,i}} \]
which are subquotients of $E_{r}$. A class in $E_{r+1}^{a-i,i}$ is represented
by a lifting to $\h^a(Filt_i/Filt_{i-r})$.\\
\ \\
When $Filt_{\sigma - 1} = 0,$ and $Filt_{\sigma } \neq 0,$ one defines
the induced filtration on $Filt_{\sigma + \rho}$ by
\[
Filt_{l} \ Filt_{\sigma + \rho} = \left\{
\begin{array}{ll}
Filt_{l} & \mbox{if} \ \ l \leq \sigma+ \rho \\
Filt_{\sigma + \rho} & \mbox{if} \ \ l \geq \sigma + \rho \ .
\end{array} \right.
\]
Then one has:
\[ E_{\infty}^{a-i,i} (\h^{a} (Filt_{\sigma + \varrho})) =
E_{\varrho + 2}^{a-i,i} \]
and the filtration on $\h^{a} (Filt_{\sigma + \varrho})$ is exhausting.
\Abschnitt{}
This shows that in general, for going from $\h^{a} (Filt_{\sigma + \varrho})$ to
$\h^{a} ({\cal F}^{\bullet})$
one has to introduce infinitely many $r$
and groups $E_{r}^{a-i,i}$,
a reason for the notation $E^{a-i,i}_{\infty}$. \\
\ \\
One says that the
{\it spectral sequence} with $E_{2}$ term $E_{2}^{a-i,i}$ and $d_{2}$
differential $d_{2}$ (simply noted $(E_{2}^{a-i,i}, d_{2})$)
{\it degenerates} in $E_{r}$ if:\index{Degeneration!of spectral sequences}\\
\ \\
The filtration on $\h^{a} ({\cal F}^{\bullet})$ is exhausting and
$E_{\infty}^{a-i,i} = E_{r}^{a-i,i}$ for all $i$, or equivalently
$d_{r+l} = 0$ for all $l \geq 0$.\\
\ \\
With this terminology, we have seen in (A.20) that a $(\varrho + 1) $-steps
filtration defines an $E_{2}$ spectral sequence which degenerates in
$E_{\varrho +2}$.
\Abschnitt{} Under the assumptions of (A.16), assume moreover that $E_{r}^{a-i,i}$
is also a free ${\cal O}$-module (for example if ${\cal O}$ is a field).
Then one has:
\[ \rank_{\cal O} \ \h^{a} ({\cal F}^{\bullet}) \leq \sum_{i} \ \rank_{\cal O}
\ E_{r}^{a-i,i} \]
for all $r$, and the spectral sequence degenerates in $E_{r}$
if and only if this is an equality.
\Abschnitt{} By (A.19), to say that the spectral {\it sequence degenerates}
in $E_{2}$ means that for all $i$, one has an exact sequence
\[ 0 \>>> \h^{a} (Filt_{i-1}) \>>> \h^{a} (Filt_{i}) \>>> \h^{a}
(Gr_{i}) \>>> 0, \]
and of course that the filtration is exhausting.
\Abschnitt{} One says that the spectral sequence $(E_{2}, d_{2})$ {\it converges}
to $\h^{a}({\cal F}^{\bullet})$ if it degenerates in $E_{r}$ for some $r$.
We sometimes write
\[ (E^{a-i,i}_{2}, d_{2}) \Longrightarrow \h^{a} ({\cal F}^{\bullet}) \]
instead of ``$(E_{2}, d_{2})$ converges to $\h^{a} ({\cal F}^{\bullet})$''.
\Abschnitt{The Hodge to de Rham spectral sequence.} On ${\cal F}^{\bullet}$,
a complex of ${\cal O}$-modules, bounded below, we define the {\it Hodge
filtration} (often called the stupid filtration) by:
\[ Filt^{i} {\cal F}^{\bullet} = {\cal F}^{\geq i} = Filt_{-i}
{\cal F}^{\bullet} \]
where
\[
({\cal F}^{ \geq i})^{l} =
\left\{
\begin{array}{ll}
0 & \mbox{if} \ \ l < i\\
{\cal F}^{l} & \mbox{if} \ \ l \geq i.
\end{array} \right.
\]
Then $Gr_{-i} = Filt_{-i}/Filt_{-i-1} = {\cal F}^{i} [-i]$ where $[\alpha ]$
means:
\[ ({\cal F}^{\bullet} [ \alpha ])^{l} = {\cal F}^{l + a}. \]
This is the so called {\it shift} by $\alpha$ to the right.
The $E_{2}$ spectral sequence
reads:
\[
E_{2}^{a-i,i} = \h^{a} (Gr_{i})  = \h^{a} ({\cal F}^{-i} [i])
= H^{a+i} ({\cal F}^{-i})
\]
where the differential $d_{2}$ goes to
\[
\h^{a+1} (Gr_{i-1}) = \h^{a+1} ({\cal F}^{-(i-1)} [i-1])
= H^{a+i} ({\cal F}^{-i+1})
\]
and is just induced by the differential in the complex.\\
\ \\
This spectral sequence is usually rewritten as an $E_{1}$ spectral
sequence by setting
\[
E_{1}^{-i,a+i}  =  E_{2}^{a-i,i}
\mbox{ \ or \ } E_{1}^{\alpha, \beta}  =  E_{2}^{\beta + 2 \alpha , - \alpha}
\]
with differentials:
\[
\begin{CD}
E_{2}^{\beta + 2 \alpha , - \alpha} \> d_{2} >> E_{2}^{\beta + 2 \alpha + 2,
- \alpha - 1}\\
\| \noarr  \| \\
E_{1}^{\alpha, \beta} \> d_{1} >> E_{1}^{\alpha + 1, \beta}
\end{CD}
\]
The $E_1$ spectral sequence obtained\index{Spectral sequence!Hodge to de Rham}
is called the Hodge to de Rham spectral sequence, at least when
${\cal F}^{\bullet}$ is some de Rham complex on $X$, possibly with some
poles, possibly with non-trivial coefficients . . . \\
\ \\
For a given $a$, one has
\[ \h^{a} ({\cal F}^{\bullet}) = \h^{a} ({\cal F}^{\leq a + 1}) \]
where
\[
({\cal F}^{\leq i})^{l} = \left\{
\begin{array}{ll}
{\cal F}^{l} & \mbox{if} \ \ l \leq i\\
0 & \mbox{if} \ \ l > i.
\end{array} \right.
\]
In particular this is a finite complex, on which the Hodge filtration induces a
finite step filtration. Therefore the $E_{1}$ Hodge to de Rham spectral
sequence always converges.\\
\ \\
To say that it degenerates in $E_{1}$ means that for any $i$ one has exact
sequences
\[ 0 \>>> \h^{a} ({\cal F}^{\geq i + 1}) \>>> \h^{a} ({\cal F}^{\geq i})
\>>> H^{a-i} ({\cal F}^{i}) \>>> 0. \]
Putting those sequences together, one obtains exact sequences
\[ 0 \>>> \h^{a} ({\cal F}^{\geq i + j }) \>>> \h^{a} ({\cal F}^{\geq i})
\>>> \h^{a} ({\cal F}^{[ i, i+j )}) \>>> 0 \]
where
\[
({\cal F}^{[ i, i + j )})^{l} = \left\{
\begin{array}{ll}
{\cal F}^{l} & \mbox{if} \ \
l \in [ i, i + j )\\
0 & \mbox{if not}.
\end{array} \right.
\]
If moreover, one knows that $\h^{a} ({\cal F}^{\bullet})$ is a free
${\cal O}$-module of finite rank, as well as $H^{a-i} ({\cal F}^{i})$,
then the $E_{1}$ Hodge to de Rham spectral sequence degenerates in $E_{1}$
if and only if
\[ \rank_{\cal O} \ \h^{a} ({\cal F}^{\bullet}) = \sum_{i} \ \rank_{\cal O}
\ H^{a-i} ({\cal F}^{i}) . \]
\Abschnitt{The conjugate spectral sequence.}
On ${\cal F}^{\bullet}$, a complex of ${\cal O}$-modules bounded below,
we defined the $\tau$-filtration:
\[
(\tau_{\leq i} {\cal F}^{\bullet})^{l} = \left\{
\begin{array}{ll}
{\cal F}^{l} &  \mbox{for} \ \
l < i\\
\ker \ d & \mbox{for} \ \ l = i\\
0 & \mbox{otherwise.}
\end{array} \right.
\]
Then $Gr_{i} = {\cal H}^{i} [-i]$ is the cohomology sheaf in degree $i$.
The $E_{2}$ spectral sequence reads
\[ E_{2}^{a-i,i} = \h^{a} (Gr_{i}) = H^{a-i} ({\cal H}^{i}) \]
where the $d_{2}$ differential goes to
\[ \h^{a +1} (Gr_{i - 1}) = H^{a-i+2} ({\cal H}^{i-1}). \]
It is called the conjugate spectral sequence.\index{Spectral sequence!conjugate}
For a given $a$, one has
\[ \h^{a} ({\cal F}^{\bullet}) = \h^{a} (\tau_{\leq a + 1}
{\cal F}^{\bullet}). \]
However $\tau_{ \leq a + 1} {\cal F}^{\bullet}$ is a finite complex on which
the $\tau$-filtration induces a finite step filtration. Therefore the
$E_{2}$-conjugate spectral sequence always converges.
Furthermore, if $\sigma : {\cal F}^{\bullet} \>>> {\cal G}^{\bullet}$
is a {\it quasi-isomorphism} then $\sigma$ induces quasi-isomorphisms
$$\tau_{\leq i}{\cal F}^{\bullet} \>>> \tau_{\leq i}{\cal G}^{\bullet}$$
for all $i$, and therefore $\sigma$ induces an isomorphism
of the conjugate spectral sequences.

To say that the conjugate spectral sequence degenerates in $E_{2}$ means that
one has exact sequences
\[ 0 \>>> \h^{a} ( \tau_{\leq i - 1}) \>>> \h^{a} (\tau_{\leq i}) \>>>
H^{a- i} ({\cal H}^{i}) \>>> 0 \]
for all $i$.
If $\h^{a} ({\cal F}^{\bullet})$ is a free ${\cal O}$-module of finite rank, as
well as $\h^{a-i} ({\cal H}^{i})$, then the degeneration in $E_{2}$ is equivalent to
\[ \rank_{\cal O} \h^{a} ({\cal F}^{\bullet}) = \sum_{i} \ \rank_{\cal O} \
H^{a-i} ({\cal H}^{i}). \]
\Abschnitt{The Leray spectral sequence.} Let $f: X \>>> Y$
be a morphism between two $k$-varieties, and let ${\cal F}^{\bullet}$ be
a complex of ${\cal O}$-modules on $X$, bounded below.

Let ${\cal F}^{\bullet} \>>> {\cal I}^{\bullet}$ be an injective
resolution. We consider the {\it direct image functor} (already used and
defined but not named in 6):
\[ (f_{*} {\cal K})_{x} = \lim_{\scriptstyle \>>> \atop x \in U}
H^{0} (f^{-1} (U), {\cal K}) \]
for any ${\cal O}$-sheaf ${\cal K}$ on $X$ and any open set $U$ in $Y$.
In particular, by definition $H^{0} (X, {\cal K}) = H^{0} (Y,
f_{*} {\cal K})$. Therefore one has
\[ \h^{a} (X, {\cal F}^{\bullet}) = \frac{\ker \ \ H^{0} (Y, f_{*} {\cal I}^{a})
\>>> H^{0} (Y, f_{*} {\cal I}^{a+1})}{\im \ H^{0} (Y, f_{*} {\cal I}^{a-1})
\>>> H^{0} (Y, f_{*} {\cal I}^{a})}. \]
One verifies immediately that, by definition,
$f_{*} {\cal I}^{i}$ is an injective sheaf as well, which allows
to write (A.10):
\[ \h^{a} (X, {\cal F}^{\bullet}) = \h^{a} (Y, (f_{*} {\cal I}^{\bullet})), \]
where $f_{*}{\cal I}^{\bullet}$ is the complex
\[ (f_{*} {\cal I}^{\bullet})^{l} = f_{*} {\cal I}^{l}. \]
One considers the conjugate spectral sequence for $(f_{*} {\cal I}^{\bullet})$:
\[ E_{2}^{a-i,i} = H^{a-i} (Y, {\cal H}^{i} (f_{*} {\cal I}^{\bullet})). \]
One defines
\[ R^{i} f_{*} {\cal F}^{\bullet}: = {\cal H}^{i} (f_{*}
{\cal I}^{\bullet}). \]
By definition
\[
 \ \ \ \ \ \ \ \ \
(R^{i} f_{*}{\cal F}^{\bullet})_{x} : \ = \
\lim_{\scriptstyle \>>> \atop \scriptstyle x \in U} \
\mbox{\large{$\frac{\ker \ H^{0} (f^{-1}(U), {\cal I}^{i} )
\rightarrow H^{0} (f^{-1} (U), {\cal I}^{i+1})}{\im \ H^{0} (f^{-1}
(U), {\cal I}^{i-1}) \rightarrow H^{0} (f^{-1} (U), {\cal I}^{i})}$}} \]
\[ = \ \lim_{\scriptstyle \>>> \atop \scriptstyle x \in U} \
\h^{i} (f^{-1} (U), {\cal F}^{\bullet}) \]
In particular, $R^{i} f_{*} {\cal F}^{\bullet}$ does not depend on the
injective resolution chosen.\\
\ \\
The $E_{2}$ spectral sequence reads
\[ E_{2}^{a-i,i} = H^{a-i} (Y, R^{i} f_{*} {\cal F}^{\bullet}), \]
with $d_{2}$ differential to $H^{a-i + 2} (Y, R^{i-1} f_{*}
{\cal F}^{\bullet})$, and is called the {\it Leray spectral sequence for}
$f$.\index{Spectral sequence!Leray}

As the conjugate spectral sequence for $(f_{*} {\cal I}^{\bullet})$ converges
to $\h^{a} (f_{*} {\cal I}^{\bullet})$ (A.26), the
{\it Leray spectral sequence for $f$ always converges to} $\h^{a}
({\cal F}^{\bullet})$.
\\
In particular, if ${\cal F}$ is just a sheaf for which $R^{i} f_{*} {\cal F}
= 0$ for $i > 0$, one has:
\[ H^{a} (X, {\cal F}) = H^{a} (Y, f_{*} {\cal F}) \ \ \mbox{for all} \ \
a. \]
\Abschnitt{} Let ${\cal G}^{\bullet}$ be a complex of ${\cal O}$-modules,
bounded below, such that $H^{i} ({\cal G}^{j}) = 0$ for $i > 0$ and
all $j$. Then the $E_{1}$ Hodge to de Rham spectral sequence $E_{1}^{ij} =
H^{j} ({\cal G}^{i})$ degenerates in $E_{2}$ and one has
\[
\begin{array}{ll}
E_{\infty}^{i,0} = E_{2}^{i,0} & = \mbox{\large{$\frac{\ker \
H^{0} ({\cal G}^{i}) \rightarrow H^{0} ({\cal G}^{i+1})}{\im \
H^{0} ({\cal G}^{i-1}) \rightarrow H^{0} ({\cal G}^{i})}$}}\\
\ \\
& = \h^{i} ({\cal G}^{\bullet}) \\
\ \\
& = \h^{i} ({\cal F}^{\bullet}) \ \ \mbox{for any quasi-isomorphism} \ \ \
{\cal F}^{\bullet} \>>> {\cal G}^{\bullet}.
\end{array}
\]
\Abschnitt{} Take for ${\cal F}^{\bullet}$ a complex of quasi-coherent
sheaves (for example some de Rham complex). We consider a collection
of very ample Cartier divisors $D_{\alpha}$ with empty intersection,
such that the open covering of $X$ defined by $U_{\alpha} : =
X - D_{\alpha}$ consists of affine varieties. Then one has:
\[ H^{a} (X, \varrho_{*} {\cal F}^{j}|_{U_{\alpha_{0} \ldots \alpha_{i}}})
= H^{a} (U_{\alpha_{0} \ldots \alpha_{i}}, {\cal F}^{j}|_{U_{\alpha_{0} \ldots
\alpha_{i}}}) \]
for all $a$ where  $\varrho : U_{\alpha_{0} \ldots \alpha_{i}} \>>>
X$ is the natural embedding of the affine set $U_{\alpha_{0} \ldots
\alpha_{i}}$.
In fact, one has
\[
(R^{i} \varrho_{*} {\cal F}^{j} )_{x} \ = \
\lim_{\scriptstyle \>>> \atop \scriptstyle x \in V} \
H^{i} (V \cap U_{\alpha_{0} \cdots \alpha_{i}}, {\cal F}^{j}) \]
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \  = \ 0 \ \ \mbox{for} \ \  i > 0 $\\
\ \\
and one applies (A.27).
By (A.28) one obtains:
\[ \h^{a} (X, {\cal F}^{\bullet}) = \frac{\ker \ \oplus C^{i} ({\cal U},
{\cal F}^{a-i}) \rightarrow \oplus \ C^{i'}({\cal U},
{\cal F}^{a+1-i'})}{\im \ \oplus C^{i'} ({\cal U}, {\cal F}^{a-1-i'})
\rightarrow \oplus \ C^{i} ({\cal U}, {\cal F}^{a-i})} \]
where
\[
\begin{array}{ll}
C^{i} ({\cal U}, {\cal F}^{j}) & = H^{0} (X, {\cal C}^{i} ({\cal U},
{\cal F}^{j}))\\
\ \\
& = \bigoplus_{\alpha_{0} < \ldots < \alpha_{i}} H^{0} (U_{\alpha_{0} \ldots
\alpha_{i}}, {\cal F}^{j}).
\end{array}
\]
\ \\
\newpage
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\normalsize
\ \\ \ \\ \ \\ \ \\ \ \\ \ \\
\small
H\'{e}l\`{e}ne Esnault and Eckart Viehweg\\
Fachbereich 6, Mathematik\\
Universit\"at GH Essen\\
Universit\"atsstr. 3\\
D-W-4300 Essen1\\
Germany
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  \addcontentsline{toc}{kapitel}{{#1}}\else
  \addcontentsline{toc}{kapitel}{{#1}}\fi%
  {\parindent 0pt \hbox{\hbox to\titlindent{\Large\bof\thekapitel\hfil}%
  \Large\bof #2}\kapitelmark{#2}{}\par}\nobreak \vskip 2.3ex\@afterheading}
 
\def\@skapitel#1{{\parindent 0pt {\Large\bof #1}
\kapitelmark{#1}{}\par}\nobreak \vskip 2.3ex\@afterheading}

\def\Kapitel{\newpage\@afterindentfalse\secdef%
\@Kapitel\@sKapitel}
 
\def\@Kapitel[#1]#2{\ifnum \c@secnumdepth >\m@ne \refstepcounter{kapitel}
  \addcontentsline{toc}{kapitel}{{#1}}\else
  \addcontentsline{toc}{kapitel}{{#1}}\fi
  {\parindent 0pt {\large\bof\Kapitel@head~\thekapitel\par\vskip 6pt 
  \Large\bof #2}\kapitelmark{#2}{}\par}\nobreak \vskip 2.3ex\@afterheading}

\def\@sKapitel#1{{\parindent 0pt {\Large\bof #1} 
  \kapitelmark{#1}{}\par}\nobreak \vskip 2.3ex\@afterheading}
 
\def\Kapitel@head{Chapter}
 
\def\paragraph{\par\addvspace{4ex}\@afterindentfalse\secdef%
\@paragraph\@sparagraph} 

\def\@paragraph[#1]#2{\ifnum\c@secnumdepth>\m@ne 
  \refstepcounter{paragraph}
  \addcontentsline{toc}{paragraph} 
  {\protect\numberline{{\S\,\theparagraph}}{{#1}}}
  \else\addcontentsline{toc}{paragraph}{{#1}}\fi% 
  {\parindent 0pt \hbox{\hbox to\titlindent{\large\bof\theparagraph\hfil}%
  \large\bof #2}\paragraphmark{#1}\par}\nobreak \vskip 1.5ex\@afterheading} 

\def\@sparagraph#1{{\parindent 0pt {\large\bof #1}} 
  \paragraphmark{#1}\par\nobreak \vskip 1.5ex\@afterheading}
 
\def\Paragraph{\par\addvspace{4ex}\@afterindentfalse\secdef%
\@Paragraph\@sParagraph} 

\def\@Paragraph[#1]#2{\ifnum\c@secnumdepth>\m@ne 
   \refstepcounter{paragraph}\else\fi%
  {\parindent 0pt  
  \hbox{\hbox to\titlindent{\large\bof\S\,\theparagraph\hfil}\large\bof #2}
  \par}\nobreak \vskip 1.5ex\@afterheading 
   \ifnum\c@secnumdepth>\m@ne
  \addcontentsline{toc}{paragraph} 
  {\protect\numberline{{\S\,\theparagraph}}{{#1}}}
  \else\addcontentsline{toc}{paragraph}{{#1}}\fi\paragraphmark{#1}} 

\def\@sParagraph#1{{\parindent 0pt {\large\bof #1}} 
 \par\nobreak \vskip 1.5ex\@afterheading\paragraphmark{#1}}
 
% LParagraph f"ur lange Titel; diese m"ussen mit
%  \LParagraph{\parbox[t]{length}{titel1 \newline titel2}} gesetzt werden 

\def\LParagraph{\par\addvspace{4ex}\@afterindentfalse\secdef% 
\@LParagraph\@sLParagraph}
 
\def\@LParagraph[#1]#2{\ifnum\c@secnumdepth>\m@ne
  \refstepcounter{paragraph}\fi% 
  {\parindent 0pt
  \hbox{\hbox to\titlindent{\large\bof\S\,\theparagraph\hfil}\large\bof #2}} 
  \nobreak \vskip 1.5ex\@afterheading}
 
\def\@sLParagraph#1{{\parindent 0pt {\large\bof #1}}
  \par\nobreak \vskip 1.5ex\@afterheading} 

 
\def\@empty{}
\def\abschnitt{\par\addvspace{1.2ex}\secdef% 
  \@abschnitt\@sabschnitt}
 
\def\@abschnitt[#1]#2{\def\tit{#2}\ifnum\c@secnumdepth>\m@ne
  \refstepcounter{abschnitt} 
  %\addcontentsline{toc}{abschnitt}{{\theabschnitt.~#1}}    % Probleme falls \cite
  \else\addcontentsline{toc}{abschnitt}{{#1}}\fi%                % in Titel 
  {\parindent 0pt \ifx\tit\@empty{\bof\theabschnitt. }\ignorespaces
  \else {\bof\theabschnitt.~#2. }\ignorespaces\fi}\ignorespaces} 

\def\@sabschnitt#1{\def\tit{#1}{\ifx\tit\@empty\noindent\ignorespaces 
  \else{\parindent 0pt {\bof #1. }}\fi}\ignorespaces}
 
\def\Abschnitt{\par\addvspace{2ex}\@afterindentfalse\secdef%
  \@Abschnitt\@sAbschnitt} 

\def\@Abschnitt[#1]#2{\ifnum\c@secnumdepth>\m@ne 
  \refstepcounter{abschnitt}
  \addcontentsline{toc}{abschnitt}{{\theabschnitt.~#1}}
  \else\addcontentsline{toc}{abschnitt}{{#1}}\fi% 
  {\parindent 0pt \hbox{\bof\theabschnitt.~#2}}
  \par\nobreak \vskip 4pt \@afterheading}

\def\@sAbschnitt#1{{\parindent 0pt {\bof #1}} 
  \par\nobreak \vskip 4pt\@afterheading}

\setcounter{secnumdepth}{3} 

 
%\def\appendix{\par
% \setcounter{kapitel}{0} 
% \setcounter{paragraph}{0}
% \setcounter{abschnitt}{0} 
% \def\theparagraph{\Alph{paragraph}}}
 
\def\appendix#1{\par\addvspace{2ex}
 \setcounter{kapitel}{0} 
 \setcounter{paragraph}{0}
 \setcounter{abschnitt}{0} 
 \def\theparagraph{A\arabic{paragraph}}
 \addcontentsline{toc}{paragraph}{APPENDIX: {#1}} 
 \paragraph*{APPENDIX: #1}}
 
\def\Appendix#1{\par\addvspace{2ex}
 \setcounter{kapitel}{0} 
 \setcounter{paragraph}{0}
 \setcounter{abschnitt}{0} 
 \def\theparagraph{A\arabic{paragraph}}
 \addcontentsline{toc}{paragraph}{APPENDIX: {#1}} 
 \paragraph*{APPENDIX:\newline{#1}}}
 

\leftmargini 25pt \leftmarginii 20pt  
\leftmarginiii 18.7pt
\leftmarginiv 17pt \leftmarginv 10pt \leftmarginvi 10pt 

\leftmargin\leftmargini 
\labelwidth\leftmargini\advance\labelwidth-\labelsep
\labelsep 5pt 
\topsep 2pt plus 0.5pt minus 0.5pt
\itemsep \topsep  
\parsep \z@
\partopsep 2pt plus 1pt minus 1pt  

\def\@listi{\leftmargin\leftmargini} 
\def\@listii{\leftmargin\leftmarginii
 \labelwidth\leftmarginii\advance\labelwidth-\labelsep 
 \topsep 2pt plus 0.5pt minus 0.5pt
 \parsep \z@ \partopsep 1pt plus 0pt minus 0.5pt 
 \itemsep \topsep}
 
\def\@listiii{\leftmargin\leftmarginiii
 \labelwidth\leftmarginiii\advance\labelwidth-\labelsep 
 \topsep 2pt plus 1pt minus 1pt
 \parsep \z@ \partopsep 1pt plus 0pt minus 1pt 
 \itemsep \topsep}
 
\def\@listiv{\leftmargin\leftmarginiv
 \labelwidth\leftmarginiv\advance\labelwidth-\labelsep} 
\def\@listv{\leftmargin\leftmarginv
 \labelwidth\leftmarginv\advance\labelwidth-\labelsep} 
\def\@listvi{\leftmargin\leftmarginvi
 \labelwidth\leftmarginvi\advance\labelwidth-\labelsep} 

 

%    **************************************** 
%    *                LISTS                 *
%    **************************************** 

\def\rom{\rm} 
\def\bof{\bf}
 
\def\labelenumi{\rom(\alph{enumi})}
\def\theenumi{\rom(\alph{enumi})} 

\def\labelenumii{\rom(\roman{enumii})} 
\def\theenumii{\rom(\roman{enumii})}
\def\p@enumii{\theenumi} 

\def\labelenumiii{\rom\arabic{enumiii}.} 
\def\theenumiii{\rom\arabic{enumiii}}
\def\p@enumiii{\theenumi(\theenumii)} 

\def\labelenumiv{\rom\Alph{enumiv}.} 
\def\theenumiv{\rom\Alph{enumiv}}
\def\p@enumiv{\p@enumiii\theenumiii} 

\def\labelitemi{$\bullet$} 
\def\labelitemii{\bf --}
\def\labelitemiii{$\ast$} 
\def\labelitemiv{$\cdot$}
 
\def\verse{\let\\=\@centercr
  \list{}{\itemsep\z@ \itemindent -1.5em\listparindent \itemindent 
         \rightmargin\leftmargin\advance\leftmargin 1.5em}\item[]}
\let\endverse\endlist 

\def\quotation{\list{}{\listparindent 1.5em 
    \itemindent\listparindent
    \rightmargin\leftmargin \parsep 0pt plus 1pt}\item[]} 
\let\endquotation=\endlist
 
\def\quote{\list{}{\rightmargin\leftmargin}\item[]}
\let\endquote=\endlist 

\def\descriptionlabel#1{\hspace\labelsep \bf #1} 
\def\description{\list{}{\labelwidth\z@ \itemindent-\leftmargin
       \let\makelabel\descriptionlabel}} 

\let\enddescription\endlist 

 
%    ****************************************
%    *         OTHER ENVIRONMENTS           * 
%    ****************************************
 
\def\endsatz{\endtrivlist\addvspace{0.5ex plus 0.2ex minus 0.1ex}}
\def\enddefn{\endtrivlist\addvspace{0.5ex plus 0.2ex minus 0.1ex}} 
\def\endbeweis{\hspace*{1em}\nobreak\qed
   \endtrivlist\addvspace{0.5ex plus 0.2ex minus 0.1ex}} 
\def\endBeweis{\hspace*{1em}\nobreak\qed
   \endtrivlist\addvspace{1ex plus 0.2ex minus 0.1ex}} 
\expandafter\def\csname endsatz*\endcsname{\endsatz}
\expandafter\def\csname enddefn*\endcsname{\enddefn} 
\def\endaufgabe{\endlist\addvspace{1.5ex plus 0.5ex minus 0.1ex}}
\def\endaufgaben{\endlist\addvspace{1.5ex plus 0.5ex minus 0.1ex}} 

\def\defn#1{\abschnitt{#1}} 
\expandafter\def\csname defn*\endcsname#1{\abschnitt*{#1}}
 
\def\satz#1{\defn{#1}\em\ignorespaces}
\expandafter\def\csname satz*\endcsname#1{\csname defn*\endcsname{#1}\em% 
\ignorespaces}
 
\def\aufgabe{\par\addvspace{2ex plus 0.5ex minus 0.1ex}
  \Abschnitt*{{\aufgabe@head}}\list{}{}\small\item[]} 

\def\aufgabe@head{Exercise} 

\newcounter{aufnum} 
\def\aufgaben{\par\addvspace{2ex plus 0.5ex minus 0.1ex}
  \Abschnitt*{{\aufgaben@head}}\paragraphmark{\aufgaben@head} 
  \list{\theparagraph.\arabic{aufnum}.}{\usecounter{aufnum}}\small}
 
\def\aufgaben@head{Exercises}
 
\def\beweis{\beweisanf{}}
\def\Beweis#1{\beweisanf{#1}} 
\def\beweisanf#1{\def\tit{#1}\par\addvspace{2ex}\ifx\tit\@empty{
\parindent 0pt  
 {\sc \beweis@head: }}\else{\parindent 0pt {\sc{#1:} }}\fi\ignorespaces}
 
\def\@empty{}
\def\beweis@head{Proof} 
\def\qqed{\hfill\rule{1.2ex}{1.2ex}}
\def\qed{{\hfill$\Box$}}     % Variante: Box am Beweisende 

 
\def\theequation{\arabic{equation}}
 
\def\titlepage{\@restonecolfalse\if@twocolumn\@restonecoltrue
\onecolumn 
     \else \newpage \fi \thispagestyle{empty}\c@page\z@}
\def\endtitlepage{\if@restonecol\twocolumn \else \newpage \fi} 

\arraycolsep 5pt 
\tabcolsep 6pt
\arrayrulewidth .4pt 
\doublerulesep 2pt
 
\tabbingsep \labelsep
 
\skip\@mpfootins = \skip\footins
 
\fboxsep = 3pt    % Space left between box and text by \fbox and \framebox.
\fboxrule = .4pt  % Width of rules in box made by \fbox and \framebox. 

 
%    ****************************************
%    *             SECTIONS                 * 
%    ****************************************
 
\newcounter{kapitel}
\newcounter{paragraph}[kapitel] 
\newcounter{abschnitt}[paragraph]
\@addtoreset{abschnitt}{kapitel} 

\def\thekapitel{\ifnum\c@kapitel>0 \Roman{kapitel}\fi} 
\def\theparagraph{\ifnum\c@paragraph>0 \arabic{paragraph}\fi}
\def\theabschnitt{\ifnum\c@paragraph>0 \theparagraph.\else\ifnum\c@kapitel>0 
  \thekapitel.\fi \fi\arabic{abschnitt}}
 

%    **************************************** 
%    *         TABLE OF CONTENTS, ETC.      *
%    **************************************** 

\def\@pnumwidth{1.55em} 
\def\@tocrmarg {2.55em}
\def\@dotsep{4.5} 
\setcounter{tocdepth}{2}
 
\newdimen\contindent \contindent\parindent     %! Einzug paragraph
\newdimen\contindentab \contindentab\parindent 
       \advance \contindentab 20pt            %! Einzug abschnitt
\newdimen\contwidth  \contwidth 10pt %! Gr"osse der box f"ur section 
                              %! number im Inhaltsverzeichnis
 
\def\tableofcontents{{\Abschnitt*{\contents@head}
  \paragraphmark{\cont@head}\parskip 0pt 
  \@starttoc{toc}\par\addvspace{100pt}}}
 
\def\contents@head{\large\bf Table of Contents\\[.5cm]}
\def\cont@head{Contents}                                      

 
\def\l@kapitel#1#2{\addpenalty{\@secpenalty}
   \addvspace{0.7em plus 1pt} \@tempdima \contwidth 
   {\begingroup
     \parindent \z@ \rightskip \@pnumwidth 
     \parfillskip -\@pnumwidth \bof \leavevmode
     #1\hfil \hbox to\@pnumwidth{\hss}\par 
    \endgroup}\addvspace{1pt}\nobreak}
 
\def\l@paragraph{\@dottedtocline{2}{\z@}{\contindent}}
\def\l@abschnitt{\@dottedtocline{3}{\contindentab}{\z@}} 

\def\listoffigures{\paragraph*{\listoffig@head 
  \@mkboth{\listoffig@head}{\listoffig@head}}\@starttoc{lof}}
 
\def\l@figure{\@dottedtocline{1}{1.5em}{2.3em}}
\def\listoffig@head{List of Figures} 
\def\listoftables{\paragraph*{\listoftab@head
   \@mkboth{\listoftab@head}{\listoftab@head}}\@starttoc{lot}} 

\let\l@table\l@figure 
\def\listoftab@head{List of Tables}
 

%    **************************************** 
%    *             BIBLIOGRAPHY             *
%    **************************************** 

 
\newdimen\refindent \refindent\parindent
\def\thebibliography{\Abschnitt*{\ref@head} 
   \addcontentsline{toc}{paragraph}{\ref@head}
   \paragraphmark{\ref@head}\small 
   %\@mkboth{\ref@head\hfill}{\ref@head}
   \list{[\arabic{enumi}]}{\labelwidth\refindent\leftmargin\refindent 
   \labelsep 9pt \itemsep 7pt \usecounter{enumi}}
   %\labelsep \z@\usecounter{enumi}} 
   \def\newblock{\hskip .11em plus .33em minus -.07em}
   \sloppy \sfcode`\.=1000\relax} 
\let\endthebibliography=\endlist
 
\def\ref@head{References}
 
\def\Thebibliography#1{\Abschnitt*{#1}
   \addcontentsline{toc}{paragraph}{#1} 
   \paragraphmark{#1}\small
   %\@mkboth{#1\hfill}{#1} 
   \list{[\arabic{enumi}]}{\labelwidth\refindent\leftmargin\refindent
   \labelsep \z@\usecounter{enumi}} 
   \def\newblock{\hskip .11em plus .33em minus -.07em}
   \sloppy \sfcode`\.=1000\relax} 
\let\endThebibliography=\endlist
 

%    **************************************** 
%    *              THE INDEX               *
%    ****************************************

\newif\if@restonecol

\def\theindex{\@restonecoltrue\if@twocolumn\@restonecolfalse\fi
\columnseprule \z@
\columnsep 35pt\twocolumn[\paragraph*{Index}]
    \@mkboth{INDEX}{INDEX}\thispagestyle{plain}\parindent\z@
    \parskip\z@ plus .3pt\relax\let\item\@idxitem}

\def\@idxitem{\vskip 1pt \par\hangindent 40pt}
\def\subitem{\par\hangindent 40pt \hspace*{8pt} - \ }
\def\subsubitem{\par\hangindent 40pt \hspace*{25pt}}
\def\endtheindex{\if@restonecol\onecolumn\else\clearpage\fi}
\def\indexspace{\par \vskip 10pt plus 5pt minus 3pt\relax}


%    ****************************************
%    *             FOOTNOTES                * 
%    ****************************************
 
\newdimen\footnoteindent \footnoteindent\parindent
\def\footnoterule{\kern-3\p@ \hrule width 71pt         %.4\columnwidth
  \kern 2.6\p@}
\long\def\@makefntext#1{\parindent 0pt\noindent 
            \hbox to \footnoteindent{\@thefnmark\hss}#1}
 

%    **************************************** 
%    *         FIGURES AND TABLES           *
%    **************************************** 

\setcounter{topnumber}{2} 
\def\topfraction{.7}
\setcounter{bottomnumber}{1} 
\def\bottomfraction{.3}
\setcounter{totalnumber}{3} 
\def\textfraction{.2}
\def\floatpagefraction{.5} 
\setcounter{dbltopnumber}{2}
\def\dbltopfraction{.7} 
\def\dblfloatpagefraction{.5}
 
\long\def\@makecaption#1#2{
   \vskip 10pt \setbox\@tempboxa\hbox{#1: #2} 
   \ifdim \wd\@tempboxa >\hsize \unhbox\@tempboxa\par
   \else \hbox to\hsize{\hfil\box\@tempboxa\hfil} 
   \fi}
 
\newcounter{figure}
\def\thefigure{\@arabic\c@figure} 

\def\fps@figure{tbp} 
\def\ftype@figure{1}
\def\ext@figure{lof} 
\def\fnum@figure{Figure \thefigure}
\def\figure{\@float{figure}} 
\let\endfigure\end@float
\@namedef{figure*}{\@dblfloat{figure}} 
\@namedef{endfigure*}{\end@dblfloat}
 
\newcounter{table}
\def\thetable{\@arabic\c@table} 

\def\fps@table{tbp} 
\def\ftype@table{2}
\def\ext@table{lot} 
\def\fnum@table{Table \thetable}
\def\table{\@float{table}} 
\let\endtable\end@float
\@namedef{table*}{\@dblfloat{table}} 
\@namedef{endtable*}{\end@dblfloat}
 

%    **************************************** 
%    *         TITLE AND ABSTRACT           *
%    **************************************** 

\def\maketitle{\par 
  \begingroup
    \def\thefootnote{\fnsymbol{footnote}} 
    \def\@makefnmark{\hbox to 0pt{$^{\@thefnmark}$\hss}}
    \if@twocolumn \twocolumn[\@maketitle] 
    \else \newpage\global\@topnum\z@\@maketitle
    \fi \thispagestyle{empty}\@thanks 
  \endgroup \setcounter{footnote}{0}
  \let\maketitle\relax \let\@maketitle\relax \let\thanks\relax 
  \gdef\@thanks{} \gdef\@author{} \gdef\@title{}}
 
%\def\@maketitle{\newpage\null \vskip 2em
%  \begin{center} 
%    {\LARGE\bof\@title \par} \vskip 1.5em
%    {\normalsize\bof\lineskip .5em 
%     \begin{tabular}[t]{c}\@author \end{tabular}\par}
%    \vskip 1em {\normalsize\rom\@date} 
%  \end{center} \par \vskip 3.0em}
 
\def\@maketitle{\newpage\null \vskip 47pt
  \begin{center} 
    {\Large\bof\@title \par} \vskip 1.5em
    {\normalsize\bof\lineskip .5em 
     \begin{tabular}[t]{c}\@author \end{tabular}\par}
     \vskip 1em {\normalsize\rom\@date} 
  \end{center} \par \vskip 26pt
} 

\def\abstract{\if@twocolumn\Abschnitt*{Abstract} 
  \else
  \begin{center} 
   {\bof Abstract\vspace{-.5em}\vspace{3pt}}
  \end{center}  
  \small\quotation\fi}
 
\def\endabstract{\if@twocolumn\else\endquotation\fi}
 

%    **************************************** 
%    *            MISCELLANEOUS             *
%    **************************************** 

\def\today{\ifcase\month\or 
  January\or February\or March\or April\or May\or June\or
  July\or August\or September\or October\or November\or December\fi 
  \space\number\day, \number\year}
 

%    **************************************** 
%    *            PAGE STYLES               *
%    **************************************** 

\mark{{}{}}   % Initializes TeX's marks 

\if@twoside 
  \def\ps@headings{\let\@mkboth\markboth
    \def\@oddfoot{} \def\@evenfoot{}%                                                                               
    \def\@evenhead{\@@line{\small\hbox
    to\titlindent{\rom\thepage\hfill}\rom\leftmark\hfil 
    \hbox to\titlindent{}}}
    \def\@oddhead{\@@line{\small 
     \hbox to\titlindent{\rom\ifnum\c@paragraph<\@ne
     \thekapitel\else\ifnum\c@kapitel>0 \thekapitel.\theparagraph 
     \else\S\,\theparagraph\fi\fi\hfil}\rom\rightmark\hfil
     \hbox to\titlindent{\hfil\rom\thepage}}} 
    \def\kapitelmark##1{\markboth{##1}{}}%
    \def\paragraphmark##1{\markright{##1}}} 
\else
  \def\ps@headings{\let\@mkboth\markboth 
    \def\@oddfoot{} \def\@evenfoot{}%
    \def\@oddhead{\@@line{\small\hbox 
      to\titlindent{\rom\ifnum\c@paragraph<\@ne
      \thekapitel\else\ifnum\c@kapitel>0 \thekapitel.\theparagraph 
      \else\S\,\theparagraph\fi\fi\hfil}\rom\rightmark\hfil
      \hbox to\titlindent{\hfil\rom\thepage}}} 
    \def\kapitelmark##1{\markright{##1}}}
\fi 

\def\ps@myheadings{\let\@mkboth\@gobbletwo 
   \def\@oddhead{\@@line{\small\hbox to\titlindent{}
% \rom\ifnum\c@paragraph<\@ne      
% \thekapitel\else\ifnum\c@kapitel>0 \thekapitel.\theparagraph
% \else\S\,\theparagraph\fi\fi\hfill 
    \rom\rightmark\hfil
      \hbox to\titlindent{\hfil\rom\thepage}}} 
   \def\@evenhead{\@@line{\small\hbox to
      \titlindent{\rom\thepage\hfil}\rom\leftmark\hfil\hbox     to\titlindent{}}}% 
   \def\@oddfoot{} \def\@evenfoot{}
   \def\kapitelmark##1{} \def\paragraphmark##1{} 
   \def\abschnittmark##1{}}
 
\def\kapitelmark#1{}
\def\paragraphmark#1{} 
\def\abschnittmark#1{}
 
\def\@@line#1{\underline{\hbox to\textwidth{\strut #1}}}
%\def\@@line#1{\hbox to\textwidth{\strut #1}} 

 
%    ****************************************
%    *           INITIALIZATION             * 
%    ****************************************
%

\ps@plain 
\pagenumbering{arabic}
\onecolumn
\if@twoside\else\raggedbottom\fi
\frenchspacing

