Hélène Esnault Eckart Viehweg Lectures on Vanishing Theorems

Hélène Esnault, Eckart Viehweg Fachbereich 6, Mathematik Universität-Gesamthochschule Essen D-45117 Essen, Germany

esnault@uni-essen.de

viehweg@uni-essen.de

ISBN 3-7643-2822-3 (Basel)

ISBN 0-8176-2822-3 (Boston)

◯ c 1992 Birkhäuser Verlag Basel, P.O. Box 133, CH-4010 Basel

We cordially thank Birkhäuser-Verlag for their permission to make this book available on the web. The page layout might be slightly different from the printed version.

Acknowledgement

These notes grew out of the DMV-seminar on algebraic geometry (Schloß 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.

Table of Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 § 1 Kodaira’s vanishing theorem, a general discussion . . . . . . . . . 4 § 2 Logarithmic de Rham complexes . . . . . . . . . . . . . . . . . . 11 § 3 Integral parts of Q l -divisors and coverings . . . . . . . . . . . . . 18 § 4 Vanishing theorems, the formal set-up. . . . . . . . . . . . . . . . 35 § 5 Vanishing theorems for invertible sheaves . . . . . . . . . . . . . 42 § 6 Differential forms and higher direct images . . . . . . . . . . . . 54 § 7 Some applications of vanishing theorems . . . . . . . . . . . . . 64 § 8 Characteristic p methods: Lifting of schemes . . . . . . . . . . . . 82 § 9 The Frobenius and its liftings . . . . . . . . . . . . . . . . . . . . 93 § 10 The proof of Deligne and Illusie [12] . . . . . . . . . . . . . . . . 105 § 11 Vanishing theorems in characteristic p. . . . . . . . . . . . . . . . 128 § 12 Deformation theory for cohomology groups . . . . . . . . . . . . 132 § 13 Generic vanishing theorems [26], [14] . . . . . . . . . . . . . . . . 137 APPENDIX: Hypercohomology and spectral sequences . . . . . . . . . . 147 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

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 ([39] and [1]).

Even if, due to J.P. Serre’s GAGA-theorems [56] 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 [52] showed that in characteristic p > 0 some additional assumptions were needed.

This was the state of the art until P. Deligne and L. Illusie [12] 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 [12] was written the two authors studied in [20] 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. Kollár’s work [41] and in the vanishing theorems M. Saito obtained as an application of his theory of mixed Hodge modules (see [54]).

It is obvious that the combination of [12] and [20] 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 [20] along with those of [12] is that they imply as well some of the known generalizations.

Generalizations have been found by D. Mumford [49], H. Grauert and O. Riemenschneider [25], C.P. Ramanujam [51] (in whose paper the method of coverings already appears), Y. Miyaoka [45] (the first who works with integral parts of Q l divisors, in the surface case), by Y. Kawamata [36] and the second author [63]. All results mentioned replace the condition “ample” in Kodaira’s result by weaker conditions. For Akizuki-Nakano type theorems A. Sommese (see for example [57]) got some improvement, as well as F. Bogomolov and A. Sommese (as explained in [6] and [57]) 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, L ⊗ ω_{X}(B)) ⟶ H^{b}(B, L ⊗ ω_{B})

where B is a divisor and L is ample or is belonging to the class of invertible sheaves considered in the generalizations.

J. Kollár [40], building up on partial results by Tankeev, studied the adjunction map directly and gave criteria for 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 [20] and [22] as corollaries of two vanishing theorems for sheaves of differential forms with values in “integral parts of Q l -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 [12] 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 [12] and the properties of the Cartier operator [34] 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 l -divisors and J. Kollár’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 [46] and of Y. Kawamata, K. Matsuda and M. Matsuki [38].

There are, of course, more subjects belonging to the circle of ideas presented in these notes which we left aside:

• L. Illusie’s generalizations of [12] to variations of Hodge structures [32].

• J.-P. Demailly’s analytic approach to generalized vanishing theorems [13].

• M. Saito’s results on “mixed Hodge modules and vanishing theorems” [54], related to J. Kollár’s program [41].

• The work of I. Reider, who used unstability of rank two vector bundles (see [6]) to show that certain invertible sheaves on surfaces are generated by global sections [53] (see however (7.23)).

• Vanishing theorems for vector bundles.

• Generalizations of the vanishing theorems for integral parts of Q l -divisors ([2], [3], [42], [43] and [44]).

However, we had the feeling that we could not pass by the generic vanishing theorems of M. Green and R. Lazarsfeld [26]. 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 [5] and Green and Lazarsfeld’s second paper [27] on this subject is missing. During the preparation of these notes C. Simpson [58] 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 [12].

§ 1 Kodaira’s vanishing theorem, a general discussion

Let X be a projective manifold defined over an algebraically closed field k and let L be an invertible sheaf on X. By explicit calculations of the Čechcohomology of the projective space one obtains:

1.1. Theorem (J. P. Serre [55]). If L is ample and F a coherent sheaf, then there is some ν_{0} ∈ IN such that

H^{b}(X, F ⊗ L^{ν}) = 0 for b > 0 and ν ≥ ν_{0}

In particular, for F = O_{X}, one obtains the vanishing of H^{b}(X, L^{ν}) for b > 0 and ν sufficiently large.

If char(k) = 0, then “ν sufficiently large” can be made more precise. For example, it is enough to choose ν such that A = L^{ν} ⊗ω_{X}^{−1} is ample, where ω_{X} = Ω_{X}^{n} is the canonical sheaf of X, and to use:

1.2. Theorem (K. Kodaira [39]). Let X be a complex projective manifold and A be an ample invertible sheaf. Then

a) H^{b}(X, ω_{X} ⊗ A) = 0 for b > 0
b) H^{b′}(X, A^{−1}) = 0  for b′ < n = dim X.

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 l , one can use flat base change to extend (1.2) to manifolds X defined over any algebraically closed field of characteristic zero.

1.3. Theorem (Y. Akizuki, S. Nakano [1]). Under the assumptions made in (1.2), let Ω_{X}^{a} denote the sheaf of a-differential forms. Then

a) H^{b}(X, Ω_{X}^{a} ⊗ A) = 0  for a + b > n
b) H^{b′}(X, Ω_{X}^{a′} ⊗ A^{−1}) = 0 for a′ + b′ < n.

§ 1  Kodaira’s vanishing theorem, a general discussion  5

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 l ) into a direct sum

⨁_{b+a=k} H^{b}(Y, Ω_{Y}^{a})

or, equivalently, of the degeneration of the “Hodge to de Rham” spectral sequence, both applied to cyclic covers π : 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.

1. Proof: With Hodge decomposition for non-compact manifolds and topological vanishing: For sufficiently large N one can find a nonsingular primedivisor H such that A^{N} = O_{X}(H). Let s ∈ H^{0}(X, A^{N}) be the corresponding section. We can regard s as a rational function, if we fix some divisor A with A = O_{X}(A) and take

s ∈ C l (X) with (s) + N · A = H.

The field L = C l (X)( ^{N} \sqrt{s}) depends only on H. Let π : 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 = (π^{∗}H)_{red} and that π : Y ⟶ X is unramified outside of D. One has

π^{∗}Ω_{X}^{a}(log H) = Ω_{Y}^{a} (log D)

where Ω_{X}^{a}(log H) denotes the sheaf of a-differential forms with logarithmic poles along H (see (2.1)). Moreover

π_{∗}O_{Y} = ⨁_{i=0}^{N−1} A^{−i} and

π_{∗}Ω_{Y}^{n} (log D) = ⨁_{i=0}^{N−1} Ω_{X}^{n}(log H) ⊗ A^{−i} = ⨁_{i=0}^{N−1} Ω_{X}^{n} ⊗ A^{N−i}

Deligne [11] has shown that

H^{k}(Y − D, C l ) ≅ ⨁_{b+a=k} H^{b}(Y, Ω_{Y}^{a} (log D)).

Since X−H is affine, the same holds true for Y −D and hence H^{k}(Y −D, C l ) = 0 for k > n. Altogether one obtains for b > 0

0 = H^{b}(Y, Ω_{Y}^{n} (log D)) = ⨁_{i=0}^{N−1} H^{b}(X, Ω_{X}^{n} ⊗ A^{N−i}).

□

In fact, a similar argument shows as well that

H^{b}(X, Ω_{X}^{a}(log H) ⊗ 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 [56] 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 : Ω_{Y}^{a} ⟶ Ω_{Y}^{a+1}

induces the zero map

d : H^{b}(Y, Ω_{Y}^{a}) ⟶ H^{b}(Y, Ω_{Y}^{a+1}).

Using this one can give another proof of (1.2):

2. Proof: Closedness of global (p, q) forms and Serre’s vanishing theorem: Let us return to the covering π : Y → X constructed in the first proof. The Galois-group G of C l (Y) over C l (X) is cyclic of order N. A generator σ of G acts on Y and D and hence on the sheaves π_{∗}Ω_{Y}^{a} and π_{∗}Ω_{Y}^{a} (log D). Both sheaves decompose in a direct sum of sheaves of eigenvectors of σ and, if we choose the N-th root of unity carefully, the i-th summand of

π_{∗}Ω_{Y}^{a} (log D) = Ω_{X}^{a}(log H) ⊗ π_{∗}O_{Y} = ⨁_{i=0}^{N−1} Ω_{X}^{a}(log H) ⊗ A^{−i}

consists of eigenvectors with eigenvalue e^{i}. For e^{i} ≠ 1 the eigenvectors of π_{∗}Ω_{Y}^{a} and of π_{∗}Ω_{Y}^{a} (log D) coincide, the difference of both sheaves is just living in the invariant parts Ω_{X}^{a} and Ω_{X}^{a}(log H). Moreover, the differential

d : O_{Y} ⟶ Ω_{Y}^{1}

is compatible with the G-action and we obtain a C l -linear map (in fact a connection)

∇_{i} : A^{−i} ⟶ Ω_{X}^{1}(log H) ⊗ A^{−i}.

Both properties follow from local calculations. Let us show first, that

π_{∗}Ω_{Y}^{a} = Ω_{X}^{a} ⊕ ⨁_{i=1}^{N−1} Ω_{X}^{a}(log H) ⊗ A^{−i} .

Since H is non-singular one can choose local parameters x_{1}, …, x_{n} such that H is defined by x_{1} = 0. Then

y_{1} = ^{N} \sqrt{x_{1}} and x_{2}, …, x_{n}

are local parameters on Y. The local generators

N · \frac{dx_{1}}{x_{1}} , dx_{2}, …, dx_{n} of Ω_{X}^{1}(log H)

lift to local generators

\frac{dy_{1}}{y_{1}} , dx_{2}, …, dx_{n} of Ω_{Y}^{1} (log D).

The a-form

ϕ = s · \frac{dy_{1}}{y_{1}} ∧ dx_{2} ∧ … ∧ 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 ∈ O_{X} · y_{1}^{i}. If ϕ 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 ∇_{i} can be described locally as well. If

s = t · y_{1}^{i} ∈ O_{X} · y_{1}^{i}

then on Y one has

ds = y_{1}^{i} · dt + t · dy_{1}^{i}

and therefore d respects the eigenspaces and ∇_{i} is given by

∇_{i}(s) = (dt + \frac{i}{N} · t\frac{dx_{1}}{x_{1}} ) · y_{1}^{i}.

If Res : Ω_{X}^{1}(log H) ⟶ O_{H} denotes the residue map, one obtains in addition that

(Res ⊗ id_{A^{−1}}) ∘ ∇_{1} : A^{−1} ⟶ O_{H} ⊗ A^{−1}

is the O_{X}-linear map

s ⟼ \frac{1}{N} s |_{H} .

Since d : H^{b}(Y, O_{Y}) ⟶ H^{b}(Y, Ω_{Y}^{1} ) is the zero map, the direct summand

∇_{1} : H^{b}(X, A^{−1}) ⟶ H^{b}(X, Ω_{X}^{1}(log D) ⊗ A^{−1})

is the zero map as well as the restriction map

N · (Res ⊗ id_{A^{−1}}) ∘ ∇_{1} : H^{b}(X, A^{−1}) ⟶ H^{b}(H, O_{H} ⊗ A^{−1}).

Hence, for all b we have a surjection

H^{b}(X, A^{−N−1}) = H^{b}(X, O_{X}(−H) ⊗ A^{−1}) ⟶ H^{b}(X, A^{−1}).

Using Serre duality and (1.1) however, H^{b}(X, A^{−N−1}) = 0 for b < n and N sufficiently large.

□

Again, the proof of (1.2) gives a little bit more:

If A is an invertible sheaf such that A^{N} = O_{X}(H) for a non-singular divisor H, then the restriction map

H^{b}(X, A^{−1}) ⟶ H^{b}(H, O_{H} ⊗ A^{−1})

is zero.

This statement is a special case of J. Kollár’s vanishing theorem ([40], 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 E with logarithmic connections

∇ : E ⟶ Ω_{X}^{1}(log H) ⊗ E

and ask which proporties of ∇ and H force cohomology groups of E to vanish. The resulting “vanishing theorems for integral parts of Q l -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, Kollár ......

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_{1}^{ab} = H^{b}(Y, Ω_{Y}^{a} (log D)) ⟹ IH^{a+b}(Y, Ω_{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 ≥ dim X, P. Deligne and L. Illusie were able to prove the degeneration (see [12]). 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

Y ⟶^{F} Y′ = Y ×_{Spec k}Spec k ⟶^{σ} Y
[figure: ↓; ↓]
Spec k  ⟶^{F_{k}} Spec k

with F_{Y} = σ ∘F. If we write D′ = (σ^{∗}D)_{red} then, roughly speaking, they show that F_{∗}(Ω_{Y}^{•} (log D)) is quasi-isomorphic to the complex

⨁_{a} Ω_{Y′}^{a}(log D′)[−a]

with Ω_{Y′}^{a}(log D′) in degree a and with trivial differentials.

By base change for σ one obtains

dim IH^{k}(Y, Ω_{Y}^{•} (log D)) = ∑_{a+b=k} dim H^{b}(Y′, Ω_{Y′}^{a}(log D′))

= ∑_{a+b=k} dim H^{b}(Y, Ω_{Y}^{a} (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 [12] 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 ≫ 0

H^{b}(Y, Ω_{Y}^{a} ⊗ A^{−p^{ν}}) = 0 for ν ≥ (m + 1)

and a + b < n, where A is ample on Y. One argues by descending induction on m: As

A^{p^{(m+1)}} = F^{∗}(A′^{p^{m}}) for A′ = σ^{∗}A

and as Ω_{Y}^{•} is a O_{Y′} complex, Ω_{Y}^{•} ⊗ A^{−p^{(m+1)}} is a complex of O_{Y′} sheaves with

IH^{k}(Y, Ω_{Y}^{•} ⊗ A^{−p^{m+1}}) = 0 for k < n.

However one has

F_{∗}(Ω_{Y}^{•} ⊗ A^{−p^{m+1}}) = ⨁_{a} Ω_{Y′}^{a} ⊗ A′^{−p^{m}}[−a]

and

0 = H^{b}(Y′, Ω_{Y′}^{a} ⊗ A′^{−p^{m}}) = H^{b}(Y, Ω_{Y}^{a} ⊗ 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 §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, N^{−1}) = 0 for a general element N ∈ P ic^{0}(X) if b is smaller than the dimension of the image of X under its Albanese map

α : X ⟶ Alb(X).

By Hodge-duality (for Hodge theory with values in unitary rank one bundles) H^{b}(X, N^{−1}) can be identified with H^{0}(X, Ω_{X}^{b} ⊗ N). If H^{b}(X, N^{−1}) ≠ 0 for all N ∈ P ic^{0}(X) the deformation theory for cohomology groups, developed by Green and Lazarsfeld, implies that for all ω ∈ H^{0}(X, Ω_{X}^{1}) the wedge product

H^{0}(X, Ω_{X}^{b} ⊗ N) ⟶ H^{0}(X, Ω_{X}^{b+1} ⊗ N)

is non-trivial. This however implies that the image of X under the Albanese map, or equivalently the subsheaf of Ω_{X}^{1} generated by global sections is small. For example, if

S^{b}(X) = {N ∈ P ic^{0}(X); H^{b}(X, N^{−1}) ≠ 0},

then the first result of Green and Lazarsfeld says that

codim_{P ic^{0}(X)}(S^{b}(X)) ≥ dim(α(X)) − b.

It is only this part of their results we include in these notes, together with some straightforward generalizations due to H. Dunio [14] (see Lectures 12 and 13). The more detailed description of S^{b}(X), due to Beauville [5], Green-Lazarsfeld [27] and C. Simpson [58] is just mentioned, without proof, at the end of Lecture

§ 2  Logarithmic de Rham complexes  11

§ 2 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 [10]. 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 = ∑_{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 τ : U = X − D ⟶ X and

Ω_{X}^{a}(∗D) = lim ; ⟶_{ν} Ω_{X}^{a}(ν · D) = τ_{∗}Ω_{U}^{a}.

Of course (Ω_{X}^{•}(∗D), d) is a complex.

2.1. Definition. Ω_{X}^{a}(log D) denotes the subsheaf of Ω_{X}^{a}(∗D) of differential forms with logarithmic poles along D, i.e.: if V ⊆ X is open, then

Γ(V, Ω_{X}^{a}(log D)) =

{ α ∈ Γ(V, Ω_{X}^{a}(∗D)); α and dα have simple poles along D}.

2.2. Properties. a)

(Ω_{X}^{•}(log D), d) ↪ (Ω_{X}^{•}(∗D), d).

is a subcomplex. b)

Ω_{X}^{a}(log D) = ⋀^{a} Ω_{X}^{1}(log D)

c) Ω_{X}^{a}(log D) is locally free. More precisely: For p ∈ X, let us say with p ∈ D_{j} for j = 1, …, s and p ∉ D_{j} for j = s+1, …, r, choose local parameters f_{1}, …, f_{n} in p such that D_{j} is defined by f_{j} = 0 for j = 1, …, s. Let us write

δ_{j} = \{ \begin{matrix}\frac{df_{j}}{f_{j}} & if j ≤ s \\ df_{j} & if j > s \end{matrix}

and for I = {j_{1}, …, j_{a}} ⊂ {1, …, n} with j_{1} < j_{2} … < j_{a}

δ_{I} = δ_{j_{1}} ∧ … ∧ δ_{j_{a}}.

Then {δ_{I}; ♯I = a} is a free system of generators for Ω_{X}^{a}(log D).

Proof: (see [10], II, 3.1 - 3.7). a) is obvious and b) follows from the explicite form of the generators given in c). Since δ_{j} is closed, δ_{I} is a local section of Ω_{X}^{a}(log D). By the Leibniz rule the O_{X}-module Ω spanned by the δ_{I} is contained in Ω_{X}^{a}(log D). Ω is locally free and, in order to show that Ω = Ω_{X}^{a}(log D) it is enough to consider the case s = 1. Each local section α ∈ Ω_{X}^{a}(∗D) can be written as

α = α_{1} + α_{2} ∧ \frac{df_{1}}{f_{1}} ,

where α_{1} and α_{2} lie in Ω_{X}^{a}(∗D) and Ω_{X}^{a−1}(∗D) and where both are in the subsheaves generated over O(∗D) by wedge products of df_{2}, …, df_{n}. α ∈ Ω_{X}^{a}(log D) implies that

f_{1} · α = f_{1} · α_{1} + α_{2} ∧ df_{1} ∈ Ω_{X}^{a} and f_{1}dα = f_{1}dα_{1} + dα_{2} ∧ df_{1} ∈ Ω_{X}^{a+1}.

Hence α_{2} as well as f_{1}α_{1} are without poles. Since

d(f_{1}α_{1}) = df_{1} ∧ α_{1} + f_{1}dα_{1} = df_{1} ∧ α_{1} + f_{1}dα − dα_{2} ∧ df_{1}

the form df_{1} ∧ α_{1} has no poles which implies α_{1} ∈ Ω_{X}^{a}.

□

Using the notation from (2.2,c) we define

α : Ω_{X}^{1}(log D) ⟶ ⨁_{j=1}^{s} O_{D_{j}}

by

α(∑_{j=1}^{n} a_{j}δ_{j}) = ⨁_{j=1}^{s} a_{j}|_{D_{j}}.

For a ≥ 1 we have correspondingly a map

β_{1} : Ω_{X}^{a}(log D) ⟶ Ω_{D_{1}}^{a−1}(log (D − D_{1})|_{D_{1}})

given by: If φ is a local section of Ω_{X}^{a}(log D), we can write

φ = φ_{1} + φ_{2} ∧ \frac{df_{1}}{f_{1}}

where φ_{1} lies in the span of the δ_{I} with 1 ∉ I and

φ_{2} = ∑_{1∈I} a_{I}δ_{I−{1}}.

Then

β_{1}(φ) = β_{1}(φ_{2} ∧ \frac{df_{1}}{f_{1}} ) = ∑ a_{I}δ_{I−\{1\}}|_{D_{1}}.

Of course, β_{i} will denote the corresponding map for the i-th component. Finally, the natural restriction of differential forms gives

γ_{1} : Ω_{X}^{a}(log (D − D_{1})) ⟶ Ω_{D_{1}}^{a}(log (D − D_{1})|_{D_{1}}).

Since the sheaf on the left hand side is generated by

{f_{1} · δ_{I}; 1 ∈ I} ∪ {δ_{I}; 1 ∉ I}

we can describe γ_{1} by

γ_{1}(∑_{1∈I} f_{1}a_{I}δ_{I} + ∑_{1∉I} a_{I}δ_{I}) = ∑_{1∉I} a_{I}δ_{I}|_{D_{1}}.

Obviously one has

2.3. Properties. One has three exact sequences: a)

0 → Ω_{X}^{1} ⟶ Ω_{X}^{1}(log D) ⟶^{α} ⨁_{j=1}^{r} O_{D_{j}} → 0.

b)

0 → Ω_{X}^{a}(log (D − D_{1})) ⟶ Ω_{X}^{a}(log D) ⟶^{β_{1}} Ω_{D_{1}}^{a−1}(log (D − D_{1})|_{D_{1}}) → 0.

c)

0 → Ω_{X}^{a}(log D)(−D_{1}) ⟶ Ω_{X}^{a}(log (D − D_{1})) ⟶^{γ_{1}} Ω_{D_{1}}^{a}(log (D − D_{1})|_{D_{1}}) → 0.

By (2.2,b) (Ω_{X}^{•}(log D), d) is a complex. It is the most simple example of a logarithmic de Rham complex.

2.4. Definition. Let E be a locally free coherent sheaf on X and let

∇ : E ⟶ Ω_{X}^{1}(log D) ⊗ E

be a k-linear map satisfying

∇(f · e) = f · ∇(e) + df ⊗ e.

One defines

∇_{a} : Ω_{X}^{a}(log D) ⊗ E ⟶ Ω_{X}^{a+1}(log D) ⊗ E

by the rule

∇_{a}(ω ⊗ e) = dω ⊗ e + (−1)^{a} · ω ∧ ∇(e).

We assume that ∇_{a+1} ∘ ∇_{a} = 0 for all a. Such ∇ will be called an integrable logarithmic connection along D, or just a connection. The complex

(Ω_{X}^{•}(log D) ⊗ E, ∇_{•})

is called the logarithmic de Rham complex of (E, ∇).

2.5. Definition. For an integrable logarithmic connection

∇ : E ⟶ Ω_{X}^{1}(log D) ⊗ E

we define the residue map along D_{1} to be the composed map

Res_{D_{1}}(∇) : E ⟶^{∇} Ω_{X}^{1}(log D) ⊗ E ⟶^{β′_{1}=β_{1}⊗id_{E}} O_{D_{1}} ⊗ E.

2.6. Lemma.

a) Res_{D_{1}}(∇) is O_{X}-linear and it factors through

E ⟶^{restr.} O_{D_{1}} ⊗ E ⟶ O_{D_{1}} ⊗ E

where restr. the restriction of E to D_{1}. By abuse of notations we will call the second map Res_{D_{1}}(∇) again.

b) One has a commutative diagram

Ω_{X}^{a}(log (D − D_{1})) ⊗ E  ⟶^{(∇_{a})∘(incl.)}  Ω_{X}^{a+1}(log D) ⊗ E
↓^{γ_{1}⊗id_{E}}  ↓^{β_{1}⊗id_{E}=β′_{1}}
Ω_{D_{1}}^{a}(log (D − D_{1}) |_{D_{1}}) ⊗ E ⟶^{((−1)^{a}·id)⊗Res_{D_{1}} (∇)} Ω_{D_{1}}^{a}(log (D − D_{1}) |_{D_{1}}) ⊗ E

Proof: a) We have

∇(g · e) = g · ∇(e) + dg ⊗ e and β′_{1}(∇(g · e)) = g · β′_{1}(∇(e)).

If f_{1} divides g then g · β′_{1}(∇(e)) = 0.

b) For ω ∈ Ω_{X}^{a}(log (D − D_{1})) and e ∈ E we have

\begin{aligned}β′_{1}(∇_{a}(ω ⊗ e)) & = & β′_{1}(dω ⊗ e + (−1)^{a} · ω ∧ ∇(e)) \\ & = & β′_{1}((−1)^{a} · ω ∧ ∇(e)). \end{aligned}

If ω = f_{1} · a_{I} · δ_{I} for 1 ∈ I, then

(−1)^{a}ω ∧ ∇(e) ∈ Ω_{X}^{a+1}(log D)(−D_{1})

and β′_{1}(∇_{a}(ω ⊗ e)) = 0. On the other hand, γ_{1}(ω) ⊗ e = 0 by definition. If ω = a_{I}δ_{I} for 1 ∉ I, then

γ_{1}(ω) ⊗ e = a_{I} · δ_{I} |_{D_{1}} ⊗e

and

β′_{1}((−1)^{a}ω ∧ ∇(e)) = (−1)^{a}ω|_{D_{1}} ⊗ Res_{D_{1}}(∇)(e).
□

2.7. Lemma. Let

B = ∑_{j=1}^{r} μ_{j}D_{i}

be any divisor and (∇, E) as in (2.4). Then ∇ induces a connection ∇^{B} with logarithmic poles on

E ⊗ O_{X}(B) = E(B)

and the residues satisfy

Res_{D_{j}}(∇^{B}) = Res_{D_{j}}(∇) − μ_{j} · id_{D_{j}}.

Proof: A local section of E(B) is of the form

σ = ∏_{j=1}^{s} f_{j}^{−μ_{j}} · e

and

∇^{B}(σ) = ∏_{j=1}^{s} f_{j}^{−μ_{j}}∇(e) + d(∏_{j=1}^{s} f_{j}^{−μ_{j}}) ⊗ e =

= ∏_{j=1}^{s} f_{j}^{−μ_{j}}∇(e) + ∑_{k=1}^{s}(∏_{j=1}^{s} f_{j}^{−μ_{j}}) · (−μ_{k})\frac{df_{k}}{f_{k}} ⊗ e.

Hence ∇^{B} : E(B) ⟶ Ω_{X}^{1}(log D) ⊗ E(B) is well defined. One obtains

Res_{D_{1}}(∇^{B}(σ)) = ∏_{j=1}^{s} f_{j}^{−μ_{j}} Res_{D_{1}}(∇(e)) + ∏_{j=1}^{s} f_{j}^{−μ_{j}}(−μ_{1}) ⊗ e |_{D_{1}} .

□

2.8. Definition. a) We say that (∇, E) satisfies the condition (∗) if for all divisors

B = ∑_{j=1}^{r} μ_{j}D_{j} ≥ D

and all j = 1 … r one has an isomorphism of sheaves

Res_{D_{j}}(∇^{B}) = Res_{D_{j}}(∇) − μ_{j} · id_{D_{j}} : E |_{D_{j}}⟶ E |_{D_{j}} .

b) We say that (∇, E) satisfies the condition (!) if for all divisors

B = ∑_{j=1}^{r} −ν_{j}D_{j} ≤ 0

and all j = 1, …, r

Res_{D_{j}}(∇^{B}) = Res_{D_{j}}(∇) + ν_{j} · id_{D_{j}} : E |_{D_{j}}⟶ E |_{D_{j}}

is an isomorphism of sheaves.

In other words, (∗) means that no μ_{j} ∈ ZZ, μ_{j} ≥ 1, is an eigenvalue of Res_{D_{j}}(∇) and (!) means the same for μ_{j} ∈ ZZ, μ_{j} ≤ 0. We will see later, that (∗) and (!) are only of interest if char (k) = 0.

2.9. Properties.

a) Assume that (E, ∇) satisfies (∗) and that B = ∑ μ_{j}D_{j} ≥ 0. Then the natural map

(Ω_{X}^{•}(log D) ⊗ E, ∇_{•}) ⟶ (Ω_{X}^{•}(log D) ⊗ E(B), ∇_{•}^{B})

between the logarithmic de Rham complexes is a quasi-isomorphism.

b) Assume that (E, ∇) satisfies (!) and that B = ∑ −μ_{j}D_{j} ≤ 0. Then the natural map

(Ω_{X}^{•}(log D) ⊗ E(B), ∇_{•}^{B}) ⟶ (Ω_{X}^{•}(log D) ⊗ E, ∇_{•})

between the logarithmic de Rham complexes is a quasi-isomorphism.

(2.9) follows from the definition of (∗) and (!) and from:

2.10. Lemma. For (E, ∇) as in (2.4) assume that

Res_{D_{1}}(∇) : E |_{D_{1}}⟶ E |_{D_{1}}

is an isomorphism. Then the inclusion of complexes

(Ω_{X}^{•}(log D) ⊗ E(−D_{1}), ∇_{•}^{−D_{1}}) ⟶ (Ω_{X}^{•}(log D) ⊗ E, ∇_{•})

is an quasi-isomorphism.

Proof: Consider the complexes E^{(ν)}:

E(−D_{1}) ⟶ Ω_{X}^{1}(log D) ⊗ E(−D_{1}) ⟶ … ⟶ Ω_{X}^{ν−1}(log D) ⊗ E(−D_{1}) ⟶

⟶ Ω_{X}^{ν}(log (D − D_{1})) ⊗ E ⟶ Ω_{X}^{ν+1}(log D) ⊗ E ⟶ … ⟶ Ω_{X}^{n}(log D) ⊗ E

We have an inclusion

E^{(ν+1)} ⟶ E^{(ν)}

and, by (2.6,b) the quotient is the complex

0 ⟶ Ω_{D_{1}}^{ν}(log (D−D_{1})|_{D_{1}})⊗E ⟶^{(−1)^{ν}⊗Res_{D_{1}} (∇)} Ω_{D_{1}}^{ν}(log (D−D_{1})|_{D_{1}})⊗E ⟶ 0

Since the quotient has no cohomology all the E^{(ν)} are quasi-isomorphic, especially E^{(0)} and E^{(n)}, as claimed.

□

2.11. The analytic case

At this point it might be helpful to consider the analytic case for a moment: E is a locally free sheaf over the sheaf of analytic functions O_{X},

∇ : E ⟶ Ω_{X}^{1}(log D) ⊗ E

is a holomorphic and integrable connection. Then ker(∇ |_{U}) = V is a local constant system. If (∗) holds true, i.e. if the residues of ∇ along the D_{j} do not have strictly positive integers as eigenvalues, then (see [10], II, 3.13 and 3.14)

(Ω_{X}^{•}(log D) ⊗ E, ∇_{•})

is quasi-isomorphic to Rτ_{∗}V. By Poincaré-Verdier duality (see [20], Appendix A) the natural map

τ_{!}V^{∨} ⟶ (Ω_{X}^{•}(log D) ⊗ E^{∨}(−D), ∇_{•}^{∨})

is a quasi-isomorphism. Hence (!) implies that the natural map

τ_{!}V ⟶ (Ω_{X}^{•}(log D) ⊗ E, ∇_{•})

is a quasi-isomorphism as well. In particular, topological properties of U give vanishing theorems for

IH^{l}(X, Ω_{X}^{•}(log D) ⊗ E)

and for some l. More precisely, if we choose r(U) to be the smallest number that satisfies:

For all local constant systems V on U one has H^{l}(U, V) = 0 for l > n + r(U),

then one gets:

2.12. Corollary.

a) If (E, ∇) satisfies (∗), then for l > n + r(U)

IH^{l}(X, Ω_{X}^{•}(log D) ⊗ E) = H^{l}(U, V) = 0.

b) If (E, ∇) satisfies (!), then for l < n − r(U)

IH^{l}(X, Ω_{X}^{•}(log D) ⊗ E) = H_{c}^{l}(U, V) = 0.

By GAGA (see [56]), (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, (∗) and (!), are satisfied, i.e. if none of the eigenvalues of Res_{D_{j}} (∇) 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:

2.13. 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 (E, ∇) a logarithmic connection with poles along D (as in (2.4)).

a) If (E, ∇) satisfies (∗), then for l > n

IH^{l}(X, Ω_{X}^{•}(log D) ⊗ E) = 0.

b) If (E, ∇) satisfies (!), then for l < n

IH^{l}(X, Ω_{X}^{•}(log D) ⊗ E) = 0.

Proof: (2.9) allows to replace E by E(N · B) in case a) or by E(−N · B) in case b) for N > 0. By Serre’s vanishing theorem (1.1) we can assume that

H^{b}(X, Ω_{X}^{a}(log D) ⊗ E) = 0

for a + b = l. The Hodge to de Rham spectral sequence (see (A.25)) implies (2.13).

□

§ 3 Integral parts of Q l -divisors and coverings

Over complex manifolds the Riemann Hilbert correspondence obtained by Deligne [10] is an equivalence between logarithmic connections (E, ∇) and representations of the fundamental group π_{1}(X −D). For applications in algebraic geometry the most simple representations, i.e. those who factor through cyclic quotient groups of π_{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.

§ 3  Integral parts of Q l -divisors and coverings  19

3.1. Notation. a) Let us write Div(X) for the group of divisors on X and

Div_{Q l}(X) = Div(X) ⊗_{ZZ} Q l .

Hence a Q l -divisor Δ ∈ Div_{Q l}(X) is a sum

Δ = ∑_{j=1}^{r} α_{j}D_{j}

of irreducible prime divisors D_{j} with coefficients α_{j} ∈ Q l .

b) For Δ ∈ Div_{Q l}(X) we write

[Δ] = ∑_{j=1}^{r}[α_{j}] · D_{j}

where for α ∈ Q l , [α] denotes the integral part of α, defined as the only integer such that

[α] ≤ α < [α] + 1.

[Δ] will be called the integral part of Δ.

c) For an invertible sheaf L, an effective divisor

D = ∑_{j=1}^{r} α_{j}Dj

and a positive natural number N, assume that L^{N} = O_{X}(D). Then we will write for i ∈ IN

L^{(i,D)} = L^{i}(−[\frac{i}{N} D]) = L^{i} ⊗ O_{X}(−[\frac{i}{N} · D]).

Usually N and D will be fixed and we just write L^{(i)} instead of L^{(i,D)}.

d) If

D = ∑_{j=1}^{r} α_{j}D_{j}

is a normal crossing divisor, we will write, for simplictiy,

Ω_{X}^{a}(log D) instead of Ω_{X}^{a}(log D_{red}).

In spite of their strange definition the sheaves L^{(i)} will turn out to be related to cyclic covers in a quite natural way. We will need this to prove:

3.2. Theorem. Let X be a projective manifold,

D = ∑_{j=1}^{r} α_{j}D_{j}

be an effective normal crossing divisor, L an invertible sheaf and N ∈ IN − {0} prime to char(k), such that L^{N} = O_{X}(D). Then for i = 0, …, N − 1 the sheaf L^{(i)^{−1}} has an integrable logarithmic connection

∇^{(i)} : L^{(i)^{−1}} ⟶ Ω_{X}^{1}(log D^{(i)}) ⊗ L^{(i)^{−1}}

with poles along D^{(i)} = ∑_{{j=1 \atop \frac{i·α_{j}}{N} ∉ZZ}}^{r} D_{j},

satisfying:

a) The residue of ∇^{(i)} along D_{j} is given by multiplication with

(i · α_{j} − N · [\frac{i · α_{j}}{N}]) · N^{−1} ∈ k.

b) Assume that either char(k) = 0, or, if char(k) = p ≠ 0, that X and D admit a lifting to W_{2}(k) (see (8.11)) and that p ≥ dim X. Then the spectral sequence

E_{1}^{ab} = H^{b}(X, Ω_{X}^{a}(log D^{(i)}) ⊗ L^{(i)^{−1}}) ⟹ IH^{a+b}(X, Ω_{X}^{•}(log D^{(i)}) ⊗ L^{(i)^{−1}})

associated to the logarithmic de Rham complex

(Ω_{X}^{•}(log D^{(i)}) ⊗ L^{(i)^{−1}}, ∇_{•}^{(i)})

degenerates in E_{1}.

c) Let A and B be reduced divisors (both having the lifting property (8.11) if char(k) = p ≠ 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 ∇^{(i)} induces a logarithmic connection

O_{X}(−B) ⊗ L^{(i)^{−1}} ⟶ Ω_{X}^{1}(log (A + B + D^{(i)}))(−B) ⊗ L^{(i)^{−1}}

and under the assumptions of b) the spectral sequence

E_{1}^{ab} = H^{b}(X, Ω_{X}^{a}(log (A + B + D^{(i)}))(−B) ⊗ L^{(i)^{−1}}) ⟹
IH^{a+b}(X, Ω_{X}^{•}(log (A + B + D^{(i)}))(−B) ⊗ L^{(i)^{−1}})

degenerates in E_{1} as well.

3.3. Remarks. a) In (3.2), whenever one likes, one can assume that i = 1. In fact, one just has to replace L by L′ = L^{i} and D by D′ = i · D .Then

L^{′N} = O_{X}(i · D) = O_{X}(D′)

and

L^{′^{(1,D′)}} = L′(−[\frac{D′}{N}]) = L^{i}(−[\frac{i}{N} D]).

b) Next, one can always assume that 0 < α_{j} < N. In fact, if α_{1} ≥ N, then

L′ = L(−D_{1}) and D′ = D − N · D_{1}

give the same sheaves as L and D:

L^{′^{(i,D′)}} = L^{i}(−i · D_{1} − [\frac{i}{N} · D′]) = L^{i}(−[\frac{i}{N} · D]).

c) In particular, for i = 1 and 0 < a_{j} < N we have

L^{(1)} = L and D^{(1)} = D.

Nevertheless, in the proof of (3.2) we stay with the notation, as started.

d) Finally, for i ≥ N one has

L^{(i,D)} = L^{i}(−[\frac{i}{N} · D]) = L^{i−N}(−[\frac{i − N}{N} · D]) = L^{(i−N,D)}.

The “L^{(i)}” are the most natural notation for “integral parts of Q l - 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:

3.4. Remarks.

a) Sometimes the integral part [Δ] is denoted by ⌊Δ⌋.

b) One can also consider the round up {Δ} = ⌈Δ⌉ given by

{Δ} = −[−Δ]

or the fractional part of Δ given by

< Δ >= Δ − [Δ].

c) For L, N and D as in (3.1,c) one can write

L = O_{X}(C)

for some divisor C. Then

Δ = C − \frac{1}{N} · D ∈ Div_{Q l}(X)

has the property that N · Δ is a divisor linear equivalent to zero. One has

L^{(i,D)} = O_{X}(i · C − [\frac{i}{N} · D]) = O_{X}(−[−i · Δ]) = O_{X}(\{i · Δ\}).

d) On the other hand, for Δ ∈ Div_{Q}(X) and N > 0 assume that N · Δ is a divisor linear equivalent to zero. Then one can choose a divisor C such that C − Δ is effective. For L = O_{X}(C) and D = N · C − N · Δ ∈ Div(X) one has

L^{N} = O_{X}(N · C) = O_{X}(D)

and

L^{(i,D)} = O_{X}(i · C − [\frac{i}{N} D]) =

O_{X}(−[−i · C + \frac{i}{N} · D]) = O_{X}(\{i · Δ\}).

e) Altogether, (3.2) is equivalent to:

For Δ ∈ Div_{Q l}(X) such that N ·Δ is a divisor linear equivalent to zero, assume that < Δ > is supported in D and that D is a normal crossing divisor. Then O_{X}({Δ}) 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.

3.5. Cyclic covers. Let L, N and

D = ∑_{j=1}^{r} α_{j}D_{j}

be as in (3.1,c) and let s ∈ H^{0}(X, L^{N}) be a section whose zero divisor is D. The dual of

s : O_{X} ⟶ L^{N} , i.e. s^{∨} : L^{−N} ⟶ O_{X},

defines a O_{X}-algebra structure on

A′ = ⨁_{i=0}^{N−1} L^{−i}.

In fact,

A′ = ⨁_{i=0}^{∞} L^{−i}/I

where I is the ideal-sheaf generated locally by

{s^{∨}(l) − l, l local section of L^{−N}}.

Let

Y′ = Spec_{X}(A′) ⟶^{π′} X

be the spectrum of the O_{X}-algebra A′, as defined in [30], page 128, for example. Let π : Y → X be the finite morphism obtained by normalizing Y′ → X. To be more precise, if Y′ is reducible, Y will be the disjoint union of the normalizations of the components of Y′ in their function fields. We will call Y the cyclic cover obtained by taking the n-th root out of s (or out of D, if L is fixed).

Obviously one has:

3.6. Claim. Y is uniquely determined by:

a) π : Y → X is finite.

b) Y is normal.

c) There is a morphism ϕ : A′ → π_{∗}O_{Y} of O_{X}-algebras, isomorphic over some dense open subscheme of X.

3.7. Notations. For D, N and L as in (3.1,c) let us write

A = ⨁_{i=0}^{N−1} L^{(i)^{−1}}.

The inclusion

L^{−i} ⟶ L^{(i)^{−1}} = L^{−i}([\frac{i}{N} · D])

gives a morphism of O_{X}-modules

ϕ : A′ ⟶ A.

3.8. Claim. A has a structure of an O_{X}-algebra, such that ϕ is a homomorphism of algebras.

Proof: The multiplication in A′ is nothing but the multiplication

L^{−i} × L^{−j} ⟶ L^{−i−j}
composed with s^{∨} : L^{−i−j} ⟶ L^{−i−j+N} ,

in case that i + j ≥ N. For i, j ≥ 0 one has

[\frac{i}{N} · D] + [\frac{j}{N} · D] ≤ [\frac{i + j}{N} · D]

and, for i + j ≥ N, one has

L^{(i+j)} = L^{i+j}(−[\frac{i + j}{N} · D]) = L^{i+j−N}(−[\frac{i + j − N}{N} · D]) = L^{(i+j−N)}.

This implies that the multiplication of sections

L^{(i)^{−1}} × L^{(j)^{−1}} ⟶ L^{−i−j}([\frac{i}{N} D] + [\frac{j}{N} D]) ⟶ L^{(i+j)^{−1}}

is well defined, and that for i + j ≥ N the right hand side is nothing but L^{(i+j−N)^{−1}}.

□

3.9.

Assume that N is prime to char(k), e a fixed primitive N-th root of unit and G =< σ > the cyclic group of order N. Then G acts on A by O_{X}-algebra homomorphisms defined by:

σ(l) = e^{i} · l for a local section l of L^{(i)^{−1}} ⊂ A.

Obviously the invariants under this G-action are

A^{G} = O_{X}.

3.10. Claim. Assume that N is prime to char(k). Then

A = π_{∗}O_{Y} or (equivalently) Y = Spec(A) .

3.11. Corollary (see [16]). The cyclic group G acts on Y and on π_{∗}O_{Y}. One has Y/G = X and the decomposition

π_{∗}O_{Y} = ⨁_{i=0}^{N−1} L^{(i)^{−1}}

is the decomposition in eigenspaces.

Proof of 3.10.: For any open subvariety X_{0} in X with codim_{X}(X − X_{0}) ≥ 2 and for Y_{0} = π^{−1}(X_{0}) consider the induced morphisms

Y_{0} ⟶^{ι′} Y
π_{0}↓  ↓^{π}
X_{0} ⟶^{ι} X

Since Y is normal one has ι′_{∗}O_{Y_{0}} = O_{Y} and π_{∗}O_{Y} = ι_{∗}π_{0_{∗}}O_{Y_{0}}. Since A is locally free, (3.10) follows from

π_{0_{∗}}O_{Y_{0}} = 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 A and π_{∗}O_{Y} follows from:

3.12. Claim. Spec (A) ⟶ X is finite and Spec(A) is normal.

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 = α_{1} · D_{1}. Let us fix isomorphisms L^{i} ≃ O_{X} for all i and assume that D_{1} is the zero set of f_{1} ∈ B. For some unit u ∈ B^{∗} the section s ∈ H^{0}(X, L^{N}) ≃ B is identified with f = u · f_{1}^{α_{1}}. For completeness, we allow D (or α_{1}) to be zero. The O_{X}-algebra A′ is given by the B-algebra

H^{0}(X, A′) = ⨁_{i=0}^{N−1} H^{0}(X, L^{−i})

which can be identified with the quotient of the ring of polynomials

A′ = B[t]/_{t^{N} −f} = ⨁_{i=0}^{N−1} B · t^{i}.

In this language

A = ⨁_{i=0}^{N−1} B · t^{i} · f_{1}^{−[ \frac{i}{N} α_{1}]} = ⨁_{i=0}^{N−1} H^{0}(X, L^{(i)^{−1}}) = H^{0}(X, A)

and ϕ : A′ → A induces the natural inclusion A′ ↪ A.

Hence (3.12) follows from the first part of the following claim.

□

3.13. Claim. Using the notations introduced above, assume that N is prime to char(k). Then one has

a) Spec A is non-singular and π : Spec A ⟶ Spec B is finite.

b) If α_{1} = 0, then Spec A ⟶ Spec B is non-ramified (hence étale).

c) if α_{1} is prime to N, we have a defining equation g ∈ A for Δ_{1} = (π^{∗}D_{1})_{red} with

g^{N} = u^{a} · f_{1} for some a ∈ IN.

d) If Γ is a divisor in Spec B such that D + Γ has normal crossings, then π^{∗}(D + Γ) has normal crossings as well.

Proof: Let us first consider the case α_{1} = 0. Then

A′ = A = B[t]/_{t^{N} −u}

for u ∈ B^{∗}. A is non-singular, as follows, for example, from the Jacobi-criterion, and A is unramified over B. Hence

Spec A ⟶ Spec B

is étale in this case and a), b) and d) are obvious.

If α_{1} = 1 , then again

A′ = A = B[t]/_{t^{N} −u·f_{1}}.

For p ∈ Spec B, choose f_{2}, …, f_{n} such that f_{1}·u, f_{2}, …, f_{n} is a local parametersystem in p. Then t, f_{2}, …, f_{n} will be a local parameter system, for q = π^{−1}(p) .

Similar, if α_{1} is prime to N, and if c) holds true, g and f_{2}, …, 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} ·t_{1}^{α_{1}} we obtain A′ as a subring of R. Spec R is non-singular over p and Spec R ⟶ Spec B is finite.

The group H =< σ_{0} > ⊕ < σ_{1} > with ord (σ_{0}) = ord (σ_{1}) = N operates on R by

σ_{ν}(t_{μ}) = \{ \begin{matrix}t_{μ} & if ν ≠ μ \\ e · t_{μ} & if ν = μ \end{matrix}

Let H′ be the kernel of the map γ : H ⟶ G =< σ > given by γ(σ_{0}) = σ and γ(σ_{1}) = σ^{α_{1}}. The quotient

Spec (R)/_{H′} = Spec R^{H′}

is normal and finite over Spec B. One has (σ_{0}^{μ}, σ_{1}^{ν}) ∈ H′, if and only if μ + να_{1} ≡ 0 mod N. Hence R^{H′} is generated by monomials t_{0}^{a} · t_{1}^{b} where a, b ∈ {0, …, N − 1} satisfy:

(∗) aμ + bν ≡ 0 mod N for all (μ, ν) with μ + α_{1}ν ≡ 0 mod N.

Obviously, (∗) holds true for (a, b) if b ≡ a · α_{1} mod N. On the other hand, choosing ν to be a unit in ZZ/_{N}, (∗) implies that b ≡ a · α_{1} mod N.

Hence, for all (a, b) satisfying (∗) we find some k with b = a · α_{1} + k · N. Since a, b ∈ {0, …, N − 1} we have

\frac{a · α_{1}}{N} ≥ −k = \frac{a · α_{1}}{N} − \frac{b}{N} > \frac{a · α_{1}}{N} − 1

or k = −[\frac{a·α_{1}}{N}].

Therefore one obtains

R^{H′} = ⨁_{a=0}^{N−1} t_{0}^{a} · t_{1}^{a·α_{1}−N·[ \frac{a·α_{1}}{N}]} · B = ⨁_{a=0}^{N−1}(t_{0} · t_{1}^{α_{1}})^{a} · f_{1}^{−[ \frac{a·α_{1}}{N}]} · B

and hence R^{H′} = ⨁_{a=0}^{N−1} t^{a} · f_{1}^{−[ \frac{a·α_{1}}{N}]} · B = A.

If α_{1} is prime to N, we can find a ∈ {0, …, N − 1} with a · α_{1} = 1 + l · N for l ∈ ZZ. Then

a · α_{1} − N · [\frac{a · α_{1}}{N}] = 1

and g = t^{a} · f_{1}^{−[ \frac{a·α_{1}}{N}]} satisfies

g^{N} = u^{a} · f_{1}^{a·α_{1}−N[ \frac{a·α_{1}}{N}]} = u^{a} · f_{1}.
□

3.14. 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)( ^{N} \sqrt{f}), where f is a rational function giving the section s.

b) π′ : Y′ ⟶ X can be as well described in the following way (see [30], p. 128-129): Let V(L^{−α}) = Spec (⨁_{i=0}^{∞} L^{−α}) be the geometric rank one vector bundle associated to L^{−α}. The geometric sections of V(L^{−α}) ⟶ X correspond to H^{0}(X, L^{α}). Hence s gives a section σ of V(L^{−N}) over X. We have a natural map

τ : V(L^{−1}) ⟶ V(L^{−N})

and Y′ = τ^{−1}(σ(X)).

The local computation in (3.13) gives a little bit more information than asked for in (3.12):

3.15. Lemma. Keeping the notations and assumptions from (3.5) assume that N is prime to char(k). Then one has

a) Y is reducible, if and only if for some μ > 1, dividing N, there is a section s′ in H^{0}(X, L^{\frac{N}{μ}}) with s = s^{′μ}.

b) π : Y → X is étale over X − D_{red} and Y is non-singular over X − Sing(D_{red}).

c) For Δ_{j} = (π^{∗}D_{j})_{red} we have

π^{∗}D = ∑_{j=1}^{r} \frac{N · α_{j}}{gcd(N, α_{j})} · Δ_{j}.

d) If Y is irreducible then the components of Δ_{j} have over D_{j} the ramification index

e_{j} = \frac{N}{gcd(N, α_{j})}.

Proof: For a) we can consider the open set Spec B ⊂ 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′ ∈ B with u = u^{′^{μ}}.

b) has been obtained in (3.13) part a) and b). For c) and d) we may assume that D = α_{1} · D_{1} and, splitting the covering in two steps, that either N divides α_{1} or that N is prime to α_{1}. In the first case, we can as well choose α_{1} to be zero (by 3.3,b) and c) as well as d) follow from part b). If α_{1} is prime to N,then π^{∗}D = e_{1} · α_{1} · Δ_{1}. Since π^{∗}D is the zero locus of f = t^{N}, N divides e_{1} · α_{1}. On the other hand, since e_{1} divides deg (Y/X) = N, one has e_{1} = N in this case.

□

3.16. 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) π^{∗}Ω_{X}^{b}(log D) = Ω_{Y}^{b} (log (π^{∗}D)).

b) The differential d on Y induces a logarithmic integrable connection

π_{∗}(d) : ⨁_{i=0}^{N−1} L^{(i)^{−1}} ⟶ π_{∗}Ω_{Y}^{1} (log (π^{∗}D)) = ⨁_{i=0}^{N−1} Ω_{X}^{1}(log D) ⊗ L^{(i)^{−1}},

compatible with the direct sum decomposition.

c) If ∇^{(i)} : L^{(i)^{−1}} ⟶ Ω_{X}^{1}(log D) ⊗ L^{(i)^{−1}} denotes the i-th component of π_{∗}(d) then ∇^{(i)} is a logarithmic integrable connection with residue

Res_{D_{j}}(∇^{(i)}) = (\frac{i · α_{j}}{N} − [\frac{i · α_{j}}{N}]) · id_{O_{D_{j}}} .

d) One has

π_{∗}(Ω_{Y}^{b}) = ⨁_{i=0}^{N−1} Ω_{X}^{b}(log D^{(i)}) ⊗ L^{(i)^{−1}} for D^{(i)} = ∑_{{j=1 \atop \frac{i·α_{j}}{N} ∈Q l −ZZ}}^{r} D_{j}.

e) The differential

π_{∗}(d) : π_{∗}O_{Y} = ⨁_{i=0}^{N−1} L^{(i)^{−1}} ⟶ π_{∗}(Ω_{Y}^{1} ) = ⨁_{i=0}^{N−1} Ω_{X}^{1}(log D^{(i)}) ⊗ L^{(i)^{−1}}

decomposes into a direct sum of

∇^{(i)} : L^{(i)^{−1}} ⟶ Ω_{X}^{1}(log D^{(i)}) ⊗ L^{(i)^{−1}}.

Proof: Again we can argue locally and assume that X = Spec B and D = α_{1}D_{1} as in (3.12). If α_{1} = 0, or if N divides α_{1}, then f_{1} is a defining equation for Δ_{1} = (π^{∗}D_{1})_{red} and the generators for Ω_{X}^{b}(log D) are generators for Ω_{Y}^{b} (log π^{∗}D) as well. For α_{1} prime to N, we have by (3.13,c) a defining equation g for Δ_{1} = (π^{∗}D_{1})_{red} with g^{N} = u^{a} · f_{1}. Hence

N · \frac{dg}{g} = \frac{df_{1}}{f_{1}} + a · \frac{du}{u}

and, since N ∈ k^{∗} and a · \frac{du}{u} ∈ Ω_{X}^{1}, one finds that \frac{df_{1}}{f_{1}} and π^{∗}Ω_{X}^{1} generate Ω_{Y}^{1} (log π^{∗}D).

We can split π in two coverings of degree N · gcd (N, α_{1})^{−1} and gcd (N, α_{1}). Hence we obtain a) for b = 1. The general case follows.

The group G acts on π_{∗}Ω_{Y}^{b} and π_{∗}Ω_{Y}^{b} (log π^{∗}D)) compatibly with the inclusion, and the action on the second sheaf is given by id ⊗ σ if one writes

π_{∗}Ω_{Y}^{b} (log (π^{∗}D)) = Ω_{X}^{b}(log D) ⊗ π_{∗}O_{Y} .

Let l be a local section of Ω_{X}^{b}(log D) ⊗ L^{(i)^{−1}} written as

l = ϕ · g_{i} for ϕ ∈ Ω_{X}^{b}(log D) and g_{i} = t^{i} · f_{1}^{−[ \frac{i·α_{1}}{N}]}.

Since

g_{i}^{N} = u^{i} · f_{1}^{i·α_{1}−N·[ \frac{iα_{1}}{N}]}

has a zero along Δ_{1} if and only if

\frac{i · α_{1}}{N} ∉ ZZ,

we find that l lies in Ω_{Y}^{b} in this case. On the other hand, if g_{i} is a unit, l lies in Ω_{Y}^{b} if and only if ϕ has no pole along D and we obtain d).

We have

N \frac{dg_{i}}{g_{i}} = i · \frac{du}{u} + (i · α_{1} − N[\frac{i · α_{1}}{N}])\frac{df_{1}}{f_{1}}

or

dg_{i} = (\frac{i}{N} \frac{du}{u} + (\frac{i}{N} α_{1} − [\frac{i · α_{1}}{N}])\frac{df_{1}}{f_{1}} ) · g_{i}.

Hence,

d(g_{i} · ϕ) ∈ Ω_{X}^{b+1}(log D) ⊗ L^{(i)^{−1}},

and (π_{∗}d) respects the direct sum decomposition. Obviously, the Leibniz rule for d implies that (π_{∗}d) as well as the components ∇^{(i)} are connections and b) and e) hold true.

Finally, for c), let ϕ ∈ O_{X}. Then by the calculations given above, we find

Res_{D_{1}}(∇^{(i)})(g_{i} · ϕ) = (\frac{i}{N} α_{1} − [\frac{i · α_{1}}{N}])g_{i} · ϕ |_{D_{1}} .

□

Proof of 3.2,a: If X is projective and

D = ∑_{j=1}^{r} α_{j}D_{j}

a normal crossing divisor we found the connection

∇^{(i)} : L^{(i)^{−1}} ⟶ Ω_{X}^{1}(log D^{(i)}) ⊗ L^{(i)^{−1}}

with the residues as given in (3.2,a) over the open submanifold X −Sing(D_{red}). Of course, ∇^{(i)} extends to X since

codim_{X}(Sing(D_{red})) ≥ 2.
□

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, Ω_{T}^{a}) ⟹ IH^{a+b}(T, Ω_{T}^{•})

for projective manifolds T by using the following covering Lemma, due to Y. Kawamata [35]:

3.17. 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

δ : T ⟶ Y

such that:

a) The degree of δ 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 (π ∘ δ)^{∗}(D + A + B) is a normal crossing divisor and (π ∘ δ)^{∗}A as well as (π ∘ δ)^{∗}B are reduced.

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} = π^{−1}(X_{0}) and T_{0} = δ^{−1}(Y_{0}). δ_{∗}Ω_{T_{0}}^{•} contains Ω_{Y_{0}}^{•} as direct summand. Since (π ∘ δ) is flat (π ∘ δ)_{∗}Ω_{T}^{•} will contain

⨁_{i=0}^{N−1} Ω_{X}^{•}(log D^{(i)}) ⊗ L^{(i)^{−1}}

as a direct summand. The E_{1}-degeneration of the spectral sequence

E_{1}^{ab} = H^{b}(T, Ω_{T}^{a}) ⟹ IH^{a+b}(T, Ω_{T}^{•})

implies (3.2,b) for each i ∈ {0, …, N − 1}. Finally, if A and B are the divisors considered in (3.2,c), A′ = (π ∘ δ)^{∗}A and B′ = (π ∘ δ)^{∗}B,

Ω_{X}^{•}(log (A + B + D^{(i)}))(−B) ⊗ L^{(i)^{−1}}

is a direct summand of

(π ∘ δ)_{∗}Ω_{T}^{•} (log (A′ + B′))(−B′)

and we can use the E_{1}-degeneration of

E_{1}^{ab} = H^{b}(T, Ω_{T}^{a} (log (A′ + B′))(−B′)) ⟹ IH^{a+b}(T, Ω_{T}^{•} (log (A′ + B′))(−B′)).

□

3.18. 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 [11]. In these lectures (see (10.23)) we will reproduce the algebraic proof of Deligne and Illusie for the degeneration.

b) If char (k) ≠ 0 and if X, 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:

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 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 §10, without using (3.17), for a field k of characteristic p ≠ 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.

To prove (3.17) we need:

3.19. Lemma (Kawamata [35]). Let X be a quasi-projective manifold, let

D = ∑_{j=1}^{r} D_{j}

be a reduced normal crossing divisor, and let

N_{1}, …, N_{r} ∈ IN − {0}

be prime to char(k). Then there exists a projective manifold Z and a finite morphism τ : Z → X such that:

a) For j = 1, … r one has τ^{∗}D_{j} = N_{j} · (τ^{∗}D_{j})_{red}.

b) τ^{∗}(D) is a normal crossing divisor.

c) The degree of τ divides some power of ∏_{j=1}^{r} N_{j}.

d) If X and D satisfy the lifting property (8.11) the same holds true for Z.

Proof: If we replace the condition that D = ∑_{j=1}^{r} D_{j} is the decomposition of D into irreducible (non-singular) components by the condition that D = ∑_{j=1}^{r} D_{j} for non-singular divisors D_{1} … D_{r} we can construct Z by induction and hence assume that N_{1} = N and N_{2} = … = N_{r} = 1.

Let A be an ample invertible sheaf such that A^{N}(−D_{1}) is generated by its global sections. Choose n = dim X general divisors H_{1}, …, H_{n} with

O_{X}(H_{i}) = A^{N}(−D_{1}).

The divisor D + ∑_{i=1}^{n} H_{i} will be a reduced normal crossing divisor. Let

τ_{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} ∩ D_{1} and τ_{i}^{∗}(D) might have non-normal crossings over H_{i} ∩ D_{1}. Let Z be the normalization of

Z_{1} ×_{X} Z_{2} ×_{X} … ×_{X} Z_{n}.

Z can inductively be constructed as well in the following way: Let Z^{(ν)} be the normalization of Z_{1} ×_{X} … ×_{X} Z_{ν} and τ^{(ν)} : Z^{(ν)} → X the induced morphism. Then, outside of the singular locus of Z^{(ν)}, the cover Z^{(ν+1)} is obtained from Z^{(ν)} by taking the N-th root out of

τ^{(ν)^{∗}}(H_{ν+1} + D_{1}) = τ^{(ν)^{∗}}(H_{ν+1}) + N · (τ^{(ν)^{∗}}D_{1})_{red}.

This is the same as taking the N-th root out of τ^{(ν)^{∗}}(H_{ν+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^{(ν+1)} lie over the singularities of Z^{(ν)}, hence inductively over H_{1} ∩ D_{1}. However, as Z is independent of the numbering of the H_{i}, the singularities of Z are lying over

⋂_{i=1}^{n}(H_{i} ∩ D_{1}) = (⋂_{i=1}^{n} H_{i}) ∩ D_{1} = ∅.

□

Proof of (3.17): Let τ : Z → X be the covering constructed in (3.19) for

D_{red} = ∑_{j=1}^{r} D_{j}

and N = N_{1} = … = N_{r}. Let T be the normalization of Z ×_{X} Y. Then T is obtained again by taking the N-th root out of τ^{∗}D. Since τ^{∗}D = N · D′ for some divisor D′ on Z, we can use (3.3,b), (3.10) and (3.15,b) to show that T is étale 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.

□

Generalizations and variants in the analytic case

(3.17) is a special case of the more general covering lemma of Kawamata:

3.20. Lemma. Let X be a projective manifold, char(k) = 0 and let π : Y → X be a finite cover such that the ramification locus D = Δ(Y/X) in X has normal crossings. Then there exists a manifold T and a finite morphism δ : T → Y. Moreover, one can assume that π ∘ δ : T → X is a Galois cover.

For the proof see [35]. As shown in [63] (3.16) can be generalized as well:

3.21. Lemma. (Generalized Hurwitz’s formula) For π : Y → X as in (3.20) let δ : Z → Y be a desingularization such that (π ∘ δ)^{∗}D = D′ is a normal crossing divisor. Then one has an inclusion

δ^{∗}π^{∗}Ω_{X}^{a}(log D) ⟶ Ω_{Z}^{a}(log D′)

giving an isomorphism over the open subscheme U in Z where (π ∘ δ) |_{Z} is finite.

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 [62] or (5.13)), i.e.:

R^{b}δ_{∗}O_{Z} = 0 for b > 0.

One can even show (see [17]):

3.22. Lemma. For Y normal and π : Y → X, δ : Z → Y as in (3.21) and τ = π ∘ δ one has:

R^{b}τ_{∗}Ω_{Z}^{a}(log D′) = \{ \begin{matrix}Ω_{X}^{a}(log D) ⊗ ⨁_{i=0}^{N−1} L^{(i)^{−1}} & for b = 0 \\ 0 & for b > 0 \end{matrix}

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′ (see Deligne [10]).

Using (3.21) and (3.22) one finds again (see [20]): The degeneration of the spectral sequence

E_{1}^{ab} = H^{b}(Z, Ω_{Z}^{a}(log D′)) ⟹ H^{a+b}(Z, Ω_{Z}^{•}(log D′))

implies (3.2,b).

Let us end this section with the following

3.23. Corollary. Under the assumptions of 3.2 assume that k = C l . Then

dim (H^{b}(X, Ω_{X}^{a}(log D^{(i)}) ⊗ L^{(i)^{−1}})) = dim (H^{a}(X, Ω_{X}^{b}(log D^{(N−i)}) ⊗ L^{(N−i)^{−1}})).

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).

□

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).

3.24. Lemma. Let X be a quasi-projective manifold, Y a normal variety and let π : Y ⟶ X be a separable finite cover. Assume that the ramification divisor

D = ∑_{j=1}^{m} D_{j} = Δ(Y/X)

of π in X is a normal crossing divisor and that for all j and all components B_{j}^{i} of π^{−1}(D_{j}) the ramification index e(B_{j}^{i}) is prime to char k. Then Y has at most quotient singularities, i.e. each point y ∈ Y has a neighbourhood of the form T/G where T is nonsingular and G a finite group acting on T.

Proof: One can assume that X is affine. For j = 1, ⋯, m define

n_{j} = lcm{e(B_{j}^{i}); B_{j}^{i} component of π^{−1}(D_{j})}.

Let τ : 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, τ is the composition of

Z = Z_{m} ⟶^{τ_{m}} Z_{m−1} ⟶^{τ_{m−1}} ⋯ ⟶ Z_{1} ⟶^{τ_{1}} Z_{0} = X

§ 4  Vanishing theorems, the formal set-up.  35

where τ_{j} : Z_{j} ⟶ Z_{j−1} is the cover obtained by taking the n_{j}-th root out of (τ_{1} ∘ τ_{2} ∘ ⋯ ∘ τ_{j−1})^{∗}(D_{j}). By (3.15,b) Z_{j} is non singular. Z is Galois over X with Galois group

G = ∏_{j=1}^{m} ZZ/n_{j} · ZZ.

Let T be the normalization of Z ×_{X} Y and δ : 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 δ_{0} = δ|_{T_{0}} is obtained by taking sucessively the \frac{n_{j}}{α_{j}} -th root out of

π^{−1}(D_{j}) = ∑_{i=1}^{r_{j}} \frac{e(B_{j}^{i})}{α_{j}} · B_{j}^{i}

for

α_{j} = gcd{e(B_{j}^{i}); B_{j}^{i} component of π^{−1}(D_{j})}.

By (3.15) all components of δ^{−1}(B_{j}^{i}) have ramification index

\frac{\frac{n_{j}}{α_{j}}}{gcd\{\frac{n_{j}}{α_{j}} , \frac{e(B_{j}^{i})}{α_{j}}\}} = \frac{n_{j}}{e(B_{j}^{i})}

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.

□

§ 4 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 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.

4.1. Assumptions. Let X be a projective manifold defined over an algebraically closed field k and let

D = ∑_{j=1}^{r} D_{j}

be a reduced normal crossing divisor. Let E be a locally free sheaf on X of finite rank and let

∇ : E ⟶ Ω_{X}^{1}(log D) ⊗ E

be an integrable connection with logarithmic poles along D. We will assume in the sequel that ∇ satisfies the E_{1}-degeneration i.e. that the Hodge to de Rham spectral sequence (A.25)

E_{1}^{ab} = H^{b}(X, Ω_{X}^{a}(log D) ⊗ E) ⟹ IH^{a+b}(X, Ω_{X}^{•}(log D) ⊗ E)

degenerates in E_{1}.

4.2. Lemma (Vanishing for restriction maps I). Assume that ∇ satisfies the condition (!) of (2.8), i.e. that for all μ ∈ IN and for j = 1, …, r the map

Res_{D_{j}}(∇) + μ · id_{O_{D_{j}}} : E |_{D_{j}}⟶ E |_{D_{j}}

is an isomorphism. Assume that ∇ satisfies the E_{1}-degeneration (4.1). Then for all effective divisors

D′ = ∑_{j=1}^{r} μ_{j}D_{j}

and all b the natural map

H^{b}(X, O_{X}(−D′) ⊗ E) ⟶ H^{b}(X, E)

is surjective.

Proof: By (2.9,b) the map

Ω_{X}^{•}(log D) ⊗ E(−D′) ⟶ Ω_{X}^{•}(log D) ⊗ E

is a quasi-isomorphism and hence induces an isomorphism of the hypercohomology groups. Let us consider the exact sequences of complexes

0 ⟶ Ω_{X}^{•^{≥1}}(log D) ⊗ E ⟶ Ω_{X}^{•}(log D) ⊗ E ⟶ E ⟶ 0
↑  ↑  ↑
0 ⟶ Ω_{X}^{•^{≥1}}(log D) ⊗ E(−D′) ⟶ Ω_{X}^{•}(log D) ⊗ E(−D′) ⟶ E(−D′) ⟶ 0.

By assumption, the spectral sequence for Ω_{X}^{•}(log D) ⊗ E degenerates in E_{1}, which implies that the morphism α in the following diagram is surjective (see (A.25)).

IH^{b}(X, Ω_{X}^{•}(log D) ⊗ E) ⟶^{α} H^{b}(X, E)
↑^{=}  ↑^{β}
IH^{b}(X, Ω_{X}^{•}(log D) ⊗ E(−D′)) ⟶ H^{b}(X, E(−D′))

Hence β is surjective as well.

□

§ 4  Vanishing theorems, the formal set-up.  37

4.3. Variant. If in (4.2)

D′ = ∑_{j=1}^{s} μ_{j}D_{j} ≥ 0 for s ≤ r,

then it is enough to assume that for j = 1, …, s and for 0 ≤ μ ≤ μ_{j} − 1

Res_{D_{j}}(∇) + μ · id_{O_{D_{j}}}

is an isomorphism. Proof: By (2.10) this is enough to give the quasi-isomorphism

Ω_{X}^{•}(log D) ⊗ E(−D′) ⟶ Ω_{X}^{•}(log D) ⊗ E

needed in the proof of (4.2).

□

4.4. Lemma (Dual version of (4.2) and (4.3)). Assume that

∇ : E ⟶ Ω_{X}^{1}(log D) ⊗ E

satisfies the E_{1}-degeneration and that for j = 1, …, s and 1 ≤ μ ≤ μ_{j}

Res_{D_{j}}(∇) − μ · id_{O_{D_{j}}}

is an isomorphism (for example, if ∇ satisfies the condition (∗) from (2.8,a)). Then for

D′ = ∑_{j=1}^{s} μ_{j}D_{j}

and all b the map

H^{b}(X, ω_{X}(D) ⊗ E) ⟶ H^{b}(X, ω_{X}(D + D′) ⊗ E)

is injective. Proof: Consider the diagram

H^{b}(X, ω_{X}(D + D′) ⊗ E) ⟶ IH^{n+b}(X, Ω_{X}^{•}(log D) ⊗ E(D′))
↑^{β} ↑^{γ}
H^{b}(X, ω_{X}(D) ⊗ E) ⟶^{α} IH^{n+b}(X, Ω_{X}^{•}(log D) ⊗ E).

α is injective by the E_{1}-degeneration (see (A.25)) , γ is an isomorphism by (2.10) and hence β is injective.

□

The lemma (4.2) or its variant (4.3) implies that for all b the natural restriction maps

H^{b}(X, E) ⟶ H^{b}(D′, O_{D′} ⊗ E)

are the zero maps. For higher differential forms this remains true, if D′ is a non-singular divisor:

4.5. Lemma (Vanishing for restriction maps II). Assume that

∇ : E ⟶ Ω_{X}^{1}(log D) ⊗ E

satisfies E_{1}-degeneration. Let D′ be a non-singular subdivisor of D and assume that for all components D_{j} of D′ the map Res_{D_{j}}(∇) is an isomorphism. (For example this follows from condition (!) in (2.8,b)). Then the restriction (see (2.3))

H^{b}(X, Ω_{X}^{a}(log (D − D′)) ⊗ E) ⟶ H^{b}(D′, Ω_{D′}^{a}(log (D − D′) |_{D′}) ⊗ E)

is zero for all a and b.

Proof: As we have seen in (2.6,b) the restriction map factors through

H^{b}(∇_{a}) : H^{b}(X, Ω_{X}^{a}(log D) ⊗ E) ⟶ H^{b}(X, Ω_{X}^{a+1}(log D) ⊗ E)

provided Res_{D_{j}}(∇) is an isomorphism on the different components D_{j} of D′. By E_{1}-degeneration, H^{b}(∇_{a}) is the zero map (see (A.25)).

□

Before we are able to state the global vanishing for E or Ω_{X}^{a}(log D) ⊗ E we need some more notations.

4.6. Definition. Let U ⊂ X be an open subscheme and let B be an effective divisor with B_{red} = X − U. Then we define the (coherent) cohomological dimension of (X, B) to be the least integer α such that for all coherent sheaves F and all k > α one finds some ν_{0} > 0 with H^{k}(X, F(ν · B)) = 0 for all multiples ν of ν_{0}. Finally, for the reduced divisor D = X − U, we write

cd(X, D) = Min{ α ; there exists some effective divisor B with B_{red} = D, such that α is the cohomological dimension of (X, B)}.

4.7. Examples.

a) For D = X − U the embedding ι : U → X is affine and for a coherent sheaf G on X we have

H^{b}(U, G |_{U}) = H^{b}(X, ι_{∗}(G |_{U})) = lim_{\overrightarrow{α∈IN}} H^{b}(X, G ⊗ O_{X}(α · B)),

where B is any effective divisor with B_{red} = D. In particular, if b > cd(X, D) we find

H^{b}(U, G |_{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 G and all multiples ν of some ν_{0} > 0 one has

dim H^{b}(X, G ⊗ O_{X}(−ν · 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).

4.8. Lemma (Vanishing for cohomology groups). Assume that X is projective and that

∇ : E ⟶ Ω_{X}^{1}(log D) ⊗ E

satisfies the E_{1}-degeneration (see (4.1)).

a) If ∇ satisfies the condition (∗) of (2.8) and if a + b > n + cd(X, D), then

H^{b}(X, Ω_{X}^{a}(log D) ⊗ E) = 0.

b) If ∇ satisfies the condition (!) of (2.8) and if a + b < n − cd(X, D), then

H^{b}(X, Ω_{X}^{a}(log D) ⊗ E) = 0.

Proof: Let us choose α ∈ ZZ with α ≥ 0 in case a) and with α ≤ 0 in case b). For B ≥ D, (2.9) tells us that

Ω_{X}^{•}(log D) ⊗ E and Ω_{X}^{•}(log D) ⊗ E(α · B)

are quasi-isomorphic. In both cases we have a spectral sequence

E_{1}^{ab} = H^{b}(X, Ω_{X}^{a}(log D) ⊗ E(α · B)) ⟹

⟹ IH^{a+b}(X, Ω_{X}^{•}(log D) ⊗ E(α · B)) = IH^{a+b}(X, Ω_{X}^{•}(log D) ⊗ E).

By assumption this spectral sequence degenerates for α = 0 and, for arbitrary α we have (see (A.16))

∑_{a+b=l} dim H^{b}(X, Ω_{X}^{a}(log D) ⊗ E) = dim IH^{l}(X, Ω_{X}^{•}(log D) ⊗ E)

≤ ∑_{a+b=l} dim H^{b}(X, Ω_{X}^{a}(log D) ⊗ E(α · 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 α > 0 in case a), or l < n − cd(X, D) and all α < 0 in case b).

□

The same argument shows:

4.9. Variant. In (4.8) we can replace a) and b) by:

c) Let D_{∗} and D_{!} be effective divisors, both smaller than D, and assume that i) For all components D_{j} of D_{∗} and all μ ∈ IN − {0}

Res_{D_{j}}(∇) − μ · id_{O_{D_{j}}}

is an isomorphism.

ii) For all components D_{j} of D_{!} and all μ ∈ IN

Res_{D_{j}}(∇) + μ · id_{O_{D_{j}}}

is an isomorphism. Then

H^{b}(X, Ω_{X}^{a}(log D) ⊗ E) = 0

for a + b > n + cd(X, D_{∗}) and for a + b < n − cd(X, D_{!}).

The analytic case As we have seen in the proof of (4.8) the condition (∗) implies that

IH^{l}(X, Ω_{X}^{•}(log D) ⊗ E) = 0 for l > n + cd(X, D).

For k = C l , this is not the best possible result. In fact, as mentioned in (2.12), (∗) implies that over C l

IH^{l}(X, Ω_{X}^{•}(log D) ⊗ E) = 0 for l > n + r(X − D)

where r(X − D) is the least number α such that H^{l}(X − D, V) = 0 for all locally constant systems V on X − D and l > n + α.

(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( )”. 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.

4.10. Definition. Let U be an algebraic irreducible variety and g : U → W a morphism. Then define

r(g) = Max{ dim Γ − dim g(Γ) − codim Γ;

Γ irreducible closed subvariety of U }

4.11. Properties. a)

r(g) = Max{ dim (generic fibre of g |_{Γ}) − codim Γ;

Γ irreducible closed subvariety of U }

b) If b denotes the maximal fibre dimension for g, then

r(g) ≤ Max{dim U − dim W; b − 1}.

c) If U′ ⊆ U is open and dense, then r(g |_{U′}) ≤ r(g).

d) If Δ ⊆ U is closed then r(g |_{Δ}) ≤ r(g) + codim_{U} (Δ).

Proof: a) and c) are obvious and b) follows from a). For d) one remarks that for Γ ⊂ Δ one has

codim_{Δ} (Γ) = codim_{U} (Γ) − codim_{U} (Δ).
□

4.12. Lemma. (Improvement of 4.8 using analytic methods) Let X be a projective manifold defined over an algebraically closed field k of characteristic zero. Assume that

∇ : E ⟶ Ω_{X}^{1}(log D) ⊗ 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 ∇ satisfies the condition (∗) of (2.8) then

H^{b}(X, Ω_{X}^{a}(log D) ⊗ E) = 0

for a + b > n + r(g).

b) If ∇ satisfies the condition (!) of (2.8) then

H^{b}(X, Ω_{X}^{a}(log D) ⊗ E) = 0

for a + b < n − r(g).

Proof: By flat base chance we can replace k by any other field k′, such that X, D, E are defined over k′. Hence, we may assume that k = C l . By GAGA (see [56]) we may assume in (4.12) that all the sheaves and ∇ are analytic. Then, by (2.12) and by the E_{1}-degeneration it is enough to show:

4.13. Lemma. Let U be an analytic manifold, W be an affine manifold and g : U → 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.

Proof (see [22]): The sheaves R^{a}g_{∗}V are analytically constructible sheaves ([61]) and their support

S_{a} = Supp(R^{a}g_{∗}V)

must be a Stein space, hence

H^{b}(W, R^{a}g_{∗}V) = 0 for b > dim S_{a}.

However, the general fibre of g |_{g^{−1}(S_{a})} must have a dimension larger than or equal to \frac{a}{2} . Hence

2 · (dim g^{−1}(S_{a}) − dim S_{a}) ≥ a

and H^{b}(W, R^{a}g_{∗}V) = 0 for

a + b > n + r(g) ≥ 2 · dim g^{−1}(S_{a}) − dim S_{a} ≥ a + dim S_{a}.

By the Leray spectral sequence (A.27)

E_{2}^{ba} = H^{b}(W, R^{a}g_{∗}V) ⟹ H^{a+b}(U, V)

one obtains (4.13).

□

4.14. Remark. If W is affine and g : U → 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.

§ 5 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 l -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 (∗) and (!) are too much to ask for in characteristic p ≠ 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 L = L^{(1)}. By (3.3) we are not losing any information.

5.1. Vanishing for restriction maps related to Q l -divisors:

Let X be a projective manifold defined over an algebraically closed field k, let L be an invertible sheaf, N ∈ IN − {0} and let

D = ∑_{j=1}^{r} α_{j}D_{j}

be a normal crossing divisor with 0 < α_{j} < N and L^{N} = O_{X}(D). Let

D′ = ∑_{j=1}^{r} μ_{j}D_{j}

be an effective divisor. Then one has:

a) If char (k) = 0 then for all b the natural morphism

H^{b}(X, L^{−1}(−D′)) ⟶ H^{b}(X, L^{−1})

§ 5  Vanishing theorems for invertible sheaves  43

is surjective and hence the morphism

H^{b}(X, ω_{X} ⊗ L) ⟶ H^{b}(X, ω_{X}(D′) ⊗ L)

injective.

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, L^{−1}(−C − D′)) ⟶ H^{b}(X, L^{−1}(−C))

is surjective and hence the morphism

H^{b}(X, ω_{X}(C) ⊗ L) ⟶ H^{b}(X, ω_{X}(D′ + C) ⊗ L)

injective.

c) If char (k) = p ≠ 0, then a) and b) hold true under the additional assumptions:

i) X and D (as well as C) satisfy the lifting property (8.11) and dim (X) ≤ p.

ii) N is prime to p.

iii) For all j and 0 ≤ μ ≤ μ_{j} − 1 one has α_{j} + μ · N ≢ 0 mod p.

Proof: By (3.2,c) O_{X}(−C) ⊗ L^{−1} has a logarithmic connection ∇ with poles along C + D_{red} satisfying E_{1}-degeneration, and Res_{D_{j}}(∇) = \frac{α_{j}}{N}. Hence (5.1) follows from (4.2) and (4.3) or (4.4).

□

5.2. Corollary (Kodaira [39], Deligne, Illusie [12]). Let X be a projective manifold and L an invertible sheaf. If char (k) = p > 0, then assume in addition that X and L admit a lifting to W_{2}(k) (8.11) and that dim X ≤ p. Then, if L is ample,

H^{b}(X, L^{−1}) = 0 for b < n = dim (X)

Proof: Choose N, prime to p = char (k), such that

H^{b}(X, L^{−N−1}) = H^{n−b}(X, ω_{X} ⊗ L^{N+1}) = 0

for b < n, and such that L^{N} is generated by global sections. If D is a general section of L^{N}, then we can apply (5.1) and find

H^{b}(X, L^{−1}(−D)) ⟶ H^{b}(X, L^{−1})

to be surjective. Since the group on the left hand side is zero, we are done. □ 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 L lifts as well is not necessary.

5.3. Definition. Let X be a projective variety and L be an invertible sheaf on X. If H^{0}(X, L^{ν}) ≠ 0, the sections of L define a rational map

ϕ_{ν} = ϕ_{L^{ν}} : X ⟶ IP(H^{0}(X, L^{ν})).

The Iitaka-dimension κ(L) of L is given by

κ(L) = \{ \begin{matrix}−∞ & if H^{0}(X, L^{ν}) = 0 for all ν \\ Max\{dim ϕ_{ν}(X); H^{0}(X, L^{ν}) ≠ 0\} & otherwise \end{matrix}

5.4. Properties. For X and L as above one has:

a) κ(L) ∈ {−∞, 0, 1, …, dim X}.

b) If H^{0}(X, L^{ν}) ≠ 0 for some ν > 0 then one can find a, b ∈ IR, a, b > 0, such that

a · μ^{κ(L)} ≤ dim H^{0}(X, L^{ν·μ}) ≤ b · μ^{κ(L)} for all μ ∈ IN − {0}.

c) If κ(L) ≠ −∞, then

κ(L) = tr.deg (⨁_{μ≥0} H^{0}(X, L^{μ})) − 1.

d) One has κ(L) = dim X, if and only if for some ν > 0 and some effective divisor C the sheaf L^{ν}(−C) is ample.

e) If A is very ample and A the zero divisor of a general section of A, then

κ(L |_{A}) ≥ Min {κ(L), dim A}.

Proof: a), b) and c) are wellknown and their proof can be found, for example, in [46], §1. For d) let A be an ample effective divisor. For n = dim X and some ν ∈ IN − {0} one finds a, b ∈ IR, a, b > 0, with a · μ^{n} < dim H^{0}(X, L^{ν·μ}) and dim H^{0}(X, L^{ν·μ}|_{A}) < b · μ^{n−1}. Hence, the exact sequence

0 ⟶ H^{0}(X, L^{ν·μ}(−A)) ⟶ H^{0}(X, L^{ν·μ}) ⟶ H^{0}(A, L^{ν·μ} |_{A})

shows that for some μ we have O_{X}(A) as a subsheaf of L^{ν·μ}. On the other hand, if A ⊂ L^{ν} is ample then

n = κ(A) ≤ κ(L^{ν}) = κ(L).

If κ(L) = n in e), then, using d) for example, κ(L |_{A}) = n − 1. If κ(L) < n, then H^{0}(X, O_{X}(−A) ⊗ L^{ν}) = 0 for all ν and b) implies

κ(L |_{A}) ≥ κ(L).
□

For our purposes we can take (5.4,b) as definition of κ(L), and we only need to know (5.4,d) and (5.4,e).

5.5. Definition. An invertible sheaf L on X is called

a) semi-ample, if for some μ > 0 the sheaf L^{μ} is generated by global sections.

b) numerically effective (nef) if for all curves C in X one has

deg (L |_{C}) ≥ 0.

The proof of (5.2) can be modified to give in characteristic zero a stronger statement:

5.6. Corollary. Let X be a projective manifold defined over a field k of characteristic zero and let L be an invertible sheaf.

a) (Kollár [40]) If L is semi-ample and B an effective divisor with H^{0}(X, L^{ν}(−B)) ≠ 0 for some ν > 0, then the natural maps

H^{b}(X, L^{−1}(−B)) ⟶ H^{b}(X, L^{−1})

are surjective for all b, or, equivalently, the adjunction map

H^{b}(X, L ⊗ ω_{X}(B)) ⟶ H^{b}(B, L ⊗ ω_{B})

is surjective for all b.

b) (Grauert-Riemenschneider [25]) If L is semi-ample and κ(L) = n = dim X, then

H^{b}(X, L^{−1}) = 0 for b < n.

Proof: Obviously a) and b) are compatible with blowing ups τ : X′ → X. In fact, using the Leray spectral sequence (A.27) we just have to remember that

R^{b}τ_{∗}τ^{∗}L = L ⊗ R^{b}τ_{∗}O_{X′} = \{ \begin{matrix}L & for b = 0 \\ 0 & for b ≠ 0 \end{matrix}

(See (5.13) for a generalization). Hence we may assume in a) that L^{ν} = O_{X}(B + C) for an effective normal crossing divisor B + C. We can choose some μ, with

[\frac{B + C}{μ}] = 0

and such that L^{μ} is generated by its global sections. If D_{1} is a general divisor of L^{μ}, i.e. the zero set of a general s ∈ H^{0}(X, L^{μ}), then D = D_{1} + B + C has normal crossings and [\frac{D}{μ}] = 0. Hence, for N = ν + μ and D′ = B the assumptions of (5.1,a) hold true and we obtain a).

For b), let us choose some divisor C and some ν such that L^{ν}(−C) = A is ample. Replacing A by some multiple, we may assume by Serre’s vanishing theorem (1.1) that

H^{b}(X, L^{−1} ⊗ A^{−1}) = H^{n−b}(X, ω_{X} ⊗ L ⊗ A) = 0

for b < n, and that A = O_{X}(B) for some divisor B. By a)

H^{b}(X, O_{X}(−B) ⊗ L^{−1}) ⟶ H^{b}(X, L^{−1})

is surjective. One obtains b), since the left hand side is zero.

□

It is not difficult to modify both parts of this proof to include in b) the case that L is nef and κ(L) = dim X. Moreover, considering very ample divisors on X and using induction on dim(X), one can as well remove the assumption “κ(L) = dim(X)” and obtain the vanishing for b < κ(L). We leave the details to the reader. Those techniques will appear in (5.12) anyway, when we prove a more general statement.

5.7. Lemma. For an invertible sheaf L on a projective manifold X the following two conditions are equivalent:

a) L is numerically effective.

b) For an ample sheaf A and all ν > 0 the sheaf L^{ν} ⊗ A is ample.

Proof: By Seshadri’s criterion A′ is ample if and only if for some ϵ > 0 and all curves C in X

deg (A′ |_{C}) ≥ ϵ · m(C)

where m(C) is the maximal multiplicity of points on C.

□

5.8. Lemma. For X, L as in (5.7), assume that L is numerically effective (and, if char (k) = p ≠ 0, that X and L satisfy the lifting property (8.11) and that dim X ≤ p). Then one has: a) κ(L) = n = dim X, if and only if c_{1}(L)^{n} > 0 (where c_{1}(L) is the Chern class of L).

b) For b ≥ 0 and for all invertible sheaves F one has a constant c_{b} > 0 with

dim H^{b}(X, F ⊗ L^{ν}) ≤ c_{b} · ν^{n−b} for all ν ∈ IN.

Proof: a) follows from b) and from the Hirzebruch-Riemann-Roch theorem which tells us that χ(X, L^{ν}) is a polynomial of deg n with highest coefficient

\frac{1}{n!} · c_{1}(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, O_{X}(H) ⊗ L^{ν} ⊗ F) = 0.

In fact, we just have to choose H such that F ⊗ ω_{X}^{−1} ⊗ O_{X}(H) is ample. Then by (5.7)

F ⊗ ω_{X}^{−1} ⊗ O_{X}(H) ⊗ L^{ν}

will be ample for all ν ≥ 0 and the vanishing required holds true by (5.2). From the exact sequence

0 ⟶ F ⊗ L^{ν} ⟶ F ⊗ L^{ν} ⊗ O_{X}(H) ⟶ F ⊗ O_{H}(H) ⊗ L^{ν} ⟶ 0

we obtain an isomorphism

H^{b−1}(H, F ⊗ O_{H}(H) ⊗ L^{ν}) ≃ H^{b}(X, F ⊗ L^{ν})

for b > 1 and a surjection

H^{0}(H, F ⊗ O_{H}(H) ⊗ L^{ν}) ⟶ H^{1}(X, F ⊗ L^{ν}).

By induction we find c_{b} for b ≥ 1 and, since H^{0}(X, F ⊗ L^{ν}) is bounded above by a polynomial of deg ν, for b = 0 as well.

□

Even if L is nef, there is in general no numerical characterisation of κ(L). For example, there are numerically effective invertible sheaves L with κ(L) = −∞. Following Kawamata [37], one defines:

5.9. Definition. Let L be a numerically effective invertible sheaf. Then the numerical Iitaka-dimension is defined as

ν(L) = Min {ν ∈ IN − {0}; c_{1}(L)^{ν} numerically trivial } − 1.

5.10. Properties. Let X, L be as in (5.8). Then one has:

a) ν(L) ≥ κ(L).

b) If L is semi-ample then

ν(L) = κ(L).

Proof: If ν(L) = dim X or κ(L) = dim X, then (5.8,a) gives

ν(L) = κ(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

κ(L |_{H}) ≥ κ(L)

and obviously

ν(L |_{H}) = ν(L).

By induction on dim (X) one obtains a). For b) we may assume that L = τ^{∗}M for a morphism τ : X → Z with M ample and with dim (Z) = κ(L). Then

c_{1}(L)^{ν} = τ^{∗}c_{1}(M)^{ν} = 0

if and only if ν > dim Z.

□

The following lemma, due to Y. Kawamata [37], is more difficult to prove, and we postpone its proof to the end of this lecture.

5.11. Lemma. For an invertible sheaf N on a projective manifold X, defined over a field k of characteristic zero, the following two conditions are equivalent:

a) N is numerically effective and ν(N) = κ(N)

b) There exist a blowing up τ : Z → X, some μ_{0} ∈ IN − {0} and an effective divisor C on Z such that

τ^{∗}N^{μ} ⊗ O_{Z}(−C)

is semi-ample for all μ ∈ IN − {0} divisible by μ_{0}.

5.12. Corollary. Let X be a projective manifold defined over a field k of characteristic zero, let L be an invertible sheaf on X, let

D = ∑_{j=1}^{r} α_{j}D_{j}

be a normal crossing divisor and N ∈ IN. Assume that

0 < α_{j} < N for j = 1, …, r.

Then one has:

a) If L^{N}(−D) is semi-ample and B an effective divisor such that

H^{0}(X, (L^{N}(−D))^{ν} ⊗ O_{X}(−B)) ≠ 0

for some ν > 0, then for all b the maps

H^{b}(X, L^{−1}(−B)) ⟶ H^{b}(X, L^{−1})

are surjective.

b) In a) one can replace “semi-ample” by the assumption that L^{N}(−D) is numerically effective and

κ(L^{N}(−D)) = ν(L^{N}(−D)).

c) (Kawamata [36] - Viehweg [63]) If L^{N}(−D) is numerically effective and

c_{1}(L^{N}(−D))^{n} > 0,

then

H^{b}(X, L^{−1}) = 0 for b < n.

d) (Kawamata [36] - Viehweg [63]) If L^{N}(−D) is numerically effective, then

H^{b}(X, L^{−1}) = 0 for b < κ(L).

e) Part d) remains true if one replaces κ(L) by κ(L ⊗ N^{−1}) for a numerically effective invertible sheaf N.

Again, the assumptions are compatible with blowing ups, except for “0 < α_{j} < N”. We need:

5.13. Claim. Let τ : X′ → X be a proper birational morphism and M = τ^{∗}L. Assume that Δ = τ^{∗}D has normal crossings. Then for

M^{(i)} = M(−[\frac{i · Δ}{N}])

one has

R^{b}τ_{∗}M^{(i)^{−1}} = \{ \begin{matrix}L^{(i)^{−1}} & for b = 0 \\ 0 & for b ≠ 0. \end{matrix}

Proof: We may assume that X is affine and that L = O_{X}. For b = 0 claim (5.13) follows from the inequality

[\frac{i · Δ}{N}] = [\frac{i · τ^{∗}D}{N}] ≥ τ^{∗}[\frac{i · D}{N}].

In general, for b ≥ 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 Δ. In fact, let Y′′ be a desingularization of Y′ and let

Y′′ ⟶^{σ} Y′ ⟶^{δ} Y
π′↓  ^{π}↓
X′ ⟶^{τ} X

be the induced morphisms. If Y′ has rational singularities, one has by definition R^{a}σ_{∗}O_{Y′′} = 0 for a > 0. Hence, if Y has rational singularities as well,

R^{a}(δ ∘ σ)_{∗}O_{Y′′} = R^{a}δ_{∗}O_{Y′} = 0

and

R^{a}τ_{∗}(π′_{∗}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 [62]). Let us recall the proof:

Let Y be any normal variety with quotient singularities and φ : Z ⟶ Y the corresponding Galois cover with Z non singular. Let δ : Y′ ⟶ Y be a desingularization such that D′ = δ^{∗}(Δ(Z/Y)) is a normal crossing divisor, where Δ(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 γ : Z′′ ⟶ Z′ be a desingularization. Altogether we obtain

Z′′ ⟶^{γ} Z′ ⟶^{δ′} Z
φ′↓  ^{φ}↓
Y′ ⟶^{δ} Y

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}}δ′_{∗}O_{Z′} = R^{a_{0}}δ′_{∗}(γ_{∗}O_{Z′′}) ↪ R^{a_{0}}(δ′ ∘ γ)_{∗}O_{Z′′}.

Since δ′ ∘ γ is a birational proper morphism of non singular varieties

R^{a_{0}}(δ′ ∘ γ)_{∗}O_{Z′′} = 0.

Since the finite morphisms φ and φ′ have no higher direct images one obtains

R^{a_{0}}δ_{∗}(φ′_{∗}O_{Z′}) = φ_{∗}(R^{a_{0}}δ′_{∗}O_{Z′}) = 0.

However, O_{Y′} is a direct summand of φ′_{∗}O_{Z′} and hence R^{a_{0}}δ_{∗}O_{Y′} = 0.

□

Proof of 5.12: let us first reduce b) to a): Applying (5.13) and replacing M by M^{(1)}, we can assume that the morphism τ : Z → X in (5.11,b), applied to N = L^{N}(−D), is an isomorphism and that for the divisor C in (5.11,b) D+C is a normal crossing divisor. L^{N·μ}(−μ·D−C) is semi-ample for all μ divisible by μ_{0}. Choosing μ large enough, the multiplicities of μ · D + C will be bounded above by N · μ. Moreover, we can assume that

H^{0}(X, L^{N·μ}(−μ · D − C)) ≠ 0

and hence, replacing μ again by some multiple, that

H^{0}(X, (L^{N·μ}(−μ·D − C))^{ν} ⊗ O_{X}(−B)) ≠ 0

for some ν > 0. Hence a) implies b).

To prove a), let us write

L^{ν·N}(−ν · D) = O_{X}(B + B′)

or

L^{ν·N} = O_{X}(ν · D + B + B′)

Blowing up, again, we can assume D + B + B′ to be a normal crossing divisor. For μ sufficiently large, we can assume that (L^{N}(−D))^{μ} is generated by global sections. If H is zero set of a general section, then

L^{(ν+μ)·N} = O_{X}(H + (ν + μ) · D + B + B′).

If μ is large enough, the multiplicities of the components of

D′ = H + (ν + μ) · D + B + B′

are smaller than (ν + μ) · N and, applying (5.1,a) for N′ = (ν + μ) · N and L^{N′} = O_{X}(D′), we obtain (5.12,a).

Let us remark next, that d), under the additional condition that κ(L) = n, implies c): In fact, c_{1}(L^{N}(−D))^{n} > 0 implies by (5.8,a) that

n = κ(L^{N}(−D)) ≤ κ(L^{N}) = κ(L).

To prove d), for κ(L) = n, we can apply (5.4,d). Hence we find a divisor C > 0 and μ > 0 with L^{μ}(−C) ample. Then by (5.7)

L^{N·ν+μ}(−ν · D − C)

is ample for all ν and, by Serre’s vanishing theorem (1.1)

H^{b}(X, L^{−1} ⊗ (L^{−N·ν−μ}(ν·D + C))^{η}) = 0 for b < n

and for η 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 ν large enough, we may again assume that the multiplicities of D′ = ν·D + C are smaller than N′ = N·ν + μ. Replacing D′ and N′ by some high multiple we can assume in addition that L^{N′}(−D′) is generated by global section and that

H^{b}(X, L^{−N′−1}(D′)) = 0 for b < n.

For an effective divisor B with O_{X}(B) = L^{′N}(−D′) we can apply a) and find

0 = H^{0}(X, L^{−1}(−B)) ⟶ H^{0}(X, L^{−1})

to be surjective. For κ(L) < dim X part d) is finally reduced to the case κ(L) = dim X by induction: Let H be a general hyperplane such that

H^{b}(X, O_{X}(−H) ⊗ L^{−1}) = 0 for b < n.

The exact sequence

0 ⟶ O_{X}(−H) ⊗ L^{−1} ⟶ L^{−1} ⟶ L^{−1} |_{H}⟶ 0

give isomorphisms

H^{b}(X, L^{−1}) ≃ H^{b}(H, L^{−1} |_{H})

for b < n − 1. Since κ(L) ≤ n − 1 we have κ(L |_{H}) ≥ κ(L) and both groups vanish for b < κ(L) by induction on dim X.

e) follows by the same argument: If κ(L ⊗ N^{−1}) = dim X, then (5.4,c) and (5.7) imply that κ(L) = dim X as well. For κ(L ⊗ N^{−1}) < dim X again

κ(L|_{H} ⊗ N^{−1}|_{H}) ≥ κ(L ⊗ N^{−1})

and by induction one obtains e).

□

Let us end this section by proving Kawamata’s lemma (5.11) which was needed to reduce (5.12,b) to (5.12,a):

Proof of (5.11): Let us assume b). Since τ^{∗}N^{μ} is nef, if and only if N is nef, and since ν(N) = ν(τ^{∗}N), we can assume that τ is an isomorphism. Moreover, we can assume μ_{0} = 1. “N^{μ}(−C) semi-ample for all μ > 0” implies that N is nef. One obtains from (5.10)

ν(N) ≥ κ(N) ≥ κ(N^{μ}(−C)) = ν(N^{μ}(−C)).

For ν = ν(N) the leading term in μ of

c_{1}(N^{μ}(−C))^{ν} = (μ · c_{1}(N) − C)^{ν}

is μ^{ν} · c_{1}(N)^{ν}. Since this term intersects H_{1} · . . . · H_{n−ν} strictly positively, for general hyperplanes H_{1}, …, H_{n−ν}, we find ν ≤ ν(N^{μ}(−C)).

To show the other direction, let ϕ_{μ_{0}} : X → Y be the rational map

X ⟶ ϕ_{μ_{0}}(X) = Y ⊂ IP(H^{0}(X, N^{μ_{0}})).

We can and we will assume that ϕ_{μ_{0}} has a connected general fibre, that dim (Y) = κ(N) and, blowing X up if necessary, that ϕ_{μ_{0}} is a morphism. For some effective divisor D we have

N^{μ_{0}}(−D) = ϕ_{μ_{0}}^{∗}L, for L ample on Y.

If F is a general fibre of ϕ_{μ_{0}}, then D |_{F} is nef.

5.14. Claim. D |_{F} is zero.

Assuming (5.14) we can blow up Y and X and assume that D = ϕ_{μ_{0}}^{∗}Δ for some divisor Δ on Y. For example, blowing up Y one can assume that ϕ_{μ_{0}} factors over a flat morphism ϕ′ : X′ → Y and that N is the pullback of a sheaf N^{′μ_{0}}(−D′) = ϕ^{′∗}L for some semi-ample sheaf L and by (5.14) D′ ⊆ ϕ^{′∗}Δ for some divisor Δ on Y. Since N′ is numerically effective D′·C ≥ 0 for all curves C in X′ contained in a fibre of ϕ′. This is only possible if D′ = ϕ^{′∗}Δ. Let us denote by τ : X → Y the morphism obtained. We have N^{μ_{0}} = τ^{∗}M for some sheaf M on Y. Of course κ(M) = dim Y and (5.12,b) holds true for M on Y, i.e. M^{μ}(−Γ) is ample for some divisor Γ > 0 and all μ >> 0. Then

N^{μ·μ_{0}}(−τ^{∗}Γ)

is semi-ample for all μ >> 0.

□

Proof of (5.14): We may assume that μ_{0} = 1. For ϕ = ϕ_{1}, one has

c_{1}(N) = c_{1}(ϕ^{∗}L) + D = ϕ^{∗}c_{1}(L) + D.

D is effective, hence c_{1}(N)^{ν_{1}} · c_{1}(ϕ^{∗}L)^{ν_{2}} · D are semi-positive cycles, i.e. for n = ν_{1} + ν_{2} + 1 + r one has

H_{1} · . . . · H_{r} · c_{1}(N)^{ν_{1}} · c_{1}(ϕ^{∗}L)^{ν_{2}} · D ≥ 0.

By definition

0 ≡ c_{1}(N)^{ν+1} = c_{1}(N)^{ν} · (c_{1}(ϕ^{∗}L) + D) for ν = ν(N).

Since c_{1}(ϕ^{∗}L) is also represented by an effective divisor, this is only possible if

0 ≡ c_{1}(N)^{ν} · c_{1}(ϕ^{∗}L) = c_{1}(N)^{ν−1} · c_{1}(ϕ^{∗}L) · (c_{1}(ϕ^{∗}L) + D).

The same argument shows that c_{1}(ϕ^{∗}L)^{2} · c_{1}(N)^{ν−1} ≡ 0 and after ν steps we get

c_{1}(ϕ^{∗}L)^{ν} · c_{1}(ϕ^{∗}L) + c_{1}(ϕ^{∗}L)^{ν}·D ≡ 0

and hence

c_{1}(ϕ^{∗}L)^{ν} · D = F · D = 0.
□

§ 6 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 l -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 Ω_{X}^{a} ⊗ 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 ≠ 0. However, since (∗) and (!) only make sense in characteristic 0, we can as well assume throughout this chapter:

6.1. Assumptions. X is a projective manifold defined over an algebraically closed field k of characteristic zero and L is an invertible sheaf on X.

Global vanishing theorems in characteristic p > 0 will appear, as far as it is possible, in Lecture 11.

6.2. Global vanishing theorem for integral parts of Q l -divisors.

For X, L as in (6.1) let

D = ∑_{j=1}^{r} α_{j}D_{i}

§ 6  Differential forms and higher direct images  55

be a normal crossing divisor, N ∈ IN with 0 < a_{j} < N for j = 1 … r and L^{N} = O_{X}(D). Then one has: a)

H^{b}(X, Ω_{X}^{a}(log D) ⊗ L^{−1}) = 0,

for a + b < n − cd(X, D) and for a + b > n + cd(X, D).

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, Ω_{X}^{a}(log (A + B + D))(−B) ⊗ L^{−1}) = 0

for a + b < n − cd(X, D + B) and for a + b > n + cd(X, D + A).

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)).

Proof: By 3.2 L^{−1}(−B) has a logarithmic integrable connection ∇ with poles along A + B + D, such that the E_{1}-degeneration holds true. Moreover, Res_{D_{j}}(∇) ∉ ZZ for j = 1, …, r. For a component A_{j} of A we have Res_{A_{j}}(∇) = 0, and for a component B_{j} of B we have Res_{B_{j}}(∇) = 1. Hence a) follows from (4.8), b) from (4.9) and finally c) from (4.13).

□

6.3. Corollary. For X, L as in (6.1), assume that L^{N} = O_{X}(D) for a normal crossing divisor D = ∑_{j=1}^{r} α_{j}D_{i} with 0 < α_{j} < N and assume that there exists an ample effective divisor B with B_{red} = D_{red}. Then H^{b}(X, L^{−1}) = 0 for b < n.

Proof: Apply (6.2,a), and (4.7,c).

□

2nd 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 κ(L^{N}(−D)) = dim X. Moreover, we may blow up, whenever we like.

We can write (replacing N and D by some multiple)

L^{N}(−D) = A(Γ)

for some effective divisor Γ and some ample sheaf A. Blowing up, we can replace everything by some high multiple and subtract some effective divisor E from the pullback of A such that the sheaf obtained remains ample. Hence one can assume D + Γ to be a normal crossing divisor. Since L^{N}(−D) is numerically effective, we can replace A by A ⊗ L^{N}(−D) and repeating this we can assume that the multiplicities of D + Γ are bounded by N. Finally, replacing again everything by some multiple we are reduced to the case that A is very ample. Writing

L^{N} = O_{X}(D′) for D′ = D + Γ + H,

H a general divisor for A, we can apply (6.3).

□

6.4. Corollary (Akizuki-Kodaira-Nakano [1]). For X, L as in (6.1) assume that L is ample. Then

H^{b}(X, Ω_{X}^{a} ⊗ L^{−1}) = 0 for a + b < n.

Moreover, if A + B is a reduced normal crossing divisor, the same holds true for

H^{b}(X, Ω_{X}^{a}(log (A + B))(−B) ⊗ L^{−1}).

Proof: We can write L^{N} = 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

… ⟶ H^{b−1}(D, Ω_{D}^{a−1}(log (A + B) |_{D})(−B |_{D}) ⊗ L^{−1}) ⟶

⟶ H^{b}(X, Ω_{X}^{a}(log (A + B))(−B) ⊗ L^{−1}) ⟶

⟶ H^{b}(X, Ω_{X}^{a}(log (A + B + D))(−B) ⊗ L^{−1}) ⟶ …

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 ≠ n.

□

If one tries to weaken “L ample” in this proof, one has to replace it by some condition compatible with the induction step.

6.5. Definition. An invertible sheaf L is called l-ample if the following two conditions hold true:

a) L^{N} is generated by global sections, for some N ∈ IN − {0}, and hence ϕ_{N} : X ⟶ IP(H^{0}(X, L^{N})) a morphism.

b) For N as in a) l ≥ Max{dim ϕ_{N}^{−1}(z); z ∈ ϕ_{N}(X)}.

6.6. Corollary (A. Sommese [57], generalized in [22]). For X, L as in (6.1) assume L to be l-ample. Then

H^{b}(X, Ω_{X}^{a} ⊗ L^{−1}) = 0

for a + b < Min {κ(L), dim X − l + 1}.

Proof: Using the notation from (6.5), we have seen in (4.11,b) that

r(ϕ_{N}) ≤ Max {dim X − κ(L), l − 1 }.

Hence (6.6) is a special case of the following more technical statement .

□

6.7. Corollary. For X, L as in (6.1) let τ : X → Y be a morphism and let E be an effective normal crossing divisor with τ^{−1}(τ(X − E)) = X − E. If for some ample sheaf A on Y and for some ν > 0 one has L^{ν} = τ^{∗}A, then

H^{b}(X, Ω_{X}^{a}(log E) ⊗ L^{−1}) = 0

for a + b < dim X − r(τ |_{X−E}).

Proof: If I is the ideal sheaf of τ(E), then A^{μ} ⊗ I will be generated by global sections for some μ ≫ 0. Hence, we may assume that L^{N}(−E) is generated by global section for N = ν ·μ. 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 L^{N}(−E), then D + E is a normal crossing divisor. We have for

τ : D ⟶ τ(D), L |_{D} and E |_{D}

the same assumptions as those asked for in 6.7. Moreover, by (4.11,d) we have

r(τ |_{X−E}) + 1 ≥ r(τ |_{D−E}).

By induction on dim X we may assume that

H^{b−1}(D, Ω_{D}^{a−1}(log E |_{D}) ⊗ L^{−1}) = 0

for

a + b < n − r(τ |_{X−E}) ≤ n + 1 − r(τ |_{D−E}).

The exact sequence (see (2.3,b))

0 → Ω_{X}^{a}(log E)⊗L^{−1} → Ω_{X}^{a}(log (D+E))⊗L^{−1} → Ω_{D}^{a−1}(log E |_{D})⊗L^{−1} → 0

implies that for those a, b the map

H^{b}(X, Ω_{X}^{a}(log E) ⊗ L^{−1}) ⟶ H^{b}(X, Ω_{X}^{a}(log (D + E)) ⊗ L^{−1})

is injective. However, since

r(τ |_{X−E}) ≥ r(τ |_{X−(D+E)})

(6.2,c) tells us that

H^{b}(X, Ω_{X}^{a}(log (D + E)) ⊗ L^{−1}) = 0

for a + b < dim X − r(τ |_{X−E}). □

6.8. 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 [2] [3] [43] [44]).

b) Sometimes it is nicer to use the dual version of (6.7). Since

⋀^{n} Ω_{X}^{1}(log E) = ω_{X} ⊗ O(E_{red})

we find the dual of Ω_{X}^{a}(log E) to be

ω_{X}^{−1} ⊗ Ω_{X}^{n−a}(log E)(−E_{red})

and by Serre duality (6.7) is equivalent to the vanishing of

H^{b}(X, Ω_{X}^{a}(log E)(−E_{red}) ⊗ L)

for

a + b > dim X + r(τ |_{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.

In [63] the second author used the Hodge duality (as in (3.23)) to reduce vanishing for H^{b}(X, 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)).

6.9. Corollary (F. Bogomolov, A. Sommese). For X, L as in (6.1) and for a normal crossing divisor B one has

H^{0}(X, Ω_{X}^{a}(log B) ⊗ L^{−1}) = 0

for a < κ(L).

Proof: (6.9) is compatible with blowing ups and we can assume that

ϕ_{N} : X ⟶ IP(H^{0}(X, L^{N}))

is a morphism. For N large enough, we can choose D such that ϕ_{N} |_{X−D} has equidimensional fibres of dimension n − κ(L) and such that L^{N} = O_{X}(D). Moreover we may assume B + D to be a normal crossing divisor. By (6.2,c)

H^{0}(X, Ω_{X}^{a}(log (B + D)) ⊗ L^{(1)^{−1}}) = 0

for a < κ(L). As Ω_{X}^{a}(log B) ⊗ L^{−1} is a subsheaf of Ω_{X}^{a}(log (B + D)) ⊗ L^{(1)^{−1}}, one obtains (6.8).

□

Global vanishing theorems always give rise to the vanishing of certain direct image sheaves.

6.10. Notations. Let X be a manifold, defined over an algebraically closed field of characteristic zero and let f : X → Z be a proper surjective morphism. Let L be an invertible sheaf on X. We will call L

a) f-numerically effective if for all curves C in X with dim f(C) = 0 one has deg (L |_{C}) ≥ 0

b) f-semi-ample if for some N > 0 the natural map f^{∗}f_{∗}L^{N} ⟶ L^{N} is surjective.

The relative Grauert-Riemenschneider vanishing theorem says, that for a birational morphism f : X → Z one has R^{b}f_{∗}ω_{X} = 0 for b > 0. As a generalization one obtains:

6.11. Corollary.

a) For f : X → Z as in (6.10) let L be an invertible sheaf such that L^{N}(−D) is f-numerically effective for a normal crossing divisor

D = ∑_{j=1}^{r} α_{j}D_{j}.

Then

R^{b}f_{∗}(L^{(1)} ⊗ ω_{X}) = 0 for b > dim X − dim Z − κ(L |_{F})

where F is a general fibre of f.

b) In particular, if f : X → Z is birational and if D = f^{∗}Δ is a normal crossing divisor for some effective Cartier divisor Δ on Z, then

R^{b}f_{∗}(ω_{X} ⊗ O_{X}(−[\frac{D}{N}])) = 0 for b > 0.

Proof: Obviously, since O_{X}(−D) = f^{∗}O_{Z}(−Δ) is f numerically effective, b) is a special case of a). As usual, in a), we can add the assumption 0 < α_{j} < N for j = 1, … r, and we will have L^{(1)} = 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 A ample invertible on Z

f^{∗}A^{ν} ⊗ L^{N}(−D)

will be numerically effective for ν >> 0 and that

κ(f^{∗}A^{ν} ⊗ L) ≥ κ(L |_{F}) + dim (Z).

Using Serre’s vanishing (1.1) we can assume that for all c > 0 and b ≥ 0

H^{c}(Z, A^{ν} ⊗ R^{b}f_{∗}(L ⊗ ω_{X})) = 0

and that H^{0}(Z, A^{ν} ⊗ R^{b}f_{∗}(L ⊗ ω_{X})) generates the sheaf A^{ν} ⊗ R^{b}f_{∗}(L ⊗ ω_{X}). By the Leray spectral sequence (A.27) we obtain

H^{b}(X, f^{∗}A^{ν} ⊗ L ⊗ ω_{X}) = H^{0}(Z, A^{ν} ⊗ R^{b}f_{∗}(L ⊗ ω_{X}))

and by (5.12,d) this group is zero for

b > dim X − dim (Z) − κ(L |_{F}) ≥ dim X − κ(f^{∗}A^{ν} ⊗ L).
□

In the special case for which L^{N}(−D) is f-semi-ample the vanishing of

R^{b}f_{∗}(L ⊗ ω_{X}) for b > dim X − dim Z

follows as well from the next statement, due to J. Kollár, [40].

6.12. Corollary of 5.12,a) (J. Kollár). In addition to the assumption of (6.11,a) we even assume that L^{N}(−D) is f-semi-ample. Then

R^{b}f_{∗}(L^{(1)} ⊗ ω_{X})

has no torsion for b ≥ 0.

Proof: As above we can assume X and Z to be projective and L^{N}(−D) to be semi-ample. Moreover, we may assume L^{N}(−D) to contain f^{∗}A for a very ample sheaf A on Z, that L = L^{(1)}, that

H^{c}(X, R^{b}f_{∗}(L ⊗ ω_{X})) = 0 for c > 0

and that R^{b}f_{∗}(L ⊗ ω_{X}) is generated by its global sections. If R^{b}f_{∗}(L ⊗ ω_{X}) has torsion for some b, then the map

R^{b}f_{∗}(L ⊗ ω_{X}) ⟶ R^{b}f_{∗}(L ⊗ ω_{X}) ⊗ O_{Z}(A)

has a non-trivial kernel K for some effective ample divisor A on Z. We may assume that O_{Z}(A) = A. Replacing L by L ⊗ f^{∗}A^{ν} again, we can assume that H^{0}(Z, K) ≠ 0. For B = f^{∗}A ,this implies that H^{0}(Z, K) lies in the kernel of

H^{b}(X, L ⊗ ω_{X}) ⟶ H^{b}(X, L ⊗ ω_{X}(B))

and hence that

H^{n−b}(X, L^{−1}(−B)) ⟶ H^{n−b}(X, L^{−1})

is not surjective, contradicting (5.12,a).

□

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 [4] and [42] by different methods, is the following lemma.

6.13. Lemma. Let Z be a projective variety in characteristic zero, U ⊆ Z be an open non-singular subvariety, δ : X → Z be a desingularization such that ι : U ≃ δ^{−1}(U) → X. Assume that X −ι(U) = E for a reduced normal crossing divisor E. Then, for a + b < dim X − dim δ(E) − 1 and all invertible sheaves M on Z one has

H^{b}(X, Ω_{X}^{a}(log E) ⊗ δ^{∗}M) = H^{b}(U, Ω_{X}^{a} ⊗ M |_{U}).

Proof: Let A be an ample invertible sheaf on Z and let E′ be an effective exceptional divisor, such that O_{X}(−E′) is relatively ample for δ. For fixed ν ≥ 0 we can choose A large enough, such that for all a, b

R^{b}δ_{∗}(Ω_{X}^{a}(log E) ⊗ O_{X}(−E − ν · E′)) ⊗ A

is generated by global sections and

H^{c}(Z, R^{b}δ_{∗}(Ω_{X}^{a}(log E) ⊗ O_{X}(−E − ν · E′)) ⊗ A) = 0

for c > 0. Moreover, for ν > 0, we may assume that τ^{∗}A(−ν · 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, Ω_{X}^{n−a}(log E) ⊗ δ^{∗}A^{−1}(ν · E′))^{∗} =
H^{b}(X, Ω_{X}^{a}(log E)(−E) ⊗ δ^{∗}A(−ν · E′)) =
H^{0}(Z, R^{b}δ_{∗}(Ω_{X}^{a}(log E) ⊗ O_{X}(−E − ν · E′)) ⊗ A).

By (6.4), for ν > 0, or by (6.7), for ν = 0, we find

R^{b}δ_{∗}(Ω_{X}^{a}(log E) ⊗ O_{X}(−E − ν · E′)) = 0

for a + b > dim X. For those a and b and for

K_{ν} = O_{X}/O_{X}(−ν · E′)

we have

R^{b}δ_{∗}(Ω_{X}^{a}(log E)(−E) ⊗ K_{ν}) = 0.

One obtains for a + b > dim X + dim τ(E) from the Leray spectral sequence (A.27) that

H^{b}(X, δ^{∗}M^{−1} ⊗ Ω_{X}^{a}(log E)(−E) ⊗ K_{ν}) = 0.

Hence for all ν ≥ 0 and a + b > dim X + dim τ(E) + 1 the map

H^{b}(X, δ^{∗}M^{−1} ⊗ Ω_{X}^{a}(log E) ⊗ O_{X}(−E − ν · E′)) ⟶

⟶ H^{b}(X, δ^{∗}M^{−1} ⊗ Ω_{X}^{a}(log E)(−E))

is bijective. By Serre duality again,

H^{b}(X, δ^{∗}M ⊗ Ω_{X}^{a}(log E)) ⟶ H^{b}(X, δ^{∗}M ⊗ Ω_{X}^{a}(log E) ⊗ O_{X}(ν · E′))

is an isomorphism for a + b < dim X − dim τ(E) − 1 and, taking the limit for ν ∈ IN, we obtain (6.13).

□

6.14. Corollary (I. Bauer, S. Kosarew [4]). Let Z be a projective variety in characteristic zero, U ⊆ Z be an open non-singular subvariety. Then, for k < n − dim (Z − U) − 1 one has

dim IH^{k}(U, Ω_{U}^{•}) = ∑_{a+b=k} dim H^{b}(U, Ω_{U}^{a}).

Proof: We can choose a desingularisation δ : X → Z and E as in (6.13). Then we have a natural map of spectral sequences

E_{1}^{ab} = H^{b}(U, Ω_{U}^{a})  ⟹  IH^{a+b}(U, Ω_{U}^{•})
↑^{φ_{a,b}}  ↑^{φ}
E_{1}^{′ab} = H^{b}(X, Ω_{X}^{a}(log E)) ⟹ IH^{a+b}(X, Ω_{X}^{•}(log E)).

Since E_{1}^{′ab} degenerates in E_{1} and since φ_{a,b} are isomorphisms for

a + b < n − dim (Z − U) − 1

the second spectral sequence has to degenerate for those a, b.

□

6.15. Corollary (see also I. Kosarew, S. Kosarew [42]). For Z and U as in (6.14) let L be an l-ample invertible sheaf on Z. Then

H^{b}(U, Ω_{U}^{a} ⊗ L^{−1} |_{U}) = 0

for

a + b < Min {κ(L), dim X − l + 1, dim X − dim (Z − U) − 1 }

Proof: For X, E as in (6.13) and M = δ^{∗}L we have by (6.7)

H^{b}(X, M^{−1} ⊗ Ω_{X}^{a}(log E)) = 0

for a + b < dim X − r(τ |_{U}) where

τ : X ⟶^{δ} Z ⟶^{ϕ_{N}} ϕ_{N}(Z)

is the composition of δ with the map given by global sections of L^{N}. However,

r(τ |_{U}) ≤ r(ϕ_{N}) ≤ Max {dim X − κ(L), l − 1}.
□

6.16. Remark. For k = C l , the reason for which certain coherent sheaves F on Z satisfy H^{b}(Z, F ⊗ H) = 0 for H ample, seems to be related to the existence of connections. This point of view, which is exploited in J. Kollár’s work on vanishing theorems [40], [41] and extended by M. Saito (see [54] and the references given there), should imply that sheaves arising as natural subquotients of O_{Z} ⊗ R^{k}f_{∗}V for a morphism f : X → Z of manifolds and a locally constant system V, sometimes have vanishing properties as the one stated above. J. Kollár proved, for example, that for a morphism f : X → Z, where X and Z are projective varieties and X non-singular, one has

H^{c}(Z, R^{b}f_{∗}ω_{X} ⊗ H) = 0

for c > 0 and H ample on Z.

Slightly more generally one has

6.17. Corollary (of (5.12,b)). Let f : X → Z be a surjective morphism of projective varieties defined over an algebraically closed field of characteristic zero, with X non-singular. Let L be an invertible sheaf on X,

D = ∑_{j=1}^{r} α_{j}D_{j}

a normal crossing divisor and N ∈ IN with

0 < α_{j} < N for j = 1, …, r.

a) If L^{N}(−D) is semi-ample and K a numerically effective invertible sheaf on Z with κ(K) = dim Z, then for c > 0 and b ≥ 0

H^{c}(Z, K ⊗ R^{b}f_{∗}(ω_{X} ⊗ L)) = 0.

b) If L^{N}(−D) is numerically effective, κ(L^{N}(−D)) = ν(L^{N}(−D)), and if (L^{N}(−D))^{μ} contains f^{∗}H for some ample sheaf H on Z and some μ > 0, then for c > 0 and all b

H^{c}(Z, R^{b}f_{∗}(ω_{X} ⊗ L)) = 0.

Proof: By (5.10,b), replacing L by L ⊗ f^{∗}K, a) follows from b). If H is the zero divisor of a general section of H^{μ} for μ ≫ 0 and if B = f^{∗}H then B is a non-singular divisor and the assumptions of (5.12,b) hold true. Hence

H^{b}(X, ω_{X} ⊗ L) ⟶ H^{b}(X, ω_{X}(B) ⊗ L)

is injective for all b. Since H is in general position we have exact sequences

0 ⟶ R^{b}f_{∗}(L ⊗ ω_{X}) ⟶ R^{b}f_{∗}(L ⊗ ω_{X}(B)) ⟶  R^{b}f_{∗}(L ⊗ ω_{B}) → 0
∥  ∥  ∥
R^{b}f_{∗}(L ⊗ ω_{X}) ⟶ O_{Z}(H) ⊗ R^{b}f_{∗}(L ⊗ ω_{X}) ⟶ O_{Z}(H) ⊗ R^{b}f_{∗}(L ⊗ ω_{X}) |_{H}

By induction on dim Z we may assume that

H^{c}(H, R^{b}f_{∗}(L ⊗ ω_{B})) = 0 for c > 0

and, if we choose μ large enough, we find by Serre’s vanishing theorem

H^{c}(Z, R^{b}f_{∗}(L ⊗ ω_{X})) = 0 for c ≥ 2.

In the Leray-spectral sequence (see A.27) all the differentials are zero, since E_{2}^{ab} ≠ 0 just for a = 0 or a = 1, and hence the upper line in the following diagram is exact.

0 ⟶ H^{1}(Z, R^{b−1}f_{∗}(L ⊗ ω_{X})) ⟶ H^{b}(X, L ⊗ ω_{X}) ⟶ H^{0}(Z, R^{b}f_{∗}(L ⊗ ω_{X}))
α↓  ↓
H^{b}(X, L ⊗ ω_{X}(B)) ⟶ H^{0}(Z, R^{b}f_{∗}(L ⊗ ω_{X}(B)))

Since α is injective and since

H^{b}(X, L ⊗ ω_{X}(B)) = H^{0}(Z, R^{b}f_{∗}(L ⊗ ω_{X}(B)))

we find H^{1}(Z, R^{b−1}f_{∗}(L ⊗ ω_{X})) = 0 for all b.

□

§ 7 Some applications of vanishing theorems

The vanishing theorems for integral parts of Q l -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 [46].

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:

§ 7  Some applications of vanishing theorems  65

All varieties are defined over an algebraically closed field k of characteristic zero.

7.1. Example: Surfaces of general type. For a projective surface S′ of general type, i.e. for a non-singular S′ with κ(ω_{S′}) = dim S′ = 2, one can blow down exceptional curves E ≃ IP^{1} with E · E = −1 ([30], p. 414). After finitely many steps one obtains a surface S without any exceptional curve, a minimal model of S′ ([30], p. 418). S is characterised by

7.2. Claim. ω_{S} is nef.

Proof: κ(S) ≥ 0 implies that ω_{S}^{N} = O_{S}(D) for D effective. A curve C with deg (ω_{S} |_{C}) < 0 must be a component of D and C^{2} < 0. However, the adjunction formula gives

−2 ≤ 2g(C) − 2 = deg (ω_{S} |_{C}) + C^{2}.

Hence the only solution is C^{2} = −1 and deg(ω_{S} |_{C}) = −1, which forces C to be exceptional.

□

D. Mumford in his appendix to [65] used the contraction of (−2) curves to show:

7.3. Theorem. If S is a minimal model and κ(ω_{S}) = 2, then ω_{S} is semi-ample.

X. Benveniste and Y. Kawamata (dim X = 3) and Y. Kawamata and V. Shokurov (see [46] 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).

Proof of 7.3 (from the Diplom-thesis of T. Nakovich, Essen): Step 1.: If p ∈ S does not lie on any curve C with deg (ω_{S} |_{C}) = 0, then for some ν ≫ 0 there is s ∈ H^{0}(S, ω_{S}^{ν}) with s(p) ≠ 0.

Proof: Let τ : S′ → S be the blowing up of p and E the exceptional curve. One has

deg (τ^{∗}ω_{S}^{μ}(−E) |_{C′}) = μ · deg (ω_{S} |_{C}) − E · C′

for curves C′ in S′ with C = τ(C′) ≠ p. Hence, for some μ ≫ 0 the sheaf L = τ^{∗}ω_{S}^{μ}(−E) will be nef. By (5.12,c) we find

H^{1}(S′, L^{−2}) ≅ H^{1}(S′, ω_{S′} ⊗ L^{2}) = H^{1}(S′, τ^{∗}ω_{S}^{2μ+1}(−E)) = 0

and hence

H^{0}(S′, τ^{∗}ω_{X}^{2μ+1}) ⟶ H^{0}(E, O_{E})

is surjective.

□

Step 2.: For ν ≫ 0 let

D = ∑_{j=1}^{r} α_{j}C_{j}

be the base locus of ω_{S}^{ν} (i.e. ω_{S}^{ν}(−D) is generated by H^{0}(S, ω_{S}^{ν}) outside of a finite number of points). Then D is a normal crossing divisor, C_{j}^{2} = −2 for j = 1, …, r and ω_{S}^{ν} is generated by H^{0}(S, ω_{S}^{ν}) outside of D.

Proof: By step 1, if for some p ∈ S there is no section s of ω_{S}^{ν} with s(p) ≠ 0, then p lies on some curve C with deg (ω_{S} |_{C}) = 0 and necessarily C is contained in the base locus. We know thereby that deg (ω_{S} |_{C_{j}}) = 0 for the components C_{j} of D. By the Hodge-index theorem ([30], p. 364) one finds for any reduced subdivisor C of D that C · C < 0. If we take C = C_{j}, then the adjunction formula shows that

C · C = −2 and C ≃ IP^{1}.

For C = (C_{1} + C_{2}) we get

C_{1} · C_{2} < 2

and C_{1} and C_{2} intersect transversally.

□

Step 3. For D as in Step 2, ω_{S}^{ν}(−D) is nef, hence ω_{S}^{N}(−D) for N ≥ ν is nef as well. We can choose N > α_{j} for j = 1, … r. For some i > 0

D′ = [\frac{i · D}{N}] = ∑_{j=1}^{r}[\frac{i · α_{j}}{N}] · C_{j}

will be reduced and non zero. By (5.12,c) again, we have for L = ω_{S}

H^{1}(S, L^{(i)^{−1}}) ≅ H^{1}(S, ω_{S} ⊗ L^{(i)}) = H^{1}(S, ω_{S}^{i+1}(−D′)) = 0

and

H^{0}(S, ω_{S}^{i+1}) ⟶ H^{0}(D′, ω_{S}^{i+1} |_{D′})

is surjective. Since the right hand side is nontrivial (in fact its dimension is just the number of connected components of D′), for i + 1 the base locus does not contain D′. After finitely many steps we are done.

□

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.

7.4. Definition. Let X be a normal variety and D be an effective Cartier divisor on X. Let τ : X′ → X be a blowing up, such that X′ is non singular and D′ = τ^{∗}D is a normal crossing divisor. We define: a)

ω_{X}\{\frac{−D}{N} \} = τ_{∗}ω_{X′}(−[\frac{D′}{N}]).

b) C_{X}(D, N) = Coker (ω_{X}\{\frac{−D}{N} \} ⟶ ω_{X}) where ω_{X} is the reflexive hull of ω_{X_{0}} for X_{0} = X − Sing (X).

c) (see [23])

e(D) = Min {N > 0; C_{X}(D, N) = 0}.

d) If X is compact and L invertible, H^{0}(X, L) ≠ 0, then

e(L) = Max {e(D); D ≥ 0 and O_{X}(D) = L}.

7.5. Properties (see [23]). 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

ω_{X}\{\frac{−D}{N} \} = ω_{X}(−[\frac{D}{N}]).

c) ω_{X}\{\frac{−D}{N} \}, C_{X}(D, N) and e(D) are independent of the blowing up τ : X′ → 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

ω_{H}\{\frac{−D |_{H}}{N} \} ⟶ ω_{X}\{\frac{−D}{N} \} ⊗ O_{X}(H) |_{H} .

e) If in d) X and H have rational sigularities, then for N ≥ e(D |_{H}), H does not meet the support of C_{X}(D, N).

Proof: a) is obvious since for N ≫ 0

τ_{∗}ω_{X′}(−[\frac{D}{N}]) = τ_{∗}ω_{X′} = ω_{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

ω_{X}\{\frac{−D}{N} \} ⟶ ω_{X}(−[\frac{D}{N}])

and it is enough to prove b) for some blowing up dominating τ. Hence, it is enough to consider the case that τ is a sequence of blowings with non-singular centers. Let us write

ω_{X′} = τ^{∗}ω_{X} ⊗ O_{X′}(∑_{i=1}^{t} α_{i} · E_{i}).

For m_{i} = codim_{X}(τ(E_{i})) one has α_{i} ≥ m_{i} − 1.

In fact, assume this to hold true for τ_{1} : X_{1} ⟶ X and

ω_{X_{1}} = τ_{1}^{∗}ω_{X} ⊗ O_{X_{1}}(∑_{i=1}^{t−1} α_{i} · E′_{i}).

If δ : 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

ω_{X′} = δ^{∗}ω_{X_{1}} ⊗ O_{X′}((m − 1) · E_{t})

(see [30], p. 188). If m_{t} > m then S lies on some E′_{ν} with m_{t} − m ≤ m_{ν} − 1. Hence

α_{t} ≥ m − 1 + α_{ν} ≥ m_{t} − 1.

On the other hand, assume that τ(E_{μ}) lies on s different components of D, let us say on D_{1}, ⋯, D_{s} but not in D_{j} for j > s. Then m_{μ} ≥ s and, if

D = ∑_{j=1}^{r} α_{j}D_{j}

one has

[∑_{j=1}^{s} \frac{α_{j}}{N}] ≤ ∑_{j=1}^{s}[\frac{α_{j}}{N}] + s − 1 ≤ ∑_{j=1}^{s}[\frac{α_{j}}{N}] + α_{μ}.

One obtains

[\frac{D′}{N}] ≤ τ^{∗}[\frac{D}{N}] + ∑_{i=1}^{t} α_{i} · E_{i}

and hence

τ^{∗}ω_{X}(−[\frac{D}{N}]) ⊂ ω_{X′}(−[\frac{D′}{N}]).

c) follows from b). Hence in d) we may assume that D′ intersects the proper transform H′ on H transversally and, of course, that H′ is non-singular. Then

[\frac{D′}{N}] |_{H′}= [\frac{D′ |_{H′}}{N}].

One has a commutative diagram

τ_{∗}ω_{X′}(−[\frac{D′}{N}] + H′) ⟶^{α}  τ_{∗}ω_{H′}(−[\frac{D′|_{H′}}{N}])  ⟶ ω_{H}
↓  ↓^{=}
τ_{∗}ω_{X′}(−[\frac{D′}{N}]) ⊗ O_{X}(H) ⟶ τ_{∗}ω_{X′}(−[\frac{D′}{N}]) ⊗ O_{X}(H) |_{H} ⟶^{γ} ω_{H}

The cokernel of α lies in R^{1}τ_{∗}ω_{X′}(−[\frac{D′}{N}]), and (6.11) shows that α is surjective. We obtain therefore a non-trivial morphism

α′ : ω_{H}\{\frac{−D |_{H}}{N} \} ⟶ ω_{X}\{\frac{−D}{N} \} ⊗ O_{X}(H) |_{H} .

Since ω_{H}\{\frac{−D|_{H}}{N} \} is torsion free d) holds true. In e) we know that ω_{H}\{\frac{−D|_{H}}{N} \} is isomorphic to ω_{H}. Hence γ is surjective. Therefore ω_{X}\{\frac{−D}{N} \} ⊗ O_{X}(H) must be isomorphic to ω_{X} ⊗ O_{X}(H) in a neighbourhood of H.

□

7.6. Remark. The diagram used to prove d) gives slightly more. Instead of assuming that D′ = τ^{∗}D it is enough to take any normal crossing divisor D′ on X′ not containing H′. Then the inclusion

τ_{∗}ω_{H′}(−[\frac{D′ |_{H′}}{N}]) ↪ τ_{∗}ω_{X′}(−[\frac{D}{N}]) ⊗ O_{X}(H) |_{H}

exists whenever H′ + D′ is a normal crossing divisor and O_{X′}(−D′) is τ-numerically effective (see (6.10)).

Up to now, we do not even know that e(L) is finite. This however follows from the first part of the next lemma, since every sheaf L lies in some ample invertible sheaf.

7.7. Lemma. Let X be a projective manifold and let L be an invertible sheaf.

a) If L is very ample and ν > 0, then

e(L^{ν}) ≤ ν · c_{1}(L)^{dim X} + 1

b) For s ∈ H^{0}(X, L) with zero-locus D assume that for some p ∈ X the section s has the multiplicity μ i.e.:

s ∈ m_{p}^{μ} ⊗ L but s ∉ m_{p}^{μ+1} ⊗ L.

Then

ω_{X}\{\frac{−D}{N} \} ⟶ ω_{X}

is an isomorphism in a neighbourhood of p for N > μ.

c) If under the assumption of b)

μ′ = [\frac{μ}{N}] − dim X + 1 ≥ 0

then ω_{X}\{\frac{−D}{N} \} is contained in m_{p}^{μ′} ⊗ ω_{X}.

70 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

Proof: a) Let D ≥ 0 be a divisor, O_{X}(D) = L^{ν}. If X is a curve then [\frac{D}{N}] = 0 for N > deg D + 1 = ν · c_{1}(L) + 1. In general, let H be the divisor of a general section of L. By induction

e(L^{ν} |_{H}) ≤ ν · c_{1}(L |_{H})^{dim H} + 1 = ν · c_{1}(L)^{dim X} + 1.

(7.5,e) tells us that C_{X}(D, N) is supported outside of H for

N ≥ ν · c_{1}(L)^{dim X} + 1

and moving H we find C_{X}(D, N) = 0. For b) and c) we may assume that the blowing up τ : X′ → X factors through the blowing up ϱ : X_{p} → X of p. For D_{p} = ϱ^{∗}D and for the exceptional divisor E of ϱ we have

Δ = D_{p} − μ · E ≥ 0

and Δ does not contain E. Assume N > μ. One has

O_{E}(Δ |_{E}) = O_{IP}^{n−1}(μ)

and, by part a), one obtains

ω_{E}\{\frac{−Δ |_{E}}{N} \} = ω_{E}.

From (7.5,e) one knows that

ω_{X_{p}}\{\frac{−Δ}{N}\} ⟶ ω_{X_{p}}

is an isomorphism in a neighbourhood of E. Hence

ω_{X_{p}}(−E) = ω_{X_{p}}\{\frac{−Δ}{N}\} ⊗ O_{X_{p}}(−E) = ω_{X_{p}}\{\frac{−Δ − N · E}{N} \}

is contained in ω_{X_{p}}\{\frac{−D_{p}}{N} \} which implies that ω_{X} = ω_{X}\{\frac{−D}{N} \} near p. If μ′ = [\frac{μ}{N}] − n + 1 ≥ 0 then

ω_{X_{p}}\{\frac{−D_{p}}{N} \} ⊂ ω_{X_{p}}\{\frac{−μ · E}{N} \} = ω_{X_{p}}(−[\frac{μ}{N}] · E)

and

ω_{X}\{\frac{−D}{N} \} = ϱ_{∗}ω_{X_{p}}\{\frac{−D_{p}}{N} \} ⊂ ϱ_{∗}ϱ^{∗}ω_{X}((n − 1 − [\frac{μ}{N}]) · E) = m_{p}^{μ′} ⊗ ω_{X}.

□ The sheaves ω_{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:

7.8. Proposition. Let X be a projective manifold, L be an invertible sheaf and D be a divisor such that L^{N}(−D) is numerically effective and

c_{1}(L^{N}(−D))^{n} > 0

for n = dim X. Then

H^{b}(X, ω_{X}\{\frac{−D}{N} \} ⊗ L) = 0 for b > 0.

Proof: This follows from (5.12,c) and (6.11,b) by using the Leray spectral sequence (A.27).

□

7.9. Remark. Demailly proved in [13] 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

ω_{X}\{\frac{−D}{N} \} = ω_{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:

7.10. Corollary. Under the assumptions of (7.8) let H be a very ample sheaf. Then

H^{dim X} ⊗ L ⊗ ω_{X}\{\frac{−D}{N} \}

is generated by global sections.

Proof.: For

F = L ⊗ ω_{X}\{\frac{−D}{N} \} ⊗ ω_{X}^{−1}

we have

H^{b}(X, F ⊗ H^{ν} ⊗ ω_{X}) = 0 for b > 0 and ν ≥ 0.

For general sections H_{1}, …, H_{n} of H passing through a given point p and for

Y_{r} = ⋂_{i=1}^{r} H_{i}

we obtain

H^{b}(Y_{r}, F ⊗ H^{ν} ⊗ ω_{Y_{r}}) = 0, for b > 0 and ν ≥ 0,

by regarding the cohomology sequence given by the short exact sequence

0 ⟶ F ⊗ H^{ν} ⊗ ω_{Y_{r}} ⟶ F ⊗ H^{ν+1} ⊗ ω_{Y_{r}} ⟶ F ⊗ H^{ν} ⊗ ω_{Y_{r+1}} ⟶ 0.

By induction we may assume that

F ⊗ H^{dim(Y_{r+1})} ⊗ ω_{Y_{r+1}}

is generated by global sections in p, and, using the cohomology sequence again one finds the same for

F ⊗ H^{dim(Y_{r})} ⊗ ω_{Y_{r}}.
□

Let us apply (7.10) for X = IP^{n} to study the behaviour of zeros of homogeneous polynomials:

7.11. Example: Zeros of polynomials . Let S be a finite set of points in IP^{n}, for n ≥ 2, and

ω_{μ}(S) = Min{ d > 0; there exists s ∈ H^{0}(IP^{n}, O_{IP}^{n}(d))

with multiplicity at least μ in each p ∈ S}.

7.12. Claim. For μ′ < μ one has

\frac{ω_{μ′}(s)}{μ′ + n − 1} ≤ \frac{ω_{μ}(s)}{μ}.

Proof.: For d = ω_{μ}(S) we have a section s ∈ H^{0}(IP^{n}, O_{IP}^{n}(d)) with divisor D′ such that s has multiplicity at least μ in each p ∈ S. Choose

d′ = [\frac{d}{μ}(μ′ + n − 1)].

Since d′ does not change if we replace d by ν · d + 1 and μ by ν · μ for ν ≫ 0, we can assume that D′ = D + H for a hyperplane H not meeting S.

For L = O_{IP}^{n}(d′ + 1), for the divisor (d′ + 1) · D, and for N = d, the assumptions of (7.8) hold true and (7.10) tells us that

O_{IP}^{n}(n + d′ + 1) ⊗ ω_{IP}^{n}\{\frac{−(d′ + 1) · D}{d} \}

is globally generated. Since d′+1 > \frac{d}{μ}(μ′+n−1) and hence μ′+n−1 ≤ \frac{(d′+1)·μ}{d} we can apply (7.7,c) and find

O_{IP}^{n}(n + d′ + 1) ⊗ ω_{IP}^{n}\{\frac{−(d′ + 1) · D}{d} \}

to be a subsheaf of

O_{IP}^{n}(d′) ⊗ ⨂_{p∈S} m_{p}^{μ′}.

□

7.13. Remark. For some generalizations and improvements and for the history of this kind of problem see [18].

Up to this point, the applications discussed are based on the global vanishing theorems for invertible sheaves. J. Kollár’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:

7.14. Example: Families of varieties over curves . Let X be a projective manifold, Z a non-singular curve and f : X → Z a surjective morphism. We call a locally free sheaf F on Z semi-positive, if for some (or equivalently: all) ample invertible sheaf A on Z and for all η > 0 the sheaf

S^{η}(F) ⊗ A

is ample. One has

7.15. Theorem (Fujita [24]). For f : X → Z as above f_{∗}ω_{X/Z} is semipositive.

Here ω_{X/Z} = ω_{X} ⊗ f^{∗}ω_{Z}^{−1} is the dualizing sheaf of X over Z. In [40] J. Kollár 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 [35]. As usually, one obtains similar results adding the L^{(i)}. For example, using the notations introduced in (7.4) one has:

7.16. Variant. Assume in addition that L is an invertible sheaf, D an effective divisor and that L^{N}(−D) is semi-ample. Then one has

a) The sheaf f_{∗}(L ⊗ ω_{X/Z}\{\frac{−D}{N} \}) is semi-positive.

b) If for a general fibre F of f one has N ≥ e(D |_{F}), then f_{∗}(L ⊗ ω_{X/Z}) is semi-positive.

7.17. Corollary. Under the assumption of (7.16) assume that D contains a smooth fibre of f. Then, if N ≥ e(D |_{F}), the sheaf f_{∗}(L ⊗ ω_{X/Z}) is ample.

Proof of (7.17): Recall that a vectorbundle F on Z is ample, if and only if τ^{∗}F is ample for some finite cover

τ : Z′ ⟶ Z.

Hence, if L′, D′, X′ and f′ are obtained by pullback from the corresponding objects over Z, it is enough to show that

τ^{∗}f_{∗}(L ⊗ ω_{X/Z}) = f′_{∗}(L′ ⊗ ω_{X′/Z′})

is ample. If, for the ramification locus Δ(Z′/Z) of Z′ over Z, the morphism f is smooth in a neighbourhood of f^{−1}(Δ(Z′/Z)) then f′ : X′ ⟶ Z′ satisfies again the assumption made in (7.14). Choosing Z′ to be ramified of order N over the point p ∈ Z with f^{−1}(p) ⊆ D, we can reduce (7.17) to the case for which D contains the N-th multiple of a fibre, say N · f^{−1}(p). We have

f_{∗}(L ⊗ ω_{X/Z}\{−\frac{D}{N} \}) ⊂ f_{∗}(L ⊗ ω_{X/Z}) ⊗ O_{Z}(−p)

and this inclusion is an isomorphism over some open set. In fact, by (7.5,e) the assumption N ≥ e(D |_{F}) implies that ω_{X/Z} and ω_{X/Z}\{−\frac{D}{N} \} are the same in a neighbourhood of a general fibre F. Hence

f_{∗}(L ⊗ ω_{X/Z}) ⊗ O_{Z}(−p)

is semi-positive.

□

Proof of (7.16): As in the proof of (7.17) the assumption made in part b) implies that

f_{∗}(L ⊗ ω_{X/Z}\{\frac{−D}{N} \}) ⊂ f_{∗}(L ⊗ ω_{X/Z})

is an isomorphism over some non-empty open subvariety and b) follows from a).

By definition of ω_{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 L = L(−[\frac{D}{N}]). Let p ∈ Z be a point in general position and F = f^{−1}(p). By (5.12,a) applied to the semiample sheaf L(F) we have a surjection

H^{0}(X, L(F) ⊗ ω_{X}(F)) ⟶ H^{0}(F, L(F) ⊗ ω_{F})

and f_{∗}(L ⊗ ω_{X}) ⊗ O_{Z}(2 · p) is generated by global sections in a neighbourhood of p.

7.18. Claim. ω_{Z}(2 · p) ⊗ ⨂^{η} f_{∗}(L ⊗ ω_{X/Z}) is semi-positive for all η > 0.

Proof: For η = 1 (7.18) holds true as we just found a trivial subsheaf ⨁^{r} O_{Z} of

f_{∗}(L ⊗ ω_{X}) ⊗ O_{Z}(2 · p)

of full rank. In general, let Y′ = X ×_{Z} … ×_{Z} X (η-times) be the fibre product and δ : Y → Y′ be a desingularization. The induced morphisms g′ : Y′ → Z and g : Y → Z satisfy:

i) Y′ is flat and Gorenstein over Z and

ω_{Y′/Z} = ⨂_{j=1}^{η} pr_{j}^{∗}ω_{X/Z}.

ii) For

M′ = ⨂_{j=1}^{η} pr_{j}^{∗}L and M = δ^{∗}M′

one has an inclusion, surjective at the general point of Z,

g_{∗}(M ⊗ ω_{Y/Z}) ⊂ g′_{∗}(M′ ⊗ ω_{Y′/Z}) = ⨂^{η} f_{∗}(L ⊗ ω_{X/Z})

iii) On a general fibre F̃ of g the divisor

Δ = ∑_{j=1}^{η} δ^{∗}pr_{j}^{∗}D

has normal crossings, [\frac{Δ}{N}]|_{F̃} = 0, and M^{N}(−Δ) is semi-ample.

In fact, i) is the compatibility of relative dualizing sheaves with pullback, ii) follows from flat base change and the inclusion δ_{∗}ω_{Y} ⊂ ω_{Y′} and iii) is obvious. Using those three properties, (7.18) follows from the case “η = 1” applied to g : Y → Z.

□

Since S^{η}( ) is a quotient of ⨂^{η}( ) and since the quotient of a semi-positive sheaf is again semipositive, one obtains (7.16).

□

7.19. Remarks.

a) If dim Z > 1, then the arguments used in the proof of (7.16,a) show that for all η > 0 and H very ample the reflexive hull G of

S^{η}(f_{∗}(L ⊗ ω_{X/Z})) ⊗ H^{dim(X)+1} ⊗ ω_{Z}

is generated by H^{0}(Z, G) over some open set. This led the second author to the definition “weakly-positive” (see [64]).

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_{∗}(L⊗ω_{X/Z}). Details have been worked out in [23]. 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 l .

The vanishing statements for higher direct images (6.11) and its corollaries (7.5,d) and (7.6) are useful to study singularities.

7.20. Example: Deformation of quotient singularities. Let X be a normal variety and f : X → S a flat morphism from X to a non-singular curve S.

7.21. Theorem. Assume that X is normal and that for some s_{0} ∈ S the variety X_{0} = f^{−1}(s_{0}) is a reduced normal surface with quotient singularities. Then the general fibre f^{−1}(η) = X_{η} has at most quotient singularities.

Proof (see [19]): If Y is a normal surface with rational singularities then Mumford [47] has shown that for p ∈ Y,

Spec (O_{Y,p}) − p = U

has only finitely many non-isomorphic invertible sheaves. Hence for some N > 0 one has ω_{U}^{N} = O_{U} and for some N > 0 the reflexive hull ω_{X}^{[N]} of ω_{Y}^{N} is invertible. Let

δ : Y′ ⟶ Y

be a desingularization. Since Y has rational singularities we have

δ_{∗}ω_{Y′} = ω_{Y}

and

δ^{∗}ω_{Y} /torsion ⊂ ω_{Y′}.

We may assume that δ^{∗}ω_{Y} /torsion = K is invertible and we write

ω_{Y′} = K ⊗ O_{Y′}(F).

With this notation we have for some effective divisor D

δ^{∗}ω_{Y}^{[N]} = K^{N} ⊗ O_{Y′}(D).

The divisors D and F can be used to characterize the quotient singularities among the rational singularities:

7.22. Claim. Y has quotient singularities if and only if [\frac{D}{N}] ≤ F.

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 ω_{Y}^{[N]} invertible and ω_{Y}^{[N]} ≃ O_{Y}.

For K^{−1} = L one has L^{N} = O_{Y′}(D) and, as in (3.5), we can consider the cyclic cover Z′ obtained by taking the N-th root out of D. Let Z be the normalization of Y in k(Z′) and

Z′ ⟶^{δ′} Z
π′↓  ↓^{π}
Y′ ⟶_{δ} Y

the induced morphism (Z is usually called the canonical covering of Y). One has

π′_{∗}ω_{Z′} = ⨁_{i=0}^{N−1} ω_{Y′} ⊗ L^{(i)}.

In fact, this follows from (3.11) by duality for finite morphisms (see [30], p.

239) or, since

L^{−i}([\frac{i · D}{N}] + D^{(i)}) = L^{−i}(D − [\frac{(N − i) · D}{N}]) = L^{(N−i)}

from (3.16,d). Recall that Z has rational singularities if and only if δ′_{∗}ω_{Z′} = ω_{Z}. Assume that Y has a quotient singularity in the point p. If Ũ is the universal cover of Y − p then the normalization Z̃ of Y in k(Ũ) is non singular and, by construction it dominates Z. Hence Z has quotient singularities and

π_{∗}δ′_{∗}ω_{Z′} = δ_{∗}π′_{∗}ω_{Z′}

is reflexive. In particular δ_{∗}ω_{Y′} ⊗ L^{(1)} is reflexive. One has

ω_{Y′} ⊗ L^{(1)} = K ⊗ L ⊗ O_{Y′}(F − [\frac{D}{N}]) = O_{Y′}(F − [\frac{D}{N}])

and the reflexivity of δ_{∗}ω_{Y′} ⊗ L^{(1)} is equivalent to F ≥ [\frac{D}{N}].

On the other hand, F ≥ [\frac{D}{N}] implies that the summand δ_{∗}O_{Y′}(F − [\frac{D}{N}]) of δ_{∗}π′_{∗}ω_{Z′} has one section without zero on Y. Hence δ′_{∗}ω_{Z′} has a section without zero on Z − π^{−1}(p), which implies that δ′_{∗}ω_{Z′} is invertible and coincides with ω_{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.

□

Proof of (7.21): Let δ : X′ → X be a desingularization. We assume that the proper transform X′_{0} of X_{0} is non-singular and write δ_{0} = δ |_{X′_{0}}. By (7.5,d), applied in the case “D = 0 ”, we have a natural inclusion

δ_{0∗}ω_{X′_{0}} ⟶ δ_{∗}ω_{X′} ⊗ O_{X}(X_{0}) |_{X_{0}} .

Since X_{0} has rational singularities one has

δ_{0∗}ω_{X′_{0}} = ω_{X_{0}} = (ω_{X} ⊗ O_{X}(X_{0}))|_{X_{0}}.

One obtains δ_{∗}ω_{X′} = ω_{X}, at least if one replaces S by a neighbourhood of s_{0}. Hence X and X_{η} have at most rational singularities.

Let us choose N > 1, such that both, ω_{X_{η}}^{[N]} and ω_{X_{0}}^{[N]} are invertible (It might

happen, nevertheless, that ω_{X}^{[N]} is not invertible). We may assume that we have choosen X such that

K = δ^{∗}ω_{X}/torsion

is an invertible sheaf. Hence

K_{0} = K ⊗ O_{X′}(δ^{∗}X_{0}) |_{X′_{0}}

is invertible and generated by global sections. Moreover, one has maps

δ_{0}^{∗}ω_{X_{0}} = δ^{∗}(ω_{X} ⊗ O_{X}(X_{0}))|_{X′_{0}} ⟶ K_{0}

and K_{0} contains δ_{0}^{∗}ω_{X_{0}}/torsion. Hence both sheaves must be the same.

Blowing up again, we can assume M = δ^{∗}ω_{X}^{[N]}/torsion to be locally free and isomorphic to K^{N}(D) where D is a divisor in the exceptional locus of δ such that X′_{0} + D has at most normal crossings.

We can choose an embedding ω_{X} ↪ O_{X} such that the zero-set does not contain X_{0}. If correspondingly K = O_{X′}(−Δ), for some Δ ≥ 0 we can choose the inclusion ω_{X} ↪ O_{X} such that

D′ = N · Δ − D ≥ 0.

Blowing up we can assume that D′ + X′_{0} is a normal crossing divisor. By definition

O_{X}(−D′) = O_{X}(−N · Δ + D) = M

and O_{X}(−D′) is δ-numerically effective.

It is our aim to use (7.6) in order to compare the sheaves K_{0} and δ_{0}^{∗}ω_{X_{0}}^{[N]} with K and with 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_{η}.

Let us write

ω_{X′_{0}} = K_{0}(F_{0}) and δ_{0}^{∗}ω_{X_{0}}^{[N]} = K_{0}^{N} ⊗ O_{X′_{0}}(D_{0}).

By (7.22) one has

F_{0} ≥ [\frac{1}{N} D_{0}].

Since

K_{0}^{N} ⊗ O_{X′_{0}}(D |_{X′_{0}}) ≃ (K^{N} ⊗ O_{X′}(D) ⊗ O_{X′}(+N · δ^{∗}X_{0})) |_{X′_{0}}

is a subsheaf of δ_{0}^{∗}ω_{X_{0}}^{[N]} one obtains that

(∗) F_{0} ≥ [\frac{1}{N} D |_{X′_{0}}] .

On the other hand, since

K_{0}^{N−1} = O_{X′_{0}}(−(N − 1)Δ |_{X′_{0}}) ⊗ O_{X′}((N − 1)δ^{∗}X_{0}) |_{X′_{0}}

and

[\frac{N − 1}{N}D′] = (N − 1)Δ − D + [\frac{1}{N} D]

one has

ω_{X′_{0}}(−[\frac{(N − 1)}{N}D′ |_{X′_{0}}]) = K_{0}(F_{0} − (N − 1)Δ |_{X′_{0}} +D |_{X′_{0}} −[\frac{1}{N} D |_{X′_{0}}]) =

K_{0}^{N}(D |_{X′_{0}} +F_{0} − [\frac{1}{N} D |_{X′_{0}}]) ⊗ O_{X′}(−(N − 1)δ^{∗}X_{0}) |_{X′_{0}}=

M |_{X′_{0}} (F_{0} − [\frac{1}{N} D |_{X′_{0}}]) ⊗ O_{X′}(δ^{∗}X_{0}) |_{X′_{0}} .

By the inequality (∗) the sheaf

δ_{0∗}(ω_{X′_{0}}(−[\frac{(N − 1)}{N}D′ |_{X′_{0}}]) ⊗ O_{X′}(−δ^{∗}X_{0}) |_{X′_{0}})

contains δ_{0∗}(M |_{X′_{0}}). If F is the divisor with ω_{X′} = K ⊗ O_{X′}(F) we get from (7.6) δ_{0∗}(M |_{X′_{0}}) as a subsheaf of

δ_{∗}ω_{X}(−[\frac{(N − 1)}{N}D′]) |_{X_{0}}= δ_{∗}(K⊗O_{X′}(F)⊗O_{X′}(−(N −1)Δ+D−[\frac{1}{N} D])) |_{X_{0}}

= δ_{∗}(K^{N} ⊗ O_{X′}(D + F − [\frac{1}{N} D])) |_{X_{0}}= δ_{∗}M(F − [\frac{1}{N} D]) |_{X_{0}} .

Of course, we have a natural morphism δ_{∗}M ⟶ δ_{0∗}(M |_{X′_{0}}) and the induced map

δ_{∗}M ⟶ δ_{∗}M(F − [\frac{1}{N} D]) |_{X_{0}}

is surjective outside of the singular locus of X_{0}. We have natural maps

ω_{X}^{[N]} ⟶ δ_{∗}δ^{∗}ω_{X}^{[N]} ⟶ δ_{∗}M ⟶ δ_{∗}M(F − [\frac{1}{N} D])|_{X_{0}} ⟶ ω_{X}^{[N]}|_{X_{0}}.

The sheaf δ_{∗}M(F − [\frac{1}{N} D]) is torsionfree and, since X_{0} is a Cartier divisor, δ_{∗}M(F − [\frac{1}{N} D])|_{X_{0}} has no torsion as well. Therefore

ω_{X}^{[N]}|_{X_{0}} = δ_{∗}M(F − [\frac{1}{N} D])|_{X_{0}}.

Since M = δ^{∗}ω_{X}^{[N]}/torsion, this is only possible if F ≥ [\frac{1}{N} ·D]. Hence F_{η} ≥ [\frac{D_{η}}{N}] where “η” denotes the restriction to the general fibre and the theorem follows from (7.22).

□

7.23. Example: Adjoint linear systems on surfaces. Studying adjoint linear systems on higher dimensional manifolds, L. Ein and R. Lazarsfeld [15] realized, that (7.7, b and c) and (7.8) can be used to reprove part of I. Reider’s theorem [53] 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.

7.24. Theorem (I. Reider, [53]). Let S be a non-singular projective surface, defined over an algebraically closed field of characteristic zero, let p ∈ S be a closed point and let L be a numerically effective invertible sheaf on S. Assume that c_{1}(L)^{2} > 4 and that for all curves C with p ∈ C ⊂ S one has c_{1}(L)·C > 1. Then there is a section σ ∈ H^{0}(S, L ⊗ ω_{S}) with σ(p) ≠ 0.

Proof, following §1 of [15]: Let H be an ample invertible sheaf on S and let m_{p} be the ideal sheaf of p. For ν ≫ 0, one has H^{2}(S, L^{ν} ⊗ H^{−1}) = 0 and by the Riemann-Roch formula one finds a, b ∈ IN with

h^{0}(S, H^{−1} ⊗ L^{ν} ⊗ m_{p}^{2·ν}) ≥ h^{0}(S, H^{−1} ⊗ L^{ν}) − h^{0}(S, O_{S}/m_{p}^{2·ν})

≥ \frac{1}{2} · c_{1}(L)^{2} · ν^{2} + a · ν + b − h^{0}(S, O_{S}/m_{p}^{2·ν}).

Since

h^{0}(S, O_{S}/m_{p}^{2·ν}) = \frac{1}{2} · (4 · ν^{2} + 2 · ν)

one finds for ν ≫ 0 a section s of H^{−1} ⊗ L^{ν} with multiplicity μ_{p} ≥ 2 · ν in p (see (7.7,b). Let

D = Δ + ∑_{i=1}^{r} ν_{i} · D_{i}

be the zero-divisor of s, where Δ is an effective divisor not containing p and p ∈ D_{i} for i = 1 ⋯ r. If D′ is any effective divisor the H^{η}(−D′) will be ample for η ≫ 0. Replacing D by η · D + D′ and ν by ν · η for a suitably choosen divisor D′, we may assume that r > 1 and that ν_{1} > ν_{2} > ⋯ > ν_{r}. Of course we can also assume that μ_{p} is even.

7.25. Claim. If μ_{p} > 2 · ν_{1} then (7.24) holds true.

Proof: Let N = \frac{μ_{p}}{2} By the choice of s one has N ≥ ν and

L^{N}(−D) = L^{N−ν} ⊗ H

is ample. By (7.8) one has

H^{1}(S, ω_{S}\{\frac{−D}{N} \} ⊗ L) = 0.

By (7.7,b), or just by definition of ω_{S}\{\frac{−D}{N} \}, one can find some open neighbourhood U of p such that

ω_{S}\{\frac{−D}{N} \} ⟶ ω_{S}

is an isomorphism on U − p. Moreover, by (7.7,c), the inclusion factors like

ω_{S}\{\frac{−D}{N} \} ⟶ ω_{S} ⊗ m_{p} ⟶ ω_{S}.

Let M be the sheaf j_{∗}j^{∗}(ω_{S}\{\frac{−D}{N} \}⊗L) where j : S−p ⟶ S is the inclusion. For some nontrivial skyscraper sheaf C supported in p, one has an exact sequence

0 ⟶ ω_{S}\{\frac{−D}{N} \} ⊗ L ⟶ M ⟶ C ⟶ 0.

Hence, M has a section σ with σ(p) ≠ 0.

□

It remains to consider the case where μ_{p} ≤ 2 · ν_{1}. If μ_{p}(D_{i}) denotes the multiplicity of D_{i} in p, then

μ_{p} = ∑_{i=1}^{r} ν_{i} · μ_{p}(D_{i}).

Since r ≥ 2 this implies that μ_{p}(D_{1}) = 1.

Let us take N = ν_{1}. Again, N ≥ ν and by (7.8)

H^{1}(S, ω_{S}\{\frac{−D}{N} \} ⊗ L) = 0.

One has an inclusion

ω_{S}\{\frac{−D}{N} \} ⊗ L ⟶ ω_{S} ⊗ L(−D_{1} − [\frac{Δ}{N}])

whose cokernel is a skyscraper sheaf. Hence

H^{1}(S, ω_{S} ⊗ L(−D_{1} − [\frac{Δ}{N}])) = 0

and the restriction map

H^{0}(S, ω_{S} ⊗ L(−[\frac{Δ}{N}])) ⟶ H^{0}(D_{1}, ω_{D_{1}} ⊗ L(−D_{1} − [\frac{Δ}{N}])|_{D_{1}})

is surjective.

The right hand side contains a section σ with σ(p) ≠ 0, since

deg(L(−D_{1} − [\frac{Δ}{N}])|_{D_{1}}) ≥ 2.

In fact one has:

N ·deg(L(−D_{1} −[\frac{Δ}{N}])|_{D_{1}}) = deg(L^{N}(−D)|_{D_{1}})+∑_{i=2}^{r} ν_{i} ·D_{i} ·D_{1} +(Δ−[\frac{Δ}{N}])·D_{1}

≥ (N − ν) · c_{1}(L) · D_{1} + c_{1}(H) · D_{1} + ∑_{i=2}^{r} ν_{i} · D_{i} · D_{1}

> (N − ν) · c_{1}(L) · D_{1} + ∑_{i=2}^{r} ν_{i} · μ_{p}(D_{i}) = (N − ν) · c_{1}(L) · D_{1} + (μ_{p} − ν_{1}).

Since c_{1}(L) · D_{1} ≥ 2 and μ_{p} ≥ 2 · ν, one obtains

N · deg(L(−D_{1} − [\frac{Δ}{N}])|_{D_{1}}) > 2 · N − 2 · ν + μ_{p} − N ≥ N.

□

7.26. Remark. It is likely that the other parts of I. Reider’s theorem [53], i.e. the lower bounds for c_{1}(L)^{2} and for c_{1}(L) · C which imply that H^{0}(S, ω_{S} ⊗ L) separates points and tangent directions, can be obtained in a similar way.

§ 8 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.

8.1.

Let X be a proper smooth variety (or a scheme) over a field k. One introduces the de Rham cohomology

H_{DR}^{b}(X/k) := IH^{b}(X, Ω_{X/k}^{•})

where Ω_{X/k}^{•} is the complex of regular differential forms, defined over k, the so called de Rham complex.

In order to compute it, one introduces the “Hodge to de Rham” spectral sequence associated to the Hodge filtration Ω_{X/k}^{≥a} (see (A.25):

E_{1}^{ab} = H^{b}(X, Ω_{X/k}^{a}) ⟹ H_{DR}^{a+b}(X/k).

If k = C l , the field of complex numbers, the classical Hodge theory tells us that the Hodge spectral sequence

E_{1 an}^{ab} = H^{b}(X_{an}, Ω_{X_{an}}^{a}) ⟹ IH^{a+b}(X_{an}, Ω_{X_{an}}^{•})

§ 8  Characteristic p methods: Lifting of schemes  83

degenerates in E_{1}, where Ω_{X_{an}}^{•} is the de Rham complex of holomorphic differential forms (see (A.25)).

In fact, one has

IH^{a+b}(X_{an}, Ω_{X_{an}}^{•}) = H^{a+b}(X_{an}, C l )

and by Hodge theory

dim H^{l}(X_{an}, C l ) = ∑_{a+b=l} dim H^{b}(X_{an}, Ω_{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 [56],

H^{b}(X_{an}, Ω_{X_{an}}^{a}) = H^{b}(X, Ω_{X/C l}^{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:

8.2. 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_{DR}^{l}(X/k) = ∑_{a+b=l} dim H^{b}(X, Ω_{X/k}^{a}).

As we have already seen in Lecture 1 and 6, theorem (8.2) implies the Akizuki - Kodaira - Nakano vanishing theorem:

AKNV: If L is ample invertible, then

H^{b}(X, Ω_{X/k}^{a} ⊗ L^{−1}) = 0 for a + b < dim X

(where, of course, char k = 0).

Mumford [47] 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 [52] 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:

8.3. Theorem (Deligne - Illusie [12]). Let X be a proper smooth variety over a perfect field k of characteristic p ≥ 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.

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:

8.4. Proposition. Theorem (8.2) and AKNV over a field k of characteristic zero are consequences of theorem (8.3).

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

8.5. Liftings of a scheme. Let S be a scheme defined over IF_{p} the field with p elements.

8.6. Definition. A lifting of S to ZZ/p^{2} is a scheme S̃, defined and flat over ZZ/p^{2}, such that S = S̃ ×_{ZZ/p^{2}} IF_{p}.

8.7. Properties.

a) S is defined by a nilpotent ideal sheaf (of square zero) in S̃. In particular the inclusion S ⊂ S̃ or, if one prefers, the projection

O_{S̃} ⟶ O_{S}

induces the identity on the underlying topological spaces (S)_{top} and (S̃)_{top}.

b) From the exact sequence of ZZ/p^{2}-modules

0 ⟶ p · ZZ/p^{2} ⟶ ZZ/p^{2} ⟶ ZZ/p ⟶ 0

one obtains, since O_{S̃} is flat over ZZ/p^{2}, the exact sequence of O_{S̃}-modules

0 ⟶ p · O_{S̃} ⟶ O_{S̃} ⟶ O_{S} ⟶ 0

and, from the isomorphism of ZZ/p^{2}-modules

p : ZZ/p ⟶ p · ZZ/p^{2},

one obtains the isomorphism of O_{S̃}-modules

p : O_{S} ⟶ p · O_{S̃}.

8.8. Example. Let k be a perfect field of characteristic p and S = Spec k. Then S̃ exists and is uniquely determined (up to isomorphism) by (8.7,b):

S̃ = Spec W_{2}(k), where W_{2}(k) is called the ring of the second Witt vectors of k.

In concrete terms, W_{2}(k) = k ⊕ k · p as additive group and the multiplication is defined by

(x + y · p)(x′ + y′ · p) = x · x′ + (x · y′ + x′ · y) · p.

8.9. Assumptions. Throughout Lectures 8 to 11 S will be a noetherian scheme over IF_{p} with a lifting S̃ to ZZ/p^{2}. X will denote a noetherian S-scheme, D ⊂ X will be a reduced Cartier divisor. X will be supposed to be smooth over S, which means that locally X is étale over the affine space A A_{S}^{n} over S (here n = dim_{S} X). D 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 étale cover such that the coordinates of A A_{S}^{n} pull back to a parameter system (t_{1}, …, t_{n}) on X, for which D is defined by

t_{1} · . . . · t_{r}, for some r ≤ n.

We allow D to be empty.

8.10. Definition. For X smooth and D a normal crossing divisor over S, we define the sheaf Ω_{X/S}^{1} (log D) of one forms with logarithmic poles along D as the O_{X}-sheaf generated locally by

\frac{dt_{i}}{t_{i}} , for i ≤ r , and by dt_{i} , for i > r

(where we use the notation from (8.9)). Ω_{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

Ω_{X/S}^{a}(log D) = ⋀^{a} Ω_{X/S}^{1}(log D).

8.11. Definition. A lifting of

D = ∑_{j=1}^{r} D_{j} ⊂ X

to S̃ consists of a scheme X̃ and subschemes D̃_{j} of X̃, all defined and flat over S̃ such that X = X̃ ×_{S̃} S and D_{j} = D̃_{j} ×_{S̃} S. We write

D̃ = ∑_{j=r}^{r} D̃_{j}.

86 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

If k is a perfect field of characteristic p and S = Spec k, we say that (X, D) admits a lifting to W_{2}(k) if liftings X̃ and D̃_{j} exist over S̃ = Spec W_{2}(k). If L is an invertible sheaf on X, we say that X and L admit a lifting to W_{2}(k) if there is a lifting X̃ of X over S̃ = Spec W_{2}(k) and an invertible sheaf L̃ on X̃ with L̃|_{X} = L. 8.12. Remark. Of course, X̃ is also a lifting of the IF_{p}-scheme X to a scheme X̃ over ZZ/p^{2}. In particular, (8.7.a) remains true and we have

(X̃)_{top} = (X)_{top} and (D̃)_{top} = (D)_{top}.

One can make (8.7,b) more precise: 8.13. Lemma. Let X be smooth over S and let X̃ be a scheme over S̃ with X̃ ×_{S̃} S = X. Then the following conditions are equivalent.

a) X̃ is smooth over S̃.

b) X̃ is a lifting of X to S̃.

c) There is an exact sequence of O_{X̃}-modules

0 ⟶ p · O_{X̃} ⟶ O_{X̃} ⟶^{r} O_{X} ⟶ 0

together with an O_{X̃}-isomorphism

p : O_{X} ⟶ p · O_{X̃}

satisfying

p(x) = p · x̃ , for x̃ ∈ O_{X̃} , and x = r(x̃).

d) If Ũ ⊆ X̃ is an open subscheme, U its image in X, π : U ⟶ A A_{S}^{n} = Spec O_{S}[t_{1}, …, t_{n}]

an étale morphism and if φ̃_{1}, …, φ̃_{n} ∈ O_{Ũ} satisfy r(φ̃_{i}) = φ_{i} = π^{∗}t_{i}, then π extends to an étale morphism

π̃ : Ũ ⟶ A A_{S̃}^{n} = Spec O_{S̃}[t_{1}, …, t_{n}]

with π̃^{∗}(t_{i}) = φ̃_{i} for i = 1, …, n.

e) For each a ≥ 0 one has an exact sequence of O_{X̃}-modules

0 ⟶ p · Ω_{_{X̃}/_{S̃}}^{a} ⟶ Ω_{_{X̃}/_{S̃}}^{a} ⟶^{r} Ω_{X/S}^{a} ⟶ 0

and an O_{X̃}-isomorphism

p : Ω_{X/S}^{a} ⟶ p · Ω_{_{X̃}/_{S̃}}^{a}

satisfying

p(ω) = p · ω̃ , for ω̃ ∈ Ω_{_{X̃}/_{S̃}}^{a} , and ω = r(ω̃).

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 X̃ = Ũ) we can, of course, define

π̃ : X̃ ⟶ A A_{S̃}^{n} with π̃^{∗}(t_{i}) = φ̃_{i}.

Given a relation ∑ λ̃_{ν} m̃_{ν} = 0 in O_{X̃} between different monomials m̃_{ν} in

φ̃_{1}, …, φ̃_{n} the exact sequence in c) implies λ̃_{ν} = p · μ̃_{ν} for μ̃_{ν} ∈ O_{X̃} and the isomorphism in c) shows that one has

∑ μ_{ν} · m_{ν} = 0 for μ_{ν} = r(μ̃_{ν}) and m_{ν} = r(m̃_{ν}).

Hence μ_{ν} = 0 as well as μ̃_{ν} = 0.

If g_{1}, …, g_{r} are locally independent generators of O_{X} as a O_{A A}_{S}^{n}-module, and if g̃_{1}, …, g̃_{r} are liftings to O_{X̃}, then each x̃ ∈ O_{X̃} verifies

x = r(x̃) = ∑_{i=1}^{r} λ_{i}g_{i}

for some λ_{i} ∈ O_{A A}_{S}^{n}. If λ̃_{1}, …, λ̃_{r} are liftings of λ_{1}, …, λ_{r} to O_{A A}n ; S̃, then

x̃ − ∑_{i=1}^{r} λ̃_{i}g̃_{i} ∈ p · O_{X̃}

and one can find μ̃_{i} ∈ O_{A A}n ; S̃ with

x̃ − ∑_{i=1}^{r} λ̃_{i}g̃_{i} = p(∑_{i=1}^{r} μ_{i}g_{i}) = ∑_{i=1}^{r} p · μ̃_{i}g̃_{i},

and

x̃ = ∑_{i=1}^{r}(λ̃_{i} + p · μ̃_{i}) · g̃_{i}.

In other terms, g̃_{1}, … g̃_{r} are generators of O_{X̃} as a O_{A A}n ; S̃-module. They are independent by the same argument which gave the independence of the m_{μ}

88 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

above. O_{X̃} as a free O_{A A}n ; S̃-module is flat. Finally, (locally in X)

Ω_{X/S}^{1} = π^{∗}Ω_{A A_{S}^{n}}^{1} = ⨁_{1=1}^{n} O_{X}dφ_{i}

and

π̃^{∗}Ω_{A An ; S̃}^{1} = ⨁_{i=1}^{n} O_{X̃}dφ̃_{i}

surjects to Ω_{_{X̃}/_{S̃}}^{1}. In fact, if ω̃ ∈ Ω_{_{X̃}/_{S̃}}^{1},

ω̃ − ∑_{i=1}^{n} λ̃_{i}dφ̃_{i} ∈ im(Ω_{X/S}^{1} ⟶^{·p} Ω_{_{X̃}/_{S̃}}^{1})

for some λ̃_{i} ∈ O_{X̃} and, as above, one can modify the λ̃_{i} to get

ω̃ = ∑_{i=1}^{n}(λ̃_{i} + p · μ̃_{i})dφ̃_{i}.

As π̃^{∗}Ω_{A An ; S̃}^{1} ⟶ Ω_{_{X̃}/_{S̃}}^{1} is injective as well, π̃ is étale.

□

8.14. Lemma. Let X be a smooth S-scheme and

D = ∑_{j=1}^{r} D_{j}

be a normal crossing divisor over S. Let X̃ be a lifting of X to S̃ and D̃_{j} ⊆ X̃ subschemes with

D̃_{j} ⊗_{S̃} S = D_{j}

for j = 1, …, r. Then the following conditions are equivalent: a)

D̃ = ∑_{j=1}^{r} D̃_{j} ⊂ X̃

is a lifting of D ⊂ X to S̃.

b) The components of D̃ are Cartier divisors in X̃.

c) If in (8.13,d) one knows that D |_{U} is the zero-set of φ_{1} · . . . · φ_{s}, then one can choose π̃ : Ũ ⟶ A A_{S̃}^{n}, such that D̃_{j} |_{Ũ} is the zero set of π̃^{∗}(t_{j}).

Proof: If D̃ ⊂ X̃ is a lifting of D ⊂ X, then the flatness of X̃ and D̃_{j} over S̃ implies that the ideal sheaf J_{_{D̃}_{j}} of D̃_{j} is flat over S̃. We have again an exact sequence

0 ⟶ p · J_{_{D̃}_{j}} ⟶ J_{_{D̃}_{j}} ⟶ J_{D_{j}} ⟶ 0

where J_{D_{j}} is the ideal sheaf of D_{j}, and an isomorphism

p : J_{D_{j}} ⟶ p · J_{_{D̃}_{j}}.

If φ̃_{j} is a lifting of φ_{j} to J_{_{D̃}_{j}}, then for any g̃ ∈ J_{_{D̃}_{j}} one has g = λ · φ_{j} and

g̃ − λ̃ · φ̃_{j} ∈ p · I_{_{D̃}_{j}}

is of the form p · μ̃ · φ̃_{j} = p(μ · φ_{j}) for some μ̃ ∈ O_{X̃}. Hence φ̃_{j} is a defining

equation for D̃_{j}. By (8.13,d) b) implies c) and obviously c) implies a).

□

8.15. Definition. Using the notations from (8.14,c) and (8.13,d) we define for a lifting D̃ ⊂ X̃ of D ⊂ X to S̃ the sheaf

Ω_{_{X̃}/_{S̃}}^{1}(log D̃)

to be the O_{X̃}-sheaf generated by

\frac{dφ̃_{j}}{φ̃_{j}} for j = 1, …, s and dφ̃_{j} for j = s + 1, …, n.

8.16. Properties.

a) For all a the sheaves

Ω_{_{X̃}/_{S̃}}^{a}(log D̃) = ⋀^{a} Ω_{_{X̃}/_{S̃}}^{1}(log D̃)

are locally free over O_{X̃}.

b) One has an exact sequence of O_{X̃}-modules

0 ⟶ p · Ω_{_{X̃}/_{S̃}}^{a}(log D̃) ⟶ Ω_{_{X̃}/_{S̃}}^{a}(log D̃) ⟶ Ω_{X/S}^{a}(log D) ⟶ 0

and an O_{X̃}-isomorphism

p : Ω_{X/S}^{a}(log D) ⟶ p · Ω_{_{X̃}/_{S̃}}^{a}(log D̃).

8.17. Proposition. Let X be a smooth S-scheme.

a) Locally in the Zariski topology X has a lifting X̃ to S̃.

b) If X̃ is a lifting of X to S̃, if X̃ is affine and Y a complete intersection in X, then there exists a lifting Ỹ of Y to S̃ and an embedding Ỹ ⊂ X̃.

c) In particular, if D is a S-normal crossing divisor on X then locally in the Zariski topology D ⊂ X has a lifting D̃ ⊂ X̃ to S̃.

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 φ ∈ O_{X}. We can choose Ỹ to be the zero set of any lifting φ̃ ∈ O_{X̃} of φ. In fact, the flatness follows easily from (8.13,c) or from the following argument. Choose

π : X ⟶ A A_{S}^{n} = Spec O_{X}[t_{1}, …, t_{n}]

with φ = π^{∗}(t_{1}). By (8.13,d) π extends to an étale map π̃ : X̃ ⟶ A A_{S̃}^{n} with φ̃ = π̃^{∗}(t_{1}).

□

8.18. Isomorphisms between liftings

Let in the sequel X be a smooth S-scheme, D ⊂ X be an S-normal crossing divisor and let, for i = 1, 2, D̃^{(i)} ⊂ X̃^{(i)} be two liftings of D ⊂ X to S̃.

8.19. Notations. A morphism u : X̃^{(1)} → X̃^{(2)} is called an isomorphism of liftings

u : (X̃^{(1)}, D̃^{(1)}) ⟶ (X̃^{(2)}, D̃^{(2)})

if u |_{X}= id_{X} and if

u^{∗}(O_{_{X̃}_{(2)}}(−D̃^{(2)})) = O_{_{X̃}_{(1)}}(−D̃^{(1)}).

8.20. Remark. We have seen in (8.12) that (X̃^{(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^{∗} : O_{_{X̃}_{(2)}} ⟶ O_{_{X̃}_{(1)}}

of sheaves of rings on (X)_{top}. The assumption u|_{X} = id_{X} forces u^{∗} to be an isomorphism.

8.21. Lemma. Locally in the Zariski topology there exists an isomorphism of liftings

u : (X̃^{(1)}, D̃^{(1)}) ⟶ (X̃^{(2)}, D̃^{(2)}).

Proof: Locally the diagonal Δ ⊂ X × X is a complete intersection and we can lift it to

Δ̃ ⊂ X̃^{(1)} × X̃^{(2)}.

For example, if φ_{1}, …, φ_{n} are local parameters on X and φ̃_{1}^{(i)}, …, φ̃_{n}^{(i)} liftings in O_{_{X̃}_{(i)}} such that D^{(i)} is the zero locus of φ_{1}^{(i)} · . . . · φ_{s}^{(i)}, then we can choose

Δ̃ to be defined by

φ̃_{j}^{(1)} ⊗ 1 − 1 ⊗ φ̃_{j}^{(2)} for j = 1, …, n.

We have isomorphisms of liftings

p_{1} : Δ̃ ⟶ X̃^{(1)}, p_{2} : Δ̃ ⟶ X̃^{(2)}

and u = p_{2} ∘ p_{1}^{−1} satisfies

u^{∗}(O_{_{X̃}_{(2)}}(−D̃^{(2)})) = O_{_{X̃}_{(1)}}(−D̃^{(1)}).

□

Let

u, v : (X̃^{(1)}, D̃^{(1)}) ⟶ (X̃^{(2)}, D̃^{(2)})

be two isomorphisms of liftings. For x̃ ∈ O_{_{X̃}_{(2)}} one has

(u^{∗} − v^{∗})(p · x̃) = p(u^{∗} − v^{∗})(x̃) = p(id − id)(x) = 0

therefore (u^{∗} − v^{∗})|_{p·O_{X̃_{(2)}}} = 0. Of course, the map

O_{X} = O_{_{X̃}_{(2)}}/p · O_{_{X̃}_{(2)}} ⟶ O_{X} = O_{_{X̃}_{(1)}}/p · O_{_{X̃}_{(1)}}

induced by (u^{∗} − v^{∗}) is zero as well, and (u^{∗} − v^{∗}) factors through

(u^{∗} − v^{∗}) : O_{X} ⟶ p · O_{_{X̃}_{(1)}} = p(O_{X}).

For x, y ∈ O_{X} with liftings x̃, ỹ ∈ O_{_{X̃}_{(2)}} one has

(u^{∗} −v^{∗})(x̃·ỹ) = u^{∗}(x̃)·u^{∗}(ỹ)−v^{∗}(x̃)·v^{∗}(ỹ) = x·(u^{∗} −v^{∗})(ỹ)+y ·(u^{∗} −v^{∗})(x̃).

Hence p^{−1} ∘ (u^{∗} − v^{∗}) : O_{X} ⟶ O_{X} is a derivation and factors through

O_{X} ⟶^{d} Ω_{X/S}^{1} ⟶ O_{X}

where we denote the second morphism by (u^{∗} − v^{∗}) again. If t̃_{j}^{(i)} is a local equation for D̃_{j}^{(i)}, i = 1, 2, with reduction t_{j} ∈ O_{X}, then

u^{∗}(t̃_{j}^{(2)}) = t̃_{j}^{(1)} · (1 + p · λ)

and

v^{∗}(t̃_{j}^{(2)}) = t̃_{j}^{(1)}(1 + p · μ).

Hence

(u^{∗} − v^{∗})(dt_{j}) = t_{j} · (λ − μ) ∈ t_{j} · O_{X}

and p^{−1} ∘ (u^{∗} − v^{∗}) even factors through

O_{X} ⟶^{d} Ω_{X/S}^{1}(log D) ⟶ O_{X}.

8.22. Proposition. Keeping the notations from (8.19) let

u : (X̃^{(1)}, D̃^{(1)}) ⟶ (X̃^{(2)}, D̃^{(2)})

be an isomorphism of liftings. Then

{v : (X̃^{(1)}, D̃^{(1)}) ⟶ (X̃^{(2)}, D̃^{(2)}); v isomorphism of liftings }

is described by the affine space

u^{∗} + Hom_{O_{X}} (Ω_{X/S}^{1}(log D), O_{X}).

Proof: It just remains to show that for

φ ∈ Hom_{O_{X}} (Ω_{X/S}^{1}(log D), O_{X})

we can find v. Define

v^{∗} : O_{_{X̃}_{(2)}} ⟶ O_{_{X̃}_{(1)}}

by

v^{∗}(x̃) = u^{∗}(x̃) − p · φ(dx).

If t̃_{j}^{(i)} is as above an equation of D̃_{j}^{(i)},

v^{∗}(t̃_{j}^{(2)}) = t̃_{j}^{(1)} · (1 + p · λ) − p · t · γ for some γ ∈ O_{X},

or

v^{∗}(t̃_{j}^{(2)}) = t̃_{j}^{(1)}(1 + p(λ − γ))

and v^{∗} satisfies the conditions posed in (8.19.).

□

8.23. Proposition. Let X be an affine scheme, smooth over S and let D be a normal crossing divisor over S. Let D̃^{(i)} ⊂ X̃^{(i)} be two liftings of D ⊂ X to S̃. Then there exists an isomorphism of liftings

u : (X̃^{(1)}, D̃^{(1)}) ⟶ (X̃^{(2)}, D̃^{(2)}).

§ 9  The Frobenius and its liftings  93

Proof: Of course, (8.22) just says that the isomorphisms of liftings over a fixed open set form a “torseur” under the group

Hom_{O_{X}} (Ω_{X/S}^{1}(log D), O_{X})

and, since X is affine and

H^{1}(X, Hom_{O_{X}} (Ω_{X/S}^{1}(log D), 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 U = {X_{α}} of X and isomorphisms of liftings

u_{α} : (X̃_{α}^{(1)}, D̃_{α}^{(1)}) ⟶ (X̃_{α}^{(2)}, D̃_{α}^{(2)}).

By (8.22) p^{−1} ∘ (u_{α}^{∗} − u_{β}^{∗}) defines a 1-cocycle with values in the sheaf

Hom_{O_{X}} (Ω_{X/S}^{1}(log D), O_{X}).

Since

H^{1}(X, Hom_{O_{X}} (Ω_{X/S}^{1}(log D), O_{X}) = 0

we find

φ_{α} ∈ Γ(X_{α}, Hom_{O_{X}} (Ω_{X/S}^{1}(log D), O_{X}))

in some possibly finer cover {X_{α}} such that

p^{−1} ∘ (u_{α}^{∗} − u_{β}^{∗}) = φ_{α} − φ_{β}.

Hence the isomorphisms of liftings u_{α}^{∗} − p · φ_{α}^{∗} glue together to u : X̃_{1} → X̃_{2}.

□

§ 9 The Frobenius and its liftings

Everything in this lecture is either elementary or taken from [12].

Let S be a noetherian scheme defined over IF_{p} and let X be a noetherian S-scheme.

9.1. Definition. The absolute Frobenius of S is the endomorphism

F_{S} : S ⟶ S

defined by the following conditions. F is the identity on the topological space and

F_{S}^{∗} : O_{S} ⟶ O_{S}

is given by F_{S}^{∗}(a) = a^{p}. In particular for x ∈ O_{X} and λ ∈ O_{S} one has

F_{X}^{∗}(λx) = λ^{p}x^{p} = F_{S}^{∗}(λ) · F_{X}^{∗}(x),

and therefore one has a commutative diagram

X ⟶^{F_{X}} X
f↓  ↓^{f}
S ⟶_{F_{S}} S

For X′ = X ×_{F_{S}} S this allows to factorize F_{X}:

X ⟶^{F} X′ ⟶^{pr_{1}} X
[figure: f] ^{f′}↓  ↓^{f}
S ⟶_{F_{S}} S

with F_{X} = pr_{1} ∘ F and f′ = pr_{2}. By abuse of notations we write

F_{S} = pr_{1} : X′ ⟶ X.

F is called the relative Frobenius (relative to S). For

x ⊗ λ ∈ O_{X′} = O_{X} ⊗_{F_{S}} O_{S} one has F^{∗}(x ⊗ λ) = x^{p} · λ

and for

x ∈ O_{X} one has F_{S}^{∗}(x) = x ⊗ 1.

9.2. Remark. The absolute Frobenius F_{S} is a morphism of schemes. In fact, F_{S}^{∗} : O_{S,s} ⟶ O_{S,s} satisfies

F_{S}^{∗^{−1}}(m_{S,s}) = {x ∈ O_{S,s}; x^{p} ∈ 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 = SpecA, then F_{S} is induced by the p-th power map A → A.

9.3. Properties.

a) Since F_{S} : (S)_{top} → (S)_{top} is the identity,

F_{S} : (X′)_{top} → (X)_{top}

is an isomorphism of topological spaces, as well as

F : (X)_{top} → (X′)_{top}.

b) If t_{1}, …, t_{m} are locally on X, generators of O_{X}, i.e.

O_{X} = O_{S}[t_{1}, …, t_{m}]/_{<f_{1},...,f_{s}>}

and f = ∑ λ\underline{i} · \underline{t}\underline{i} , for \underline{t}\underline{i} = t_{1}^{i_{1}} · . . . · t_{m}^{i_{m}}, and λ\underline{i} ∈ O_{S}, then one has

F_{S}^{∗}(f) = ∑ λp ; \underline{i}\underline{t}\underline{i} .

Hence

O_{X′} = O_{X} ⊗_{F_{S}} O_{S} = O_{S}[t_{1}, …, t_{m}]/_{<F_{S}^{∗}(f_{1}),...F_{S}^{∗}(f_{s})>} .

For g = ∑ μ\underline{i}\underline{t}\underline{i} ∈ O_{X′} one has

F^{∗}(g) = ∑ μ\underline{i}t^{p·\underline{i}} where t^{p·\underline{i}} = t_{1}^{p·i_{1}} · . . . · t_{m}^{p·i_{m}}.

c) If X is smooth over S, one has locally étale morphisms π : X → A A_{S}^{n}, hence a diagram, where the right hand squares are by definition cartesian:

X ⟶^{F} X′ ⟶^{F_{S}} X
π↓  ↓_{π′}  ↓^{π}
A A_{S}^{n} ⟶^{F} (A A_{S}^{n})′ ⟶^{F_{S}} A A_{S}^{n}
[figure: ↓; ↓]
S ⟶^{F_{S}} S

For

A A_{S}^{n} = SpecO_{S}[t_{1} … t_{n}]

we have

(A A_{S}^{n})′ = SpecO_{S}[t_{1} … t_{n}]

and F^{−1}O_{(A A_{S}^{n})′} is the subsheaf of O_{A A}_{S}^{n} given by O_{S}[t_{1}^{p}, …, t_{n}^{p}]. Hence F_{∗}O_{A A}_{S}^{n} is freely generated over O_{(A A_{S}^{n})′} by

F_{∗}(t_{1}^{a_{1}} · . . . · t_{n}^{a_{n}}) for 0 ≤ a_{i} < p.

We have an isomorphism f : X ⟶ X′ ×_{(A A_{S}^{n})′} A A_{S}^{n} and the left upper square in the diagram is cartesian as well. In fact, for x ∈ X we may assume that the maximal ideal m_{X,x} ⊂ O_{X,x} is generated by t_{1}, … t_{n} and, if O_{x} is the local ring of x in X′ ×_{(A A_{S}^{n})′} A A_{S}^{n}, then the maximal ideal m_{x} of O_{x} has the same generators. Hence

f^{∗} : O_{x} ⟶ O_{X,x}

is a local homomorphism inducing a surjection

m_{x} ⟶ m_{X,x}/m_{X,x}^{2}.

d) The sheaf F_{∗}O_{X} is again a locally free O_{X}-module. Using the notation from part c) it is generated by

F_{∗}π^{∗}(t_{1}^{a_{1}} … t_{n}^{a_{n}}) for 0 ≤ a_{i} < p.

Therefore, for any locally free sheaf F on X, the sheaf F_{∗}F is locally free over O_{X′}. For example, if D is a normal crossing divisor on X, then F_{∗}Ω_{X/S}^{a}(log D) is locally free.

9.4. Definition. When S has a lifting S̃ to ZZ/p^{2} (see (8.5)), a lifting F̃_{S̃} of F_{S} is a finite morphism

F̃_{S̃} : S̃ ⟶ S̃

whose restriction to S is F_{S}.

Similarly, if X and X′ have liftings X̃ and X̃′ to S̃, a lifting F̃ of the relative Frobenius F is a finite morphism

F̃ : X̃ ⟶ X̃′

which restricts to F.

In particular, (8.13,c) gives rise to an exact sequence of O_{_{X̃}_{′}}-modules

0 ⟶ F̃_{∗}p · O_{X̃} ⟶ F̃_{∗}O_{X̃} ⟶ F_{∗}O_{X} ⟶ 0

together with an O_{_{X̃}_{′}}-isomorphism

p : F_{∗}O_{X} ⟶ F̃_{∗}p · O_{X̃} = p · F̃_{∗}O_{X̃}.

9.5. Assumptions. For the rest of this lecture we assume S to be a scheme over IF_{p} with a lifting S̃ to ZZ/p^{2} and a lifting

F̃_{S̃} : S̃ ⟶ S̃

of the absolute Frobenius.

Moreover we keep the assumptions made in (8.9). Hence X is supposed to be smooth over S and D ⊂ X is a normal crossing divisor over S. We write D′ = F_{S}^{∗}(D) for F_{S} : X′ → X.

9.6. Example. If k is a perfect field, S = Spec k and S̃ = Spec W_{2}(k), then one takes

F̃_{S̃}^{∗}(x + y · p) = x^{p} + y^{p} · 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 S̃ = Spec W_{2}(k) if and only if X′ does.

9.7. Proposition. Let D ⊂ X be an S-normal crossing divisor of the smooth S-scheme X. Let

D̃′ ⊂ X̃′ be a lifting of D′ ⊂ X′

to S̃. Then locally in the Zarisky topology

D ⊂ X has a lifting D̃ ⊂ X̃

to S̃ such that F lifts to

F̃ : X̃ ⟶ X̃′ with F̃^{∗}O_{_{X̃}_{′}}(−D̃′) = O_{X̃}(−p · D̃).

Proof: By (8.17,c) we know that a lifting D̃ ⊂ X̃ exists locally. Let

π : X ⟶ A A_{S}^{n} = Spec O_{S}[t_{1}, …, t_{n}]

be étale and D_{j} be the zero set of φ_{j} = π^{∗}(t_{j}). By (8.14,c) we can choose liftings of φ_{j} to φ̃_{j} ∈ O_{X̃} and of

φ′_{j} = F_{S}^{∗}(φ_{j}) = φ_{j} ⊗ 1

to φ̃′_{j} such that D̃_{j} is defined by φ̃_{j} and D̃′_{j} by φ̃′_{j}. We can define

F̃^{∗} by F̃^{∗}(φ̃′_{j}) = φ̃_{j}^{p}.

By the explicit description of F in (9.3,c) F̃ restricts to F and

F̃^{∗}O_{_{X̃}_{′}}(−D̃′) = O_{X̃}(−p · D̃).

□

9.8. Remark. We have seen in (8.12) that (X̃)_{top} = (X)_{top} and (X̃′)_{top} = (X′)_{top}. By (9.3,a) we have (X)_{top} ≃ (X′)_{top} and hence we can regard a lifting F̃ as a morphism

F̃^{∗} : O_{_{X̃}_{′}} ⟶ O_{X̃}

of sheaves of rings over (X′)_{top}.

Similarly to (8.22) we have

9.9. Proposition. Keeping notations and assumptions from (9.7) assume that

D ⊂ X has a lifting D̃ ⊂ X̃

to S̃ and let F̃_{0} : X̃ → X̃′ be one lifting of F with

F̃_{0}^{∗}O_{_{X̃}_{′}}(−D̃′) = O_{X̃}(−p · D̃).

98 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

Then

{F̃ : X̃ ⟶ X̃′; F̃^{∗}O_{_{X̃}_{′}}(−D̃′) = O_{X̃}(−p · D̃), F̃ lifting of F }

is described by the affine space

F̃_{0}^{∗} + Hom_{O_{X′}} (Ω_{X′/S}^{1}(log D′), F_{∗}O_{X}).

Proof: As in (8.22) for isomorphisms of liftings one finds that F̃^{∗} −F̃_{0}^{∗} is zero on p · O_{X̃} and induces the zero map from O_{X′} to O_{X}. Hence F̃^{∗} − F̃_{0}^{∗} induces

F̃^{∗} − F̃_{0}^{∗} : O_{X′} ⟶ p · O_{X}.

For x̃′, ỹ′ ∈ O_{_{X̃}_{′}} one has

(F̃^{∗} − F̃_{0}^{∗})(x̃′ · ỹ′) = F (x′)(F̃^{∗} − F̃_{0}^{∗})(ỹ′) + F (y′)(F̃^{∗} − F̃_{0}^{∗})(x̃′) and

p^{−1}(F̃^{∗} − F̃_{0}^{∗}) : O_{X′} ⟶ O_{X}

factorizes through

O_{X′} ⟶^{d} Ω_{X′/S}^{1} ⟶ O_{X}

where the right hand side morphism is O_{X′}-linear and is again denoted by (F̃^{∗} − F̃_{0}^{∗}). For φ̃_{j} ∈ O_{X̃} and φ̃′_{j} ∈ O_{_{X̃}_{′}} , local parameters for D̃_{j} and D̃′_{j} respectively, which lift φ_{j} and φ′_{j} = φ_{j} ⊗ 1, one has

F̃^{∗}(φ̃′_{j}) = φ̃_{j}^{p} · (1 + p · λ̃)

and

F̃_{0}^{∗}(φ̃′_{j}) = φ̃_{j}^{p} · (1 + p · λ̃_{0})

for some λ̃, λ̃_{0} ∈ O_{X̃}. Therefore

(F̃^{∗} − F̃_{0}^{∗})(φ̃′_{j}) = φ̃_{j}^{p}(p(λ̃ − λ̃_{0}))

and

(F̃^{∗} − F̃_{0}^{∗})(dφ̃′_{j}) = p^{−1} ∘ (F̃^{∗} − F̃_{0}^{∗})(φ̃′_{j}) ∈ O_{X}(−p · D_{j}).

Hence

(F̃^{∗} − F̃_{0}^{∗}) : Ω_{X′/S}^{1} ⟶ O_{X}

extends to

(F̃^{∗} − F̃_{0}^{∗}) : Ω_{X′/S}^{1}(log D′) ⟶ O_{X}.

Conversely, for

φ ∈ Hom_{O_{X′}} (Ω_{X′/S}^{1}(log D′), O_{X})

we define

F̃^{∗} = F̃_{0}^{∗} + φ^{∗} by F̃^{∗}(x̃) = F̃_{0}^{∗}(x̃) − p(φ(dx)).

We have

O_{X′}(−D′) ⟶^{d} Ω_{X′/S}^{1}(log D′)(−D′) ⟶^{φ} O_{X}(−p · D)

and

F̃^{∗}(O_{_{X̃}_{′}}(−D̃′)) = O_{X̃}(−p · D̃).

□

9.10. Corollary. Under the assumption of (9.7) assume that X is affine and that

D ⊂ X has a lifting D̃ ⊂ X̃

to S̃. Then there is a lifting F̃ : X̃ → X̃′ of F with

F̃^{∗}O_{_{X̃}_{′}}(−D̃′) = O_{X̃}(−p · D̃).

Proof: One repeats the argument used to prove (8.23), replacing X̃^{(1)} by X̃ and X̃^{(2)} by X̃′ and using (9.7) and (9.9) instead of (8.21) and (8.22).

□

9.11. Remark. Let X be a smooth S-scheme, X̃′ a lifting of X′ and X̃^{(i)}, for i = 1, 2, two liftings of X to S̃. Assume that we have a lifting

F̃_{2} : X̃^{(2)} ⟶ X̃′

and isomorphisms

u : X̃^{(1)} ⟶ X̃^{(2)} and v : X̃^{(1)} ⟶ X̃^{(2)},

both lifting the identity. Then, considering again O_{_{X̃}_{′}}, O_{_{X̃}_{(1)}} and O_{_{X̃}_{(2)}} as sheaves of rings on (X′)_{top}, one has

O_{_{X̃}_{′}} \underrightarrow{_{F̃}_{2}^{∗}} O_{_{X̃}_{(2)}} ⟶^{(u^{∗}−v^{∗})} O_{_{X̃}_{(1)}}

and

(u^{∗} − v^{∗}) ∘ F̃_{2}^{∗}(x̃′) = (u^{∗} − v^{∗})(d(F^{∗}(x′))) = 0

since F^{∗}(x′) is a p-th power. We find:

(F̃_{2} ∘ u)^{∗} = (F̃_{2} ∘ v)^{∗} : O_{_{X̃}_{′}} ⟶ O_{_{X̃}_{(1)}}.

In other words, (F̃_{2} ∘ u)^{∗} does not depend on the choice of u.

100 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

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

F̃ : X̃ ⟶ X̃′

of F. We have more generally:

9.12. Corollary. Let X be a smooth scheme and D ⊂ X be a normal crossing divisor over S. Given liftings

D̃ ⊂ X̃ and D̃′ ⊂ X̃′ of D ⊂ X and D′ ⊂ X′

(respectively) to S̃, the exact obstruction for lifting F to

F̃ : X̃ ⟶ X̃′ with F̃^{∗}O_{_{X̃}_{′}}(−D̃′) = O_{X̃}(−p · D̃)

is a class

[F_{_{X̃}′,_{D̃}′}] ∈ H^{1}(X′, Hom_{O_{X′}} (Ω_{X′/S}^{1}(log D′), F_{∗}O_{X}))

which does not depend on (X̃, D̃).

Proof: By (9.7) or (9.10) one can cover X̃ by affine X̃_{α} such that F lifts to F̃_{α} on X̃_{α} with the required property for D. Then by (9.9) (F̃_{α}^{∗}−F̃_{β}^{∗}) describes a 1-cocycle with values in

Hom_{O_{X′}} (Ω_{X′/S}^{1}(log D′), F_{∗}O_{X}).

Changing the F̃_{α} corresponds to changing the cocycle by a coboundary. We define [F_{_{X̃}′,_{D̃}′}] to be the cohomology class of this cocycle. If [F_{_{X̃}′,_{D̃}′}] = 0 one finds for a possibly finer cover {X′_{α}}

φ_{α} ∈ Γ(X′_{α}, Hom_{O_{X′}} (Ω_{X′/S}^{1}(log D′), F_{∗}O_{X}))

such that the F̃_{α}^{∗} + φ_{α} glue together to give F̃ : X̃ → X̃′.

If X̃^{(i)} are two liftings, X̃_{α}^{(i)} coverings and F̃_{α}^{(i)} liftings of F, for i = 1, 2 we can apply (8.21) or (8.23) to get isomorphisms of liftings

u_{α} : X̃_{α}^{(1)} ⟶ X̃_{α}^{(2)}.

By (9.11) we have on X̃′_{α} ∩ X̃′_{β}

(F̃_{α}^{(2)} ∘ u_{α})^{∗} − (F̃_{β}^{(2)} ∘ u_{β})^{∗} = (F̃_{α}^{(2)} ∘ u_{α})^{∗} − (F̃_{β}^{(2)} ∘ u_{α})^{∗} =
u_{α}^{∗} ∘ (F̃_{α}^{(2)∗} − F̃_{β}^{(2)∗}) ∈ Γ(X′_{α} ∩ X′_{β}, Hom_{O_{X′}} (Ω_{X′/S}^{1}(log D′), F_{∗}O_{X}))

As u_{α}^{∗} is the identity on X′_{α} the cocycle defined by F̃_{α}^{(2)} and F̃_{α}^{(2)} ∘ u_{α} are the same and F̃_{α}^{(2)} and F̃_{α}^{(1)} define the same cohomology class.

□

9.13. The Cartier operator

Let X be a smooth scheme over S and F : X → X′ be the Frobenius relative to S. The key observation is that the differential d in the de Rham complex Ω_{X/S}^{•} is O_{X′}-linear as, using the notations of (9.1),

dF^{∗}(x ⊗ 1) = dx^{p} = 0.

If D is a normal crossing divisor over S, the homology sheaves

H^{a} = H^{a}(F_{∗}Ω_{X/S}^{•}(log D))

are O_{X′}-modules computed by the following

9.14. Theorem (Cartier, see [9] [34]). One has an isomorphism of O_{X′}-modules

C^{−1} : Ω_{X′/S}^{1}(log D′) ⟶ H^{1}(F_{∗}Ω_{X/S}^{•}(log D))

such that:

a) For x ∈ O_{X} one has

C^{−1}(d(x ⊗ 1)) = x^{p−1}dx in H^{1}.

b) If t is a local parameter defining a component of D, then

C^{−1}(\frac{d(t ⊗ 1)}{t ⊗ 1}) = \frac{dt}{t} in H^{1}.

c) C^{−1} is uniquely determined by a) and b).

d) For all a ≥ 0 one has an isomorphism

⋀^{a} C^{−1} : Ω_{X′/S}^{a}(log D′) ⟶ H^{a}(F_{∗}Ω_{X/S}^{•}(log D))

obtained by wedge product from C^{−1}.

Proof: c) is obvious since Ω_{X′/S}^{1}(log D′) is generated by elements of the form

d(x ⊗ 1) and \frac{d(t ⊗ 1)}{t ⊗ 1}.

For the existence of C^{−1} let us first assume that D = ∅. Then

(x + y)^{p−1}(dx + dy) − x^{p−1}dx − y^{p−1}dy = df

where for

γ_{i} ∈ IF_{p} with γ_{i} ≡ \frac{1}{p} ( \begin{matrix}p \\ i \end{matrix} ) mod p

102 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

we take

f = ∑_{i=1}^{p−1} γ_{i} · x^{i} · y^{p−i} = \frac{1}{p}[(x + y)^{p} − x^{p} − y^{p}].

Moreover, one has

(y · x)^{p−1}d(y · x) = x^{p} · y^{p−1}dy + y^{p} · x^{p−1}dx

and d(x^{p−1}dx) = 0. Hence, the property a) defines C^{−1}. For D ≠ ∅, b) is compatible with the definition of C^{−1} on Ω_{X′/S}^{1}. In fact,

C^{−1}(t ⊗ 1 · \frac{d(t ⊗ 1)}{t ⊗ 1}) = F^{∗}(t ⊗ 1)C^{−1}(\frac{d(t ⊗ 1)}{t ⊗ 1}) = t^{p} \frac{dt}{t}.

Having defined C^{−1}, we can define ⋀^{a} C^{−1} as well. As in (9.3,c) we have locally a cartesian square

X ⟶^{F} X′
π↓ ↓_{π′}
A A_{S}^{n} ⟶^{F} (A A_{S}^{n})′

with π and π′ étale. Hence to show that ⋀^{a} C^{−1} is an isomorphism it is enough to consider the case

X = A A_{S}^{n} = Spec O_{S}[t_{1}, … t_{n}]

and D to be the zero set of t_{1} · . . . · t_{r}. If B^{a} is the IF_{p}-vector space freely generated by

t_{1}^{i_{1}} · . . . · t_{n}^{i_{n}} · ω_{α_{1}} ∧ … ∧ ω_{α_{a}}

for 0 ≤ i_{ν} < p for ν = 1, …, n 1 ≤ α_{1} < α_{2} … < α_{a} ≤ n

ω_{ν} = \{ \begin{matrix}\frac{dt_{ν}}{t_{ν}} & ν = 1, …, r \\ dt_{ν} & ν = r + 1, …, n \end{matrix}

then B^{•}, with the usual differential is a subcomplex of F_{∗}Ω_{X/S}^{•}(log D). One has F_{∗}Ω_{X/S}^{•}(log D) = O_{X′} ⊗_{IF_{p}} B^{•} and (9.14) follows from the following claim.

□

9.15. Claim. One has

i) H^{0}(B^{•}) = IF_{p} . ii) H^{1}(B^{•}) has the basis {ω_{1}, …, ω_{r}, t_{r+1}^{p−1}ω_{r+1}, …, t_{n}^{p−1}ω_{n}} . iii) H^{a}(B^{•}) = ⋀^{a} H^{1}(B^{•}).

Proof: For n = 1, this is shown easily: Obviously ker (d : B^{0} ⟶ B^{1}) = IF_{p}. For D = ∅ let us write K^{•} = B^{•}. One has

K^{1} =< t^{i}dt; i = 0, …, p − 1 >_{IF_{p}}

and

dK^{0} =< dt^{i+1} = (i + 1) · t^{i}dt; i = 0, …, p − 2 >_{IF_{p}} .

For D ≠ ∅ write L^{•} = B^{•}. One has

L^{1} =< t^{i} \frac{dt}{t}; i = 0, …, p − 1 >_{IF_{p}}

and

dL^{0} =< dt^{i} = i · t^{i} · \frac{dt}{t}; i = 1, … p − 1 >_{IF_{p}} .

In both cases (9.15) is obvious. For n > 1 one can write

B^{•} = \underbrace{L ⊗_{IF_{p}} L^{•} ⊗ … ⊗_{IF_{p}} L^{•}}_{r times} ⊗_{IF_{p}} \underbrace{K^{•} ⊗ … ⊗_{IF_{p}} K^{•}}_{n−r times.} .

By the Künneth formula (A.8)

H^{a}(B^{•}) = ∑ ; ∑_{i=1}^{n} ε_{i}=a = H^{ε_{1}}(L^{•}) ⊗ … ⊗ H^{ε_{r}}(L^{•}) ⊗ … ⊗ H^{ε_{n}}(K^{•}).

which implies a), b) and c).

□

9.16. Notation. Following Deligne-Illusie, we define

Ω_{X/S}^{•}(A, B) = Ω_{X/S}^{•}(log (A + B))(−A)

where A + B is a normal crossing divisor over S. 9.17. Corollary. The Cartier operator induces an isomorphism

Ω_{X′/S}^{a}(A′, B′) ⟶ H^{a}(F_{∗}Ω_{X/S}^{•}(A, B))

Proof: By (2.7) the residues of

d : O_{X}(−A) ⟶ Ω_{X/S}^{1}(log (A + B))(−A) = Ω_{X/S}^{1}(A, B)

along the components of A are all 1 and by (2.10)

Ω_{X/S}^{•}(log (A + B))(−p · A) ⟶ Ω_{X/S}^{•}(A, B)

is a quasi isomorphism. Since

F_{∗}Ω_{X/S}^{•}(log (A + B))(−p · A) = F_{∗}Ω_{X/S}^{•}(log (A + B)) ⊗_{O_{X′}} O_{X′}(−A′) we can apply (9.14).

□

9.18. Duality

Let us keep the notation from (9.16). The wedge product

Ω_{X/S}^{n−i}(log D) ⊗ Ω_{X/S}^{i}(log D) ⟶^{∧} Ω_{X/S}^{n} ⊗ O_{X}(D)

is a perfect duality of locally free sheaves. Hence one obtains:

9.19. Lemma. Ω_{X/S}^{n−i}(A, B) ⊗ Ω_{X/S}^{i}(B, A) ⟶^{∧} Ω_{X/S}^{n} is a perfect duality.

9.20. Lemma. One has a perfect duality

F_{∗}Ω_{X/S}^{n−i}(A, B) ⊗ F_{∗}Ω_{X/S}^{i}(B, A) ⟶ Ω_{X′/S}^{n}

given by

F_{∗}Ω_{X/S}^{n−i}(A, B) ⊗ F_{∗}Ω_{X/S}^{i}(B, A) ⟶_{∧} F_{∗}Ω_{X/S}^{n} ⟶ H^{n} ⟶_{C} Ω_{X′/S}^{n}

where C is the Cartier operator.

Proof: In fact, this is nothing but duality for finite flat morphisms ([30], p 239). One has

\begin{aligned}F_{∗}Ω_{X/S}^{n−i}(A, B) & = & F_{∗}Hom_{O_{X}} (Ω_{X/S}^{i}(B, A), Ω_{X/S}^{n}) \\ & ≃ & Hom_{O_{X′}} (F_{∗}Ω_{X/S}^{i}(B, A), Ω_{X′/S}^{n}) \end{aligned}

and (9.20) is just saying that F_{∗}Ω_{X/S}^{n} ⟶ Ω_{X′/S}^{n} 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 A_{S}^{n}, A the zero set of t_{1} · . . . · t_{s} and B the zero set of t_{s+1} … t_{r}. Define, for a > 0, B^{a}(A, B) to be the IF_{p}-vector space generated by all

φ = t_{1}^{i_{1}} . . . · t_{n}^{i_{n}} · ω_{α_{1}} ∧ … ∧ ω_{α_{a}}

with

ω_{ν} = \{ \begin{matrix}\frac{dt_{ν}}{t_{ν}} & for ν = 1, … s, …, r \\ dt_{ν} & for ν = r + 1, …, n \end{matrix}

where the indices are given by

0 < i_{ν} ≤ p for ν = 1, …, s
0 ≤ i_{ν} < p for ν = s + 1, …, r, …, n
and by  1 ≤ α_{1} < α_{2} < … < α_{a} ≤ n.

Similarly we have B^{a}(B, A) by taking as index set

0 ≤ i_{ν} < p for ν = 1, …, s and ν = r + 1, …, n
0 < i_{ν} ≤ p for ν = s + 1, …, r
and  1 ≤ α_{1} < α_{2} < … < α_{a} ≤ n.

§ 10 The proof of Deligne and Illusie [12] 105

For a = n − i the only generator δ of B^{i}(B, A) with C(φ ∧ δ) ≠ 0 is

δ = t_{1}^{j_{1}} · . . . · t_{n}^{j_{n}}ω_{β_{1}} ∧ … ∧ ω_{β_{i}}

with

{β_{1}, …, β_{i}} ∪ {α_{1}, …, α_{n−i}} = {1, …, n}

and

i_{ν} + j_{ν} = \{ \begin{matrix}p & for ν = 1, …, r \\ p − 1 & for ν = r + 1, …, n. \end{matrix}

□

9.21. Remark. For φ ∈ B^{n−i−1}(A, B) and δ ∈ B^{i}(B, A) the explicit description of the duality in the proof of (9.20) shows that (up to sign)

C(dφ ∧ δ) = C(φ ∧ dδ).

Hence we obtain as well:

9.22. Corollary. Under the duality in (9.20) the transposed of the differential d is again d (up to sign).

§ 10 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 ⊂ X a S-normal crossing divisor, and S is a noetherian scheme over ZZ/p which admits a lifting S̃ to ZZ/p^{2} as well as a lifting F̃_{S̃} : S̃ → S̃ of the absolute Frobenius F_{S}.

10.1. The two term de Rham complex

is defined as

τ_{≤1} F_{∗}Ω_{X/S}^{•} (log D).

Hence, as explained in (A.26), it is the complex

F_{∗}O_{X} ⟶ Z^{1}

where

Z^{1} = Ker (F_{∗}Ω_{X/S}^{1} (log D) ⟶ F_{∗}Ω_{X/S}^{2} (log D)).

One has a short exact sequence of complexes

0 ⟶ H^{0} ⟶ τ_{≤1} F_{∗}Ω_{X/S}^{•} (log D) ⟶ H^{1}[−1] ⟶ 0

given by

0 ⟶ H^{0} ⟶ F_{∗}O_{X}
↓
Z^{1} ⟶ H^{1} ⟶ 0

where H^{0} = O_{X′} and H^{1} is O_{X′}- isomorphic to Ω_{X′/S}^{1}(log D) via the Cartier operator (9.14).

10.2. Definition. A splitting of τ_{≤1}F_{∗}Ω_{X/S}^{•}(log D) is a diagram

τ_{≤1}F_{∗}Ω_{X/S}^{•}(log D) ⟶^{σ}  K^{•}
↑^{θ}
H^{0} ⊕ H^{1}[−1]

where K^{•} is the Čech complex

C^{•}(U, τ_{≤1}F_{∗}Ω_{X/S}^{•}(log D))

associated to some affine open cover U of X′, where σ is the induced morphism, hence a quasi-isomorphism (see (A.6)), and where θ is a quasi-isomorphism. We may assume, of course, that

H^{i} ⟶^{σ} H^{i} ⟶^{θ^{−1}} H^{i}

is the identity for i = 0, 1.

10.3. Example. Assume that D ⊂ X and D′ ⊂ X′ both lift to

D̃ ⊂ X̃ and D̃′ ⊂ X̃′

on S̃ and that F lifts to F̃ : X̃ → X̃′ in such a way that

F̃^{∗}O_{_{X̃}_{′}}(−D̃′) = O_{X̃}(−p · D̃).

For example, if S = Spec k for a perfect field k and if D ⊂ X has a lifting D̃ ⊂ X̃ then as we have seen in (9.6) D′ ⊂ X′ has a lifting as well. By (9.12) the existence of F̃ is equivalent to

[F_{_{X̃}′,_{D̃}′}] = 0 in H^{1}(X′, Hom_{O_{X′}} (Ω_{X′/S}^{1}(log D′), F_{∗}O_{X})).

For example it automatically exists if this group vanishes.

Anyway, if the liftings D̃, X̃, D̃′, X̃′ and F̃ exist, the morphism

F̃^{∗} : Ω_{_{X̃}′/_{S̃}}^{1}(log D̃′) ⟶ Ω_{_{X̃}/_{S̃}}^{1}(log D̃)

verifies

F̃^{∗}|_{p·Ω_{X̃′/S̃}^{1}(log _{ D̃}′)} = 0.

In fact,

F^{∗} : Ω_{X′/S}^{1}(log D′) ⟶ Ω_{X/S}^{1}(log D)

is given by F^{∗}(d(t ⊗ 1)) = d(t^{p}) and hence it is the zero map. We have a commutative diagram

Ω_{X′/S}^{1}(log D′) ⟶_{≃}^{p} p · Ω_{_{X̃}′/_{S̃}}^{1}(log D̃′)
↓_{F^{∗}}  ↓_{F̃}^{∗}
Ω_{X/S}^{1}(log D) ⟶_{≃}^{p} p · Ω_{_{X̃}/_{S̃}}^{1}(log D̃)

and hence the vertical morphisms are both zero. The same argument shows that the factorization

F̃^{∗} : Ω_{X′/S}^{1}(log D′) ⟶ Ω_{_{X̃}/_{S̃}}^{1}(log D̃)

takes values in

p · Ω_{_{X̃}/_{S̃}}^{1}(log D̃) = p · Ω_{X/S}^{1}(log D).

The induced map

p^{−1} ∘ F̃^{∗} : Ω_{X′/S}^{1}(log D′) ⟶ Ω_{X/S}^{1}(log D)

can be written in coordinates as follows. For x ∈ O_{X} let x̃ ∈ O_{X̃} be a lifting of x and let x̃′ ∈ O_{_{X̃}_{′}} be a lifting of x′ = x ⊗ 1. One writes

F̃^{∗}(x̃′) = x̃^{p} + p(u(x̃, x̃′))

for some u(x̃, x̃′) ∈ O_{X}. Then

p^{−1} ∘ F̃^{∗}(dx̃′) = x^{p−1}dx + du(x̃, x̃′).

In particular the image of p^{−1} ∘ F̃^{∗} lies in

Z^{1} ⊂ F_{∗}Ω_{X/S}^{1}(log D)

and the composition with Z^{1} ⟶ H^{1} gives back the Cartier operator.

In this example, i.e. if the liftings D̃, X̃, D̃′, X̃′ and F̃ all exist, we can take U = {X′} and define

θ : O_{X′} ⊕ Ω_{X′/S}^{1}(log D′)[−1] ⟶ τ_{≤1}F_{∗}Ω_{X/S}^{•}(log D)

by

O_{X′} ⟶ F_{∗}O_{X}
↓^{0}  ↓^{d}
Ω_{X′/S}^{1} ⟶_{p^{−1}∘_{F̃}^{∗}} Z^{1}

and, by (9.14), θ is a quasi-isomorphism.

10.4. Notation. We call a cohomology class

φ ∈ IH^{1}(X′, Hom_{O_{X′}} (H^{1}, F_{∗}O_{X}) → Hom_{O_{X′}} (H^{1}, Z^{1}))

a splitting cohomology class, if φ maps to the identity in

H^{0}(X′, Hom_{O_{X′}} (H^{1}, H^{1})) = IH^{1}(X′, Hom_{O_{X′}} (H^{1}, H^{1})[−1]).

10.5. Proposition. The splittings of

τ_{≤1}F_{∗}Ω_{X/S}^{•}(log D)

are in one to one correspondence with the splitting cohomology classes

φ ∈ IH^{1}(X′, Hom_{O_{X′}} (H^{1}, F_{∗}O_{X}) ⟶ Hom_{O_{X′}} (H^{1}, Z^{1})).

Proof: Let φ be a splitting cohomology class, realized as cocycle

φ_{αβ} ∈ Γ(X′_{αβ}, Hom_{O_{X′}} (H^{1}, F_{∗}O_{X}))

and

ψ_{α} ∈ Γ(X′_{α}, Hom_{O_{X′}} (H^{1}, Z^{1}))

for some affine open cover U = {X′_{α}} of X′. Hence, using the notations from (A.6) for the differential in the Čech complex, δφ = 0 and dφ − δψ = 0. By assumption ψ_{α} induces the identity in

Γ(X′_{α}, Hom_{O_{X′}} (H^{1}, H^{1})).

Then θ = (id, (φ_{αβ}, ψ_{α})) is the map wanted, i.e.

O_{X′}  ⟶^{id}  ⨁_{U} ϱ_{α∗}F_{∗}O_{X}|_{X′_{α}}
0↓  ↓^{δ⊕d}
H^{1} ⟶^{(φ_{αβ},ψ_{α})} ⨁_{U} ϱ_{αβ∗}F_{∗}O_{X}|_{X′_{αβ}} ⊕ ⨁_{U} ϱ_{α∗} Z^{1}|_{X′_{α}}
↓  ↓^{(−δ⊕d, 0⊕−δ)}
0 ⟶ ⨁_{U} ϱ_{αβγ∗}F_{∗}O_{X}|_{X′_{αβγ}} ⊕ ⨁_{U} ϱ_{αβ∗} Z^{1}|_{X′_{αβ}}.

where

ϱ_{α_{1},···α_{r}} : X′_{α_{1},···α_{r}} ⟶ X′

denotes the embedding, and where

(−δ ⊕ d, 0 ⊕ −δ)(x_{αβ}, z_{α}) = (−δ(x), d(x) − δ(z)).

Conversely, let for some U

τ_{≤1}(F_{∗}Ω_{X/S}^{•}(log D)) ⟶^{σ} K^{•} ⟵^{θ} H^{0} ⊕ H^{1}[−1]

be a splitting. As H^{1} is O_{X′}-locally free, σ⊗id and θ⊗id are quasi-isomorphisms of the corresponding complexes tensored with Hom_{O_{X′}} (H^{1}, O_{X′}). We obtain therefore maps

H^{0}(X′, Hom_{O_{X′}} (H^{1}, H^{1})) = IH^{1}(X′, H^{1} ⊗ Hom_{O_{X′}} (H^{1}, O_{X′})[−1])
↓
IH^{1}(X′, [H^{0} ⊕ H^{1}[−1]] ⊗ Hom_{O_{X′}} (H^{1}, O_{X′}))
↓^{≃}
IH^{1}(X′, τ_{≤1}F_{∗}Ω_{X/S}^{•}(log D) ⊗ Hom_{O_{X′}} (H^{1}, O_{X′}))
↓^{=}
IH^{1}(X′, Hom_{O_{X′}} (H^{1}, F_{∗}O_{X}) → Hom_{O_{X′}} (H^{1}, Z^{1}))
↓
H^{0}(X′, Hom_{O_{X′}} (H^{1}, H^{1}))

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_{H^{1}} ∈ H^{0}(X′, Hom_{O_{X′}} (H^{1}, H^{1}))

is a splitting cohomology class φ. Obviously, the both constructions are inverse to each other.

□

10.6. Remark. In (10.2) one could have replaced in the definition of a splitting the Čech complex by any complex K^{•} bounded below and quasi-isomorphic to τ_{≤1}F_{∗}Ω_{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 σ and θ define a map from

IH^{0}(X′, Hom_{O_{X′}} (H^{1}, O_{X′}) ⊗ K^{•})

to

IH^{1}(X′, Hom_{O_{X′}} (H^{1}, O_{X′}) ⊗ τ_{≤1}F_{∗}Ω_{X/S}^{1}(log D)).

This is of course the case when K^{•} is a complex of 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.

10.7. Main theorem. Let X be a smooth scheme over S and D ⊂ X be a S-normal crossing divisor. Then

a) A lifting D̃′ ⊂ X̃′ of D′ ⊂ X′ to S̃ defines a splitting cohomology class

φ = φ_{(_{X̃}′,_{D̃}′)}.

b) Every splitting cohomology class φ is of the shape φ = φ_{(_{X̃}′,_{D̃}′)} for some

lifting D̃′ ⊂ X̃′ of D′ ⊂ X′ to S̃.

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 Čech-cohomology.

Proof: a) Let U = {X_{α}} be an affine cover of X, such that the X_{αβ} are affine, and such that (8.17) and (9.10) give liftings

D̃_{α} ⊂ X̃_{α} of D_{α} ⊂ X_{α}

to S̃ and liftings

F̃_{α} : X̃_{α} ⟶ X̃′ of F : X ⟶ X′

satisfying

F̃_{α}^{∗}O_{_{X̃}_{′}}(−D̃′) = O_{_{X̃}_{α}}(−p · D̃_{α}).

For X̃_{αβ} = X̃_{α}|_{X_{αβ}} (8.23) implies the existence of isomorphisms of liftings

u_{αβ} : X̃_{αβ} ⟶ X̃_{βα}

As in the proof of (9.12) one uses (9.9) to define

φ_{αβ} = p^{−1} ∘ (F̃_{α}^{∗} − (F̃_{β} ∘ u_{αβ})^{∗}) ∈ Γ(X′_{αβ}, Hom_{O_{X′}} (Ω_{X′/S}^{1}(log D′), F_{∗}O_{X}))

and by (9.11) φ_{αβ} is a cocycle. On X̃_{α} we obtained in (10.3) a map

ψ_{α} = p^{−1} ∘ F̃_{α}^{∗} ∈ Γ(X′_{α}, Hom_{O_{X′}} (Ω_{X′/S}^{1}(log D′), Z^{1}))

lifting the Cartier operator C^{−1}.

□

10.8. Claim. One has

p^{−1} ∘ F̃_{β}^{∗}|_{X′_{βα}} = p^{−1} ∘ (F̃_{β} ∘ u_{αβ})^{∗}

in

Γ(X′_{βα}, Hom_{O_{X′}} (Ω_{X′/S}^{1}(log D′), Z′)).

Proof: For x̃′ ∈ O_{_{X̃}_{′}} and x̃_{β} ∈ O_{_{X̃}_{β}} we can write

F̃_{β}^{∗}(x̃′) = x̃_{β}^{p} + p · u(x̃_{β}, x̃′).

Since u_{αβ}^{∗}|_{p·O_{X_{αβ}}} is the identity, one obtains

u_{αβ}^{∗}F̃_{β}^{∗}(x̃′) = u_{αβ}^{∗}(x̃_{β})^{p} + p · u(x̃_{β}, x̃′).

By (10.3) we have

p^{−1} ∘ F̃_{β}^{∗}(dx̃′) = x^{p−1}dx + du(x̃_{β}, x̃′) = p^{−1} ∘ (F̃_{β} ∘ u_{αβ})^{∗}.
□

Now (10.8) is just saying that δψ = dφ and therefore (φ_{αβ}, ψ_{β}) defines a cohomology class φ_{(_{X̃}′,_{D̃}′)} in

IH^{1}(X′, Hom_{O_{X′}} (Ω_{X′/S}^{1}(log D′), F_{∗}O_{X}) ⟶ Hom_{O_{X′}} (Ω_{X′/S}^{1}(log D′), Z^{1})).

By construction its image in

H^{0}(X′, Hom_{O_{X′}} (Ω_{X′/S}^{1}(log D′), H^{1}))

is given by (ψ_{β}) and hence it is the Cartier operator.

b) Conversely, let (φ_{αβ}, ψ_{α}) be the cocycle giving the splitting cohomology class φ for some affine covering U′ = {X′_{α}}. First we want to add some coboundary to get a new representative of φ.

By (8.17), (9.10) and (8.21) we can assume that we have:

i) Liftings to S̃ :

D̃_{α} ⊂ X̃_{α} of D_{α} = D|_{X_{α}} ⊂ X_{α} = X|_{X′_{α}},
D̃′_{α} ⊂ X̃′_{α} of D′_{α} = D′|_{X′_{α}} ⊂ X′_{α}

and

F̃_{α} : X̃_{α} ⟶ X̃′_{α} of F_{α} = F |_{X_{α}} : X_{α} ⟶ X′_{α}

with

F̃_{α}^{∗} : O_{_{X̃}_{′}_{α}}(−D̃′_{α}) = O_{_{X̃}_{α}}(−p · D̃_{α}).

ii) Isomorphisms of liftings:

u′_{αβ} : X̃′_{αβ} ⟶ X̃′_{βα} and u_{αβ} : X̃_{αβ} ⟶ X̃_{βα}

were we keep the notation X̃′_{αβ} = X̃′_{α}|_{X_{αβ}}.

For x̃′ ∈ O_{_{X̃}{′ \atop αβ}} we can write u_{αβ}^{′∗}(x̃′) = x̃′ + p · λ_{αβ}(x̃′). Then

F̃_{α}^{∗}u_{αβ}^{′∗}(x̃′) = F̃_{α}^{∗}(x̃′) + p · F^{∗}λ_{αβ}(x̃′) = F̃_{α}^{∗}(x̃′) + p · λ_{αβ}(x̃′)^{p}.

Since d(λ_{αβ}(x̃′)^{p}) = 0 the explicit description of p^{−1} ∘ F̃_{α}^{∗} in (10.3) gives

10.9. Claim. One has p^{−1} ∘ F̃_{α}^{∗}|_{X′_{αβ}} = p^{−1} ∘ (u′_{αβ} ∘ F̃_{α})^{∗} in

Γ(X′_{αβ}, Hom_{O_{X′}} (Ω_{X′/S}^{1}(log D′), Z^{1})).

Define

θ_{α} = p^{−1} ∘ F̃_{α}^{∗} ∈ Γ(X′_{α}, Hom_{O_{X′}} (Ω_{X′/S}^{1}(log D′), F_{∗}O_{X})).

Replacing U′ by some finer cover if necessary we find

f_{α} ∈ Γ(X′_{α}, Hom_{O_{X′}} (Ω_{X′/S}^{1}(log D′), F_{∗}O_{X}))

such that df_{α} = θ_{α} − ψ_{α}.

10.10. Claim. For σ′_{αβ} = φ_{αβ} + δf_{α} the cohomology class φ is represented by the cocycle (σ′_{αβ}, θ_{α}).

Proof: This is obvious since

(σ′_{αβ}, θ_{α}) = (φ_{αβ} + δf_{α}, ψ_{α} + df_{α}).
□

The main advantage of σ′_{αβ} is that it comes in geometric terms: Write, using (9.9)

σ_{αβ} = p^{−1} ∘ ((u′_{αβ} ∘ F̃_{α})^{∗} − (F̃_{β} ∘ u_{αβ})^{∗})

in

Γ(X′_{αβ}, Hom_{O_{X′}} (Ω_{X′/S}^{1}(log D′) , F_{∗}O_{X})).

Applying (10.9) to the first and (10.8) to the second summand one gets

dσ_{αβ} = θ_{α} − θ_{β} = df_{α} − df_{β} + ψ_{α} − ψ_{β} = d(f_{α} − f_{β}) + dφ_{αβ}.

§ 10 The proof of Deligne and Illusie [12] 113

Hence g_{αβ} = σ_{αβ} − φ_{αβ} − (f_{α} − f_{β}) is closed and lives in

Γ(X′_{αβ}, Hom_{O_{X′}} (Ω_{X′/S}^{1}(log D′), O_{X′})).

By (8.22) u_{αβ}^{′∗} − g_{αβ} defines a new isomorphism of liftings

v′_{αβ} : X̃′_{αβ} ⟶ X̃′_{βα}.

As F̃_{α}^{∗} = pF^{∗} on p · O_{X′_{α}}, one has p^{−1} ∘ F̃_{α}^{∗} ∘ g_{αβ} = g_{αβ} and

\begin{aligned}σ′_{αβ} = & φ_{αβ} + (f_{α} − f_{β}) = σ_{αβ} − g_{αβ} = \\ & p^{−1} ∘ [(u′_{αβ} ∘ F̃_{α})^{∗} − (F̃_{β} ∘ u_{αβ})^{∗} − F̃_{α}^{∗} · g_{αβ}] . \end{aligned}

One obtains 10.11. Claim. σ′_{αβ} = p^{−1} ∘ [(v′_{αβ} ∘ F̃_{α})^{∗} − (F̃_{β} ∘ u_{αβ})^{∗}]. The proof of (10.7) ends with 10.12. Claim. The cocycle condition for σ′_{αβ} allows the glueing of X̃′_{α} to X̃′ using v′_{αβ}. Proof: One has to show that

v′_{αγ} = v′_{βγ} ∘ v′_{αβ}

or, by (8.22), that the homomorphism defined there verifies

v_{αβ}^{′∗} ∘ v_{βγ}^{′∗} − v_{αγ}^{′∗} = 0.

Since F̃_{α}^{∗} is injective, it is enough to show that

p^{−1} ∘ F̃_{α}^{∗} ∘ [v_{αβ}^{′∗} ∘ v_{βγ}^{′∗} − v_{αγ}^{′∗}] = 0

as homomorphism in

Hom_{O_{X′}} (Ω_{X′/S}^{1}(log D′), F_{∗}O_{X}).

The cocycle condition for σ′ is p^{−1} ∘ [F̃_{α}^{∗} ∘ v_{αβ}^{′∗} − u_{αβ}^{∗} ∘ F̃_{β}^{∗} − F̃_{α}^{∗} ∘ v_{αγ}^{′∗} + u_{αγ}^{∗} ∘ F̃_{γ}^{∗} + F̃_{β}^{∗} ∘ v_{βγ}^{′∗} − u_{βγ}^{∗} ∘ F̃_{γ}^{∗}] = 0. Since

F̃_{α}^{∗} ∘ v_{αβ}^{′∗} − u_{αβ}^{∗} ∘ F̃_{β}^{∗}

is a homomorphism from O_{X′} to p · O_{X}, we can replace it by [F̃_{α}^{∗} ∘ v_{αβ}^{′∗} − u_{αβ}^{∗} ∘ F̃_{β}^{∗}] ∘ v_{βγ}^{∗} as v_{βγ}^{∗} is the identity on O_{X′_{βγ}}. Similarly, we can add some u_{αβ}^{∗} at the right hand side (see (9.11)) and get

\begin{matrix}0 = p^{−1} ∘ [F̃_{α}^{∗} ∘ v_{αβ}^{′∗} ∘ v_{βγ}^{′∗} − u_{αβ}^{∗} ∘ F̃_{β}^{∗} ∘ v_{βγ}^{′∗} − F̃_{α}^{∗} ∘ v_{αγ}^{′∗}+ \\ +u_{αγ}^{∗} ∘ F̃_{γ}^{∗} − u_{αβ}^{∗} ∘ F̃_{β}^{∗} ∘ v_{βγ}^{′∗} − u_{αβ}^{∗}u_{βγ}^{∗} ∘ F̃_{γ}^{∗}] \end{matrix}

where all the summands are morphisms from O_{_{X̃}_{′}_{γ}} ⟶ O_{_{X̃}_{α}}. This is the same as

0 = p^{−1} ∘ F̃_{α}^{∗} ∘ (v_{αβ}^{′∗} ∘ v_{βγ}^{′∗} − v_{αγ}^{′∗}) + p^{−1} ∘ (u_{αγ}^{∗} − u_{αβ}^{∗}u_{βγ}^{∗})F̃_{γ}^{∗}.

By (9.11) the term on the right is zero and

0 = p^{−1} ∘ F̃_{α}^{∗} ∘ (v_{αβ}^{′∗} ∘ v_{βγ}^{′∗} − v_{αγ}^{′∗}).
□

10.13. Splittings of the de Rham complex.

Let us generalize (10.2) to τ_{≤i}F_{∗}Ω_{X/S}^{•}(log D) for i > 1. As remarked in (10.6) one can, using the derived category, replace the complex K^{•} in the following definition by any complex K^{•} bounded below and quasi-isomorphic to τ_{≤i}F_{∗}Ω_{X/S}^{•}(log D).

10.14. Definition. A splitting of τ_{≤i}F_{∗}Ω_{X/S}^{•}(log D) is a diagram

τ_{≤i}F_{∗}Ω_{X/S}^{•}(log D) ⟶^{σ}  K^{•}
↑^{θ}
⨁_{j≤i} H^{j}[−j]

where K^{•} is the Čech complex

C^{•}(U, τ_{≤i}F_{∗}Ω_{X/S}^{•}(log D))

associated to some affine cover U of X (and hence σ a quasi-isomorphism) and where θ is a quasi-isormorphism. Here again,

⨁_{j≤i} H^{j}[−j]

is the complex with zero differential and with H^{j} in degree j and τ_{≤i} is the filtration explained in (A.26).

10.15. Example. Let us return to the assumptions made in (10.3), i.e. that the liftings D̃, X̃, D̃′, X̃′ and especially F̃ exist. We had defined there a morphism

ψ = p^{−1} ∘ F̃^{∗} : Ω_{X′/S}^{1}(log D′) ⟶ Z^{1}

which was a lifting of C^{−1}.

We define

ψ^{j}(ω_{1} ∧ … ∧ ω_{j}) = ψ(ω_{1}) ∧ ψ(ω_{2}) ∧ … ∧ ψ(ω_{j})

where ω_{l} ∈ Ω_{X′/S}^{1}(log D′). Since ψ(ω_{l}) is closed, the image of ψ^{j} lies in Z^{j}

and, since the Cartier operator was defined as ⋀^{j} C^{−1} the map ψ^{j} induces the Cartier operator on in

Hom_{O_{X′}} (Ω_{X′/S}^{j}(log D′), H^{j}).

10.16. Theorem. Let X be a smooth S-scheme, D ⊂ X be a normal crossing divisor over S and let

D̃′ ⊂ X̃′ be a lifting of D′ ⊂ X′

to S̃. Then the splitting cohomology class φ_{(_{X̃}′,_{D̃}′)} of (10.7,a) induces a splitting of

τ_{≤i}F_{∗}Ω_{X/S}^{•}(log D) for i < p = char (S).

In particular, if p > dim_{S} X, it induces a splitting of the whole de Rham complex

F_{∗}Ω_{X/S}^{•}(log D).

Proof: Let (φ_{αβ}, ψ_{α}) be a Čech cocycle for (φ_{_{X̃}′,_{D̃}′}) where we regard (φ_{αβ}, ψ_{α}) as an O_{X′}-homomorphism:

(φ, ψ) : Ω_{X′/S}^{1}(log D′) ⟶ C^{1}(F_{∗}O_{X}) ⊕ C^{0}(Z^{1}).

We define

(φ, ψ)^{⊗j}(ω_{1}, ⊗ ⋯ ⊗ ω_{j}) ∈ C^{j}(τ_{≤i}F_{∗}Ω_{X/S}^{•}(log D))

for all 0 < j ≤ i and all

ω_{1} ⊗ ⋯ ⊗ ω_{j} ∈ ⨂_{1}^{j} Ω_{X′/S}^{1}(log D′)

by the following inductive formula: For any cocycle

\underline{b} := (b_{j}, …, b_{0}), b_{l} ∈ C^{l}(F_{∗}Ω_{X/S}^{j−l}(log D)),

with

db_{j−s} + (−1)^{j−s}δb_{j−s−1} = 0 for all 0 ≤ s ≤ j,

we define

\underline{b} ⊗ (φ, ψ)(ω_{j+1}) = (a_{j+1}, …, a_{0}) =: \underline{a},

116 H. Esnault, E. Viehweg: Lectures on Vanishing Theorems

with

a_{l} ∈ C^{l}(F_{∗}Ω_{X/s}^{j+1−l}(log D)),

by the rule

a_{j+1−s} := (−1)^{s}b_{j−s} ∪ φ + b_{j+1−s} ∪ ψ

where

(b_{j−s} ∪ φ)_{α_{0}...α_{j+1−s}} := (b_{j−s})_{α_{0}...α_{j−s}} · φ_{α_{j−s},α_{j+1−s}}(ω_{j+1})

and

(b_{j+1−s} ∪ ψ)_{α_{0}...α_{j+1−s}} := (b_{j+1−s})_{α_{0}...α_{j+1−s}} · ψ_{α_{j+1−s}}(ω_{j+1}).

One has

da_{j+1−s} + (−1)^{j+1−s}δa_{j−s} = 0

and therefore \underline{a} is a Čech cocycle. We have, for j ≤ i, a diagram

(Ω_{X′/S}^{1}(log D′))^{⊗ j} ⟶^{(φ,ψ)^{⊗ j}} C^{j}(τ_{≤i}F_{∗}Ω_{X/S}^{1}(log D))closed
↓_{π^{j}} ↓
Ω_{X′/S}^{j}(log D′) \underrightarrow{⋀^{j} C^{−1}}  H^{j}

and any section δ^{j} of π^{j} allows to define

θ_{j} = (⋀^{j} C^{−1})^{−1} ∘ δ^{j} ∘ (φ, ψ)^{⊗j}.

The splitting

θ : ⨁_{j≤i} H^{j}[−j] ⟶ K^{•} is θ = ⨁_{j≤i} θ_{j}[−j].

Such sections δ^{j} exist for j ≤ i < char (S):

δ^{j}(ω_{1} ⊗ … ⊗ ω_{j}) = \frac{1}{j!} ∑_{s∈Σ_{j}} sign (s) · ω_{s(1)} ∧ … ∧ ω_{s(j)},

where Σ_{j} denotes the symmetric group.

□

10.17. Corollary. Let D = A + B in (10.16). Then for i < p the splitting cohomology class φ_{(_{X̃},_{D̃})} induces a splitting of

τ_{≤i}F_{∗}Ω_{X/S}^{•}(A, B)

§ 10 The proof of Deligne and Illusie [12] 117

as well, i.e. a quasi-isomorphism

θ : ⨁_{j≤i} Ω_{X′/S}^{j}(A′, B′)[−j] ⟶ K^{•}(A, B)

where K^{•}(A, B) is the Čech complex of

τ_{≤i}F_{∗}Ω_{X/S}^{•}(A, B).

Proof: As in (9.17) one obtains from (2.7) and (2.10) a quasi-isomorphism

F_{∗}Ω_{X/S}^{1}(log (A + B)) ⊗_{O_{X′}} O_{X′}(−A′)

∥
F_{∗}(Ω_{X/S}^{•}(log (A + B))(−p · A))
↓
F_{∗}Ω_{X/S}^{•}(A, B) .

For K^{•}, the Čech complex of τ_{≤i}F_{∗}Ω_{X/S}^{•}(log D), we have a quasi-isomorphism

K^{•} ⊗_{O_{X′}} O_{X′}(−A) ⟶ K^{•}(A, B)

and the existence of θ follows from (10.16).

□

10.18. Remark. In (10.3) and (10.15) we have seen that if both D′ ⊂ X′ and D ⊂ X lift to D̃′ ⊂ X̃′ and D̃ ⊂ X̃, and if F lifts to: F̃ : X̃ → X̃′ with

F̃^{∗}O_{_{X̃}_{′}}(−D̃′) = O_{X̃}(−p · D̃),

then

ψ = p^{−1} ∘ F̃^{∗} : Ω_{X′/S}^{1}(log D′) ⟶ Z^{1} ⊂ Ω_{X/S}^{1}(log D)

lifts the Cartier operator, and gives an especially nice splitting of

τ_{≤1}F_{∗}Ω_{X/S}^{•}(log D).

This defines

⋀^{l} ψ : Ω_{X′/S}^{l}(log D′) ⟶ Z^{l} ⊂ Ω_{X/S}^{l}(log D)

lifting the Cartier operator, and therefore one obtains a quasi-isomorphism:

⨁_{j} Ω_{X′/S}^{j}(log D′)[−j] \underrightarrow{⋀^{•} ψ} F_{∗}Ω_{X/S}^{•}(log D),

which gives via (10.17) a quasi-isomorphism

⨁_{j} Ω_{X′/S}^{j}(A′, B′)[−j] \underrightarrow{⋀^{•} ψ} F_{∗}Ω_{X/S}^{•}(A, B)

if D = A + B. In particular, there is here no restriction on dim_{S} X in this case.

In general one has

10.19. Proposition. Let X, A and B be as in (10.17). Then the splitting cohomology class φ_{(_{X̃}′,_{D̃}′)} induces a splitting of

F_{∗}Ω_{X/S}^{•}(A, B)

when dim_{S} X ≤ p and S is affine.

Proof: Of course this is nothing but (10.17) if dim_{S} X < p. For dim_{S} X ≥ p, we observe first that whenever j : U → X is the embedding of an open set such that (X − U) is a divisor, then for coherent sheaves F on U and G on X one has

Hom_{O_{X}} (j_{∗}F, G)_{x} = \{ \begin{matrix}Hom_{O_{X}} (F, G|_{U})_{x} & for x ∈ U \\ 0 & for x ∈ (X − U), \end{matrix}

that is

Hom_{O_{X}} (j_{∗}F, G) = j_{!}Hom_{O_{X}} (F, G|_{U}).

Let us consider the O_{X′}-maps defined in (10.17) for 0 ≤ l ≤ p − 1:

Ω_{X′/S}^{l}(B′, A′) ⟶ C^{l}(F_{∗}O_{X}) + ⋯ + C^{0}(Z^{l})
(φ_{α_{0}...α_{l}}, …, φ_{α_{0}})

with the cocycle condition

dφ_{α_{0}...α_{k}} + (−1)^{l}δφ_{α_{0}...α_{k−1}} = 0.

The composite map

Ω_{X′/S}^{l}(B′, A′) ⟶ C^{0}(Z^{l}) ⟶ C^{0}(H^{l})

is just ⋀^{l} C^{−1}. Applying for 1 ≤ k ≤ l ≤ p − 1 the functor Hom_{O_{X′}} (−, Ω_{X′/S}^{n}), where n = dim_{S} X, one obtains O_{X′}-linear maps (see (9.19) and (9.20))

(j_{α_{0}...α_{k}})_{!}F_{∗}Ω_{X/S}^{n−(l−k)}(A, B) ⟶^{φ_{α_{0}...α_{k}}^{∨}} Ω_{X′/S}^{n−l}(A′, B′)

and therefore O_{X′}-maps

F_{∗}Ω_{X/S}^{n−(l−k)}(A, B) ⟶^{φ_{α_{0}...α_{k}}^{∨}} C^{k}(Ω_{X′/S}^{n−l}(A′, B′)).

For k = 0, one has an exact sequence

0 ⟶ Z^{l} ⟶ F_{∗}Ω_{X/S}^{l}(B, A) ⟶ F_{∗}Ω_{X/S}^{l+1}(B, A),

and applying (9.20) again, one obtains that

Hom_{O_{X′}} (Z^{l}, Ω_{X′/S}^{n}) = \frac{F_{∗}Ω_{X/S}^{n−l}(A, B)}{dF_{∗}Ω_{X/S}^{n−l−1}(A, B)}

which gives similarly a O_{X′}-linear map

\frac{F_{∗}Ω_{X/S}^{n−l}(A, B)}{dF_{∗}Ω_{X/S}^{n−l−1}(A, B)} ⟶^{φ_{α_{0}}^{∨}} C^{0}(Ω_{X′/S}^{n−l}(A′, B′)).

The cocycle condition tells us that

φ_{α_{0}...α_{k}}^{∨} ∘ d + (−1)^{l}δφ_{α_{0}...α_{k−1}}^{∨} = 0.

This means that φ^{∨} defines a map of complexes

τ_{≥n−l}F_{∗}Ω_{X/S}^{•}(A, B)[(n − l)] ⟶ τ_{≤l}C^{•}(Ω_{X′/S}^{n−l}(A′, B′)),

where

τ_{≥n−l}F_{∗}Ω_{X/S}^{•}(A, B)[(n − l)] :=

\frac{F_{∗}Ω_{X/S}^{n−l}(A, B)}{dF_{∗}Ω_{X/S}^{n−l}(A, B)} ⟶ F_{∗}Ω_{X/S}^{n−l+1}(A, B) ⟶ ⋯ ⟶ F_{∗}Ω_{X/S}^{n}(A, B).

The composite map

H^{n−l} ⟶ τ_{≥n−l}F_{∗}Ω_{X/S}^{•}(A, B)[(n − l)] ⟶ C^{•}(Ω_{X′/S}^{n−l}(A′, B′))

is given by ⋀^{n−l} C. As τ_{≥n−l} maps to τ_{≥n−l+1} , we find in this way a O_{X′}-map

τ_{≥n−p+1}F_{∗}Ω_{X/S}^{•}(A, B) ⟶^{φ^{∨}} ⨁_{i=n−p+1}^{n} C^{•}(Ω_{X′/S}^{i}(A′, B′))[−i]

which is a quasi-isomorphism. In particular, for any open set U′ ⊂ X′ and any O_{X′}-sheaf F′ one has

IH^{l}(U′, τ_{≥n−p+1}F_{∗}Ω_{X/S}^{•}(A, B) ⊗ F′) = ⨁_{i=n−p+1}^{n} H^{l−i}(U′, Ω_{X′/S}^{i}(A′, B′) ⊗ F′).

Consider now n = p as needed to finish the proof of (10.19). From the exact sequence

0 ⟶ H^{0} ⟶ F_{∗}Ω_{X/S}^{•}(A, B) ⟶ τ_{≥1}F_{∗}Ω_{X/S}^{•}(A, B) ⟶ 0

we obtain an exact sequence of hypercohomology groups

IH^{p}(F_{∗}Ω_{X/S}^{•}(A, B) ⊗ H^{p^{∨}}) ⟶ ⨁_{a=1}^{p} H^{p−a}(Ω_{X′/S}^{a}(A′, B′) ⊗ H^{p^{∨}}) ⟶

⟶ H^{p+1}(H^{0} ⊗ H^{p^{∨}})

where H^{p^{∨}} := Hom_{O_{X′}} (H^{p}, O_{X′}).

As dim_{S} X = p and S is affine, one has H^{p+1}(H^{0} ⊗ H^{p^{∨}}) = 0, and therefore

⋀^{p} C ∈ H^{0}(Ω_{X′/S}^{p}(A′, B′) ⊗ H^{p^{∨}})

lifts to some

Γ ∈ IH^{p}(F_{∗}Ω_{X/S}^{•}(A, B) ⊗ H^{p^{∨}}).

Representing Γ by a Čech cocycle

H^{p} ⟶^{[Γ]} C^{p}(F_{∗}O_{X}) + ⋯ + C^{0}(F_{∗}Ω_{X/S}^{p}(A, B)),

and taking a common refinement U of the Čech covers defining φ and Γ, one obtains altogether a quasi-isomorphism

(φ, [Γ] ∘ ⋀^{p} C^{−1}) : ⨁_{j} Ω_{X′/S}^{j}(A′, B′)[−j] ⟶ C^{•}(F_{∗}Ω_{X/S}^{•}(A, B)).

□

10.20. Remark. If dim_{S} X = n > p, and S is affine, one considers the exact sequence

0 ⟶ τ_{≤n−p}F_{∗}Ω_{X/S}^{•}(A, B) ⟶ F_{∗}Ω_{X/S}^{•}(A, B) ⟶ τ_{≥n−p+1}Ω_{X/S}^{•}(A, B) ⟶ 0

giving the short exact sequences

IH^{q}(F_{∗}Ω_{X/S}^{•}(A, B) ⊗ H^{q^{∨}}) ⟶ ⨁_{a=n−p+1}^{n} H^{q−a}(Ω_{X′/S}^{a}(A′, B′) ⊗ H^{q^{∨}}) ⟶

⟶ H^{q+1}(τ_{≤n−p}F_{∗}Ω_{X/S}^{•}(A, B) ⊗ H^{q^{∨}})

for all p ≤ q ≤ n. If n − p ≤ p − 1, then

H^{q+1}(τ_{≤n−p}F_{∗}Ω_{X/S}^{•}(A, B) ⊗ H^{q^{∨}}) = ⨁_{a=0}^{n−p} H^{q+1−a}(Ω_{X′/S}^{a}(A′, B′) ⊗ H^{q^{∨}}).

If those groups are vanishing, then the same argument as above shows that one obtains a splitting of F_{∗}Ω_{X/S}^{•}(A, B).

For example, take D = ∅. For n = p + 1, one requires the vanishing of

H^{p+1}(H^{p^{∨}}), H^{p}(Ω_{X′/S}^{1} ⊗ H^{p^{∨}}) and of H^{p+1}(Ω_{X′/S}^{1} ⊗ H^{p+1^{∨}}),

that is, via duality, the vanishing of

H^{0}(Ω_{X′/S}^{p} ⊗ Ω_{X′/S}^{p+1}) and H^{1}(Ω_{X′/S}^{p⊗2}).

Using (10.19) it is now quite easy to prove theorem (8.3) and some generalizations.

10.21. Theorem. Let f : X → S be a smooth proper S-scheme, dim_{S} X ≤ p, and let D ⊂ X be a S-normal crossing divisor. Assume that there exists a lifting

D̃′ ⊂ X̃′ of D′ ⊂ X′

to S̃. Let D = A + B. Then one has:

a) The O_{S}-sheaves

E_{1}^{ab} = R^{b}f_{∗}Ω_{X/S}^{a}(A, B)

are locally free and compatible with arbitrary base change.

b) The Hodge to de Rham spectral sequence

E_{1}^{ab} ⟹ IR^{a+b}f_{∗}Ω_{X/S}^{•}(A, B)

degenerates in E_{1}.

Proof: Assuming a), part b) follows if one knows that IR^{l}f_{∗}Ω_{X/S}^{•}(A, B) is a locally free O_{S}-module of rank

∑_{a+b=l} rank_{O_{S}}(E_{1}^{ab}).

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

IR^{l}f_{∗}Ω_{X/S}^{•}(A, B) = IR^{l}f′_{∗}(F_{∗}Ω_{X/S}^{•}(A, B)) = ⨁_{a} R^{l−a}f′_{∗}Ω_{X′/S}^{a}(A′, B′).

Hence, if a) holds true for f : X → S, then

IR^{l}f_{∗}Ω_{X/S}^{•}(A, B) = ⨁_{a} F_{S}^{∗}R^{l−a}f_{∗}Ω_{X/S}^{a}(A, B)

is locally free of the right rank and one obtains b).

By “cohomology and base change” ([50], II §5, for example), there exist for a = 0, …, l bounded complexes E_{a}^{•} of vector bundles on S, such that

H^{l}(E_{a}^{•}) = R^{l−a}f_{∗}Ω_{X/S}^{a}(A, B)

and, for any affine map φ : T ⟶ S,

H^{l}(φ^{∗}E_{a}^{•}) = R^{l−a}f_{T^{∗}}Ω_{X_{T} /T}^{a} (A_{T} , B_{T})

where X_{T} , A_{T} , B_{T} , X′_{T} , f_{T} : X_{T} → T and f′_{T} : X′_{T} → T are obtained by pullback from X, A, B, X′, f and f′ : X′ → S.

For example, R^{l−a}f′_{∗}Ω_{X′/S}^{a}(A′, B′) is given by H^{l}(F_{S}^{∗}E_{a}^{•}) and hence IR^{l}f_{∗}Ω_{X/S}^{•}(A, B) by the l-th homology of the complex

F_{S}^{∗}E^{•} for E^{•} = ⨁_{a} E_{a}^{•}.

To prove a), we have to show that for all l, the sheaf

H^{l}(E^{•}) = ⨁_{a} H^{l}(E_{a}^{•})

is locally free. If this is wrong, then we take l_{0} to be the maximal l with H^{l}(E^{•}) not locally free. Hence, if ∂_{•} denotes the differential in E^{•}, ker ∂_{l_{0}} is a vector bundle, let us say of rank r, but the image of

∂_{l_{0}−1} : E^{l_{0}−1} ⟶ ker ∂_{l_{0}}

is not a subbundle.

For some closed point s ∈ S one finds an infinitesimal neighbourhood Ŝ, for example one of the form

Ŝ = Spec(O_{S,s}/m_{S,s}^{μ}) for μ >> 0,

such that im(∂_{l_{0}−1}|Ŝ) is not a subbundle of ker(∂_{l_{0}}|Ŝ). Let us write Ŝ = SpecR where R is an Artin ring. For any R-module M let lg(M) denote the lenght.

∂_{l_{0}−1} is represented by a matrix Δ_{l_{0}−1}. For the point s ∈ Ŝ one has

h = rank_{k(s)}H^{l_{0}}(E^{•} ⊗ k(s)) > \frac{lg(H^{l_{0}}(E^{•} ⊗ R))}{lg(R)} .

Let n_{0} ⊂ R be the ideal generated by the (r − h + 1) minors of Δ_{l_{0}−1}. Then

H^{l_{0}}(E^{•} ⊗ R/n_{0})

is free of rank h as an R/n_{0}-module. If n′_{0} ⊂ n_{0} ⊂ R is another ideal with n′_{0} ≠ n_{0}, then

lg(H^{l_{0}}(E^{•} ⊗ R/n′_{0})) < h · 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

H^{l}(E^{•} ⊗ R/n)

are free of rank h(l) over R/n for all l, but for some l′ and for all ideals n′ ⊂ n with n′ ≠ n one has

lg(H^{l′}(E^{•} ⊗ R/n′)) < h(l′) · lg(R/n′) .

In particular, this holds true for the ideal n′ of R generated by F ∗ ; Ŝn. Let us write T′ = Spec(R/n′) and T = Spec(R/n). We have affine morphisms

T′ ⟶^{δ} T ⟶^{j} T′
↓  ↓  ↓
S ⟶^{F_{S}} S ⟶^{=} S .

H^{l}(E^{•}|_{T}) is free of rank h(l) over R/n for all l and hence

H^{l}(δ^{∗}j^{∗}E^{•}|_{T′}) = H^{l}((F_{S}^{∗}E^{•})|_{T′})

is free of rank h(l) over R/n′. The Hodge to de Rham spectral sequence implies that

lg(H^{l}(E^{•}|_{T′})) = ∑_{a} lg(R^{l−a}f_{T′∗}Ω_{X_{T′} /T′}^{a}(A_{T′}, B_{T′})) ≥

≥ lg(IR^{l}f_{T′∗}Ω_{X_{T′} /T′}^{•}(A_{T′}, B_{T′})).

On the other hand,

IR^{l}f_{T′∗}Ω_{X_{T′} /T′}^{•}(A_{T′}, B_{T′}) = IR^{l}f′_{∗}(F_{∗}Ω_{X/S}^{•}(A, B) ⊗ f^{′∗}(R/n′)) =

= ⨁_{a} R^{l−a}f′_{∗}(Ω_{X′/S}^{a}(A′, B′) ⊗ f^{′∗}(R/n′)).

We can apply base change and find the latter to be

⨁_{a} H^{l}(δ^{∗}E_{a}^{•}|_{T}) = H^{l}(δ^{∗}E^{•}|_{T}) .

Since the sheaves H^{l}(E^{•}|_{T}) are locally free, we have altogether

lg(H^{l}(E^{•}|_{T′})) ≥ lg(H^{l}(δ^{∗}E^{•}|_{T})) =
= lg(δ^{∗}H^{l}(E^{•}|_{T})) = h(l) · lg(R/n′) .

For l = l′ this contradicts the choice of n and n′. Hence H^{l}(E^{•}) is locally free. □ 10.22. Remark. As shown in [12], 4.1.2. it is enough to assume that for all a and b the dimension of H^{b}(X_{s}, Ω_{X_{s}}^{a}(A_{s}, B_{s})) is finite for closed points s ∈ S and that the conjugate spectral sequence

cE_{2}^{ij} = R^{i}f′_{T ∗}H^{j}(F_{∗}Ω_{X_{T} /T}^{•} (A_{T} , B_{T})) ⟹ IR^{i+j}f_{T ∗}Ω_{X_{T} /T}^{•} (A_{T} , B_{T})

satisfies

cE_{2}^{ij} = _{c}E_{∞}^{ij} for i + j = l

and for all T, in order to obtain (10.21).

10.23. 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_{1}^{ab} = H^{b}(X, Ω_{X}^{a}(A, B)) ⟹ IH^{a+b}(X, Ω_{X}^{•}(A, B))

degenerates in E_{1}.

Proof: By flat base change we may assume that K is of finite type over Q l . Hence we find a ring R, of finite type over ZZ, such that K is the quotient field over R. Let f : X → SpecR be a proper morphism with X = X ×_{R} K, and A and B divisors on X with A = A|_{X} and B = B|_{X}. Replacing SpecR by some open affine subscheme, we may assume that f is smooth, that D = A + B is a normal crossing divisor over SpecR and that

IR^{l}f_{∗}Ω_{X /SpecR}^{•}(A, B) and R^{b}f_{∗}Ω_{X /Spec R}^{a}(A, B)

are locally free for all l, a and b. Take a closed point s ∈ SpecR with

char k(s) = p > dim_{S} X.

Hence

X_{s} = X ×_{R} k(s) ⟶ S = k(s) , A_{s} = A|_{X_{s}} and B_{s} = B|_{X_{s}}

satisfy the assumptions made in (10.21) and

∑_{a+b=l} rank R^{b}f_{∗}Ω_{X /Spec R}^{a}(A, B) = rank IR^{l}f_{∗}Ω_{X /Spec R}^{•}(A, B).

□

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 ≠ 0 as well.

Proof of (3.2,b) and c) in characteristic p ≠ 0: Recall, that on the projective manifold X we considered the invertible sheaves

L^{(i)} = L^{(i,D)} = L^{i}(−[\frac{i · D}{N}]) where

D = ∑_{j=1}^{r} α_{i}D_{j}

is a normal crossing divisor and L an invertible sheaf with L^{N} = O_{X}(D). For N prime to char k, we constructed in §3, for i = 0, …, N − 1, integrable logarithmic connections

∇_{(i)} : L^{(i)^{−1}} ⟶ Ω_{X}^{1}(log D^{(i)}) ⊗ L^{(i)^{−1}}

with poles along

D^{(i)} = ∑_{{j=1 \atop \frac{i·α_{j}}{N} ∉ZZ}}^{r} D_{j}.

The residue of ∇_{(i)} along D_{j} ⊂ D^{(i)} is given by multiplication with

(i · α_{j} − N · [\frac{i · α_{j}}{N}]) · 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:

□

10.24. Claim. Let k be a perfect field, N prime to char k = p and assume that p ≥ 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, Ω_{X}^{a}(log (A + B + D^{(i)}))(−B) ⊗ L^{(i)^{−1}}) ⟹
IH^{a+b}(X, Ω_{X}^{•}(log (A + B + D^{(i)}))(−B) ⊗ L^{(i)^{−1}})

degenerates in E_{1}.

Proof: Let F : X → X′ be the relative Frobenius morphism and L′, D′, A′ and B′ be the sheaf and the divisors on X′ obtained by field extensions. Then

F^{∗}L^{′(i,D′)} = F^{∗}L^{′(i)} = L^{p·i}(−p · [\frac{i · D}{N}])

contains

L^{(p·i)} = L^{p·i}(−[\frac{p · i · D}{N}]).

Since p is prime to N one has D^{(i)} = D^{(p·i)}. The connection

∇_{(p·i)^{−1}} : O_{X}(−B) ⊗ L^{(p·i)^{−1}} ⟶ Ω_{X}^{1}(log (A + B + D^{(i)}))(−B) ⊗ L^{(p·i)^{−1}}

induces a connection with the same poles on

O_{X}(−B) ⊗ F^{∗}L^{′(i)^{−1}}

whose residues along D_{j} ⊂ D^{(i)} are given by multiplication with

N^{−1} · p · i · α_{j} − [\frac{p · i · α_{j}}{N}] + ([\frac{p · i · α_{j}}{N}] − p[\frac{i · α_{j}}{N}]) .

Obviously this number is zero modulo p. Since

[\frac{p · i · α_{j}}{N}] ≤ \frac{p · i · α_{j}}{N} < p · ([\frac{i · α_{j}}{N}] + 1) = p · [\frac{i · α_{j}}{N}] + p

(2.10) implies that the complexes

Ω_{X}^{•} (log (A + B + D^{(i)}))(−B) ⊗ F^{∗}L^{′(i)^{−1}}

and

Ω_{X}^{•}(log (A + B + D^{(i)}))(−B) ⊗ L^{(p·i)^{−1}}

are quasi-isomorphic.

10.25. Claim. The complex

F_{∗}(Ω_{X}^{•}(log (A + B + D^{(i)}))(−B) ⊗ F^{∗}L^{′(i)^{−1}})

is isomorphic to the complex

F_{∗}(Ω_{X}^{•}(log (A + B + D^{(i)}))(−B))

tensorized with L^{′(i)^{−1}}.

Proof: Let π : Y → X be the cyclic cover obtained by taking the N-th root out of D, let F : Y → Y′ be the relative Frobenius of Y. We have the induced diagram

Y ⟶^{F} Y′
π↓  ↓_{π′}
X ⟶^{F} X′

and, on X′ − Sing(D_{red}),

π′_{∗}O_{Y′} = Ker(d : π′_{∗}F_{∗}O_{Y} ⟶ π′_{∗}F_{∗}Ω_{Y}^{1} ) =

Ker(d : F_{∗}(⨁_{j=0}^{N−1} L^{(j)^{−1}}) ⟶ F_{∗}(⨁_{j=0}^{N−1} Ω_{X}^{1}(log D^{(j)}) ⊗ L^{(j)^{−1}})).

Since F^{∗} of the i-th eigenspace L^{′(i)} of π′_{∗}O_{Y′} lies in the p · i-th eigenspace L^{(i·p)} of π_{∗}O_{Y} one has

L^{′(i)^{−1}} = Ker(∇_{(p·i)} : F_{∗}L^{(p·i)^{−1}} ⟶ F_{∗}(Ω_{X}^{1}(log D^{(i)}) ⊗ L^{(p·i)^{−1}}))

= Ker(∇_{(p·i)} : (F_{∗}O_{X}) ⊗ L^{′(i)^{−1}} ⟶ (F_{∗}Ω_{X}^{1}(log D^{(i)})) ⊗ L^{′(i)^{−1}})

By the Leibniz rule ∇_{(p·i)} restricted to (F_{∗}O_{X}) ⊗ L^{′(i)^{−1}} is nothing but d ⊗ id as claimed.

□

From (10.19) and by base change

dim IH^{l}(X, Ω_{X}^{•}(log (A + B + D^{(i)}))(−B) ⊗ L^{(p·i)^{−1}}) =

dim IH^{l}(X′, F_{∗}(Ω_{X}^{•}(log (A + B + D^{(i)}))(−B) ⊗ L^{(p·i)^{−1}})) =

∑_{a+b=l} dim H^{b}(X′, ω_{X′}^{a}(log (A′ + B′ + D^{′(i)}))(−B′) ⊗ L′^{(i)^{−1}} =

∑_{a+b=l} dim H^{b}(X, ω_{X}^{a}(log (A + B + D^{(i)}))(−B) ⊗ L^{(i)^{−1}} ≥

dim IH^{l}(X, Ω_{X}^{•}(log (A + B + D^{(i)}))(−B) ⊗ L^{(i)^{−1}}).

For some ν > 0 one has p^{ν} ≡ 1 mod N. Repeating the argument ν − 1 times one finds

dim IH^{l}(X, Ω_{X}^{•}(log (A + B + D^{(i)}))(−B) ⊗ L^{(i)^{−1}}) ≥

dim IH^{l}(X, Ω_{X}^{•}(log (A + B + D^{(i)}))(−B) ⊗ L^{(p^{ν−1}·i)^{−1}}) ≥ ⋯

⋯ ≥ ∑_{a+b=l} dim H^{b}(X, Ω_{X}^{a}(log (A + B + D^{(i)}))(−B) ⊗ L^{(i)^{−1}}) ≥

dim IH^{l}(X, Ω_{X}^{•}(log (A + B + D^{(i)}))(−B) ⊗ L^{(i)^{−1}}) .

Hence all the inequalities must be equalities and one obtains (10.24).

□

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).

□

§ 11 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 [12]. 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.

11.1. 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 D̃ ⊂ X̃ to W_{2}(k). Let M be a locally free O_{X}-module. Then, for l < char(k) one has

∑_{a+b=l} dim H^{b}(X, Ω_{X}^{a}(log D) ⊗ M) ≤ ∑_{a+b=l} dim H^{b}(X, Ω_{X}^{a}(log D) ⊗ F_{X}^{∗}M).

Proof: By (10.16) we have

dim IH^{l}(X, Ω_{X}^{•}(log D) ⊗ F_{X}^{∗}M) = ∑_{a+b=l} dim H^{b}(X′, Ω_{X′}^{a}(log D′) ⊗ M′)

for the sheaf M′ = pr_{1}^{∗}M on X′ = X ×_{F_{S}} S and D′ = D ×_{F_{S}} S. By base change the right hand side is

∑_{a+b=l} dim H^{b}(X, Ω_{X}^{a}(log D) ⊗ M).

The Hodge to de Rham spectral sequence implies that the left hand side is smaller than or equal to

∑_{a+b=l} dim H^{b}(X, Ω_{X}^{a}(log D) ⊗ F_{X}^{∗}M).

□

11.2. Corollary. Under the assumptions of (11.1) assume that M is invertible. Then

∑_{a+b=l} dim H^{b}(X, Ω_{X}^{a}(log D) ⊗ M) ≤ ∑_{a+b=l} dim H^{b}(X, Ω_{X}^{a}(log D) ⊗ M^{p}).

§ 11 Vanishing theorems in characteristic p. 129

11.3. Corollary (Deligne, Illusie, Raynaud, see [12]). For a+b < Min {char(k), dim X} and L ample and invertible, one has under the assumptions of (11.1.)

H^{b}(X, Ω_{X}^{a}(log D) ⊗ L^{−1}) = 0.

Proof: For ν large enough, and L^{−1} = M one has

H^{b}(X, Ω_{X}^{a}(log D) ⊗ M^{p^{ν}}) = 0

for b < dim X. By (11.2) one has

H^{b}(X, Ω_{X}^{a}(log D) ⊗ M^{p^{ν−1}}) = 0

and after finitely many steps one obtains (11.3)

□

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).

11.4. Corollary. Let k be a perfect field, let X be a proper smooth k-scheme with dim X ≤ char k and let L be a numerically effective sheaf (see (5.5)). Assume that A is a very ample sheaf, such that X and A lift to X̃ and Ã over W_{2}(k), with Ã very ample over S̃. Then one has:

a) A^{dim X+1} ⊗ L ⊗ ω_{X} is generated by global sections.
b) A^{dim X+2} ⊗ L ⊗ ω_{X} is very ample.

Proof: By (11.3) and Serre duality H^{1}(X, A^{dim X} ⊗ L ⊗ ω_{X}) = 0. Hence one has a surjection

H^{0}(X, A^{dim X+1} ⊗ L ⊗ ω_{X}) ⟶ H^{0}(H, A^{dim X} ⊗ L ⊗ ω_{H})

where H is a smooth zero divisor of a general section of A. Since 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

A^{dim X} ⊗ L ⊗ ω_{H}

is generated by global sections and, moving H, we obtain a). Part b) follows directly from a) (see [30], II. Ex.7.5).

□

11.5. 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 L be an invertible sheaf on X. Assume that (X, D) and L admit liftings to W_{2}(k) and that one has:

(∗) For some ν_{0} ∈ IN and all ν ≥ 0 the sheaf L^{ν_{0}+ν} ⊗ O_{X}(−D) is ample.

Then, for a + b < dim X ≤ char k one has

H^{b}(X, Ω_{X}^{a}(log D) ⊗ L^{−1}) = 0.

Proof: By (5.7), the assumption (∗) implies that L is numerically effective. Let us choose μ_{0} such that L^{μ_{0}·ν_{0}}(−μ_{0} · D) is very ample and

L^{μ_{0}·ν_{0}}(−μ_{0} · D) ⊗ ω_{X}^{−1} and L^{μ_{0}·ν_{0}}(−μ_{0} · D + D_{red}) ⊗ ω_{X}^{−1}

are both ample. From (11.4) we find for n = dim X that both,

L^{μ_{0}·ν_{0}(n+3)+ν}(−μ_{0}(n + 3) · D) and L^{μ_{0}·ν_{0}(n+3)+ν}(−μ_{0}(n + 3) · D + D_{red})

are very ample for all ν ≥ 0. Hence the assumption (∗) in (11.5) can be replaced by

(∗∗) For some ν_{0} ∈ IN and all ν ≥ 0 the sheaves L^{ν_{0}+ν} ⊗ O_{X}(−D) and L^{ν_{0}+ν} ⊗ O_{X}(−D + D_{red}) are very ample.

Choose η ∈ IN − \{0\} such that N = p^{η} + 1 > ν_{0} and [\frac{D}{N}] = 0. Let H be the zero set of a general section of L^{N} ⊗ O_{X}(−D). In §3 we constructed an integrable logarithmic connection ∇_{(i)} on the sheaf

L^{(i)^{−1}} = L^{−i}([\frac{i · (D + H)}{N}]) .

Let F : X → X′ be the relative Frobenius morphism and L′, D′, H′ the sheaf and the divisors on X′, obtained by field extension via F_{Spec k} : k → k from 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_{∗}Ω_{X}^{•}(log (D + H))) ⊗ L^{′(i)^{−1}} ⟶ F_{∗}(Ω_{X}^{•}(log (D + H)) ⊗ L^{(p·i)^{−1}}).

This inclusion is a quasi-isomorphism and as in (10.24) one obtains from (10.19) and base change the inequalities

dim IH^{l}(X, Ω_{X}^{•}(log (D + H)) ⊗ L^{(1)^{−1}}) ≤

∑_{a+b=l} dim H^{b}(X, Ω_{X}^{a}(log (D + H)) ⊗ L^{(1)^{−1}}) =

∑_{a+b=l} dim H^{b}(X′, Ω_{X′}^{a}(log (D′ + H′)) ⊗ L′^{(1)^{−1}}) =

dim IH^{l}(X, Ω_{X}^{•}(log (D + H)) ⊗ L^{(p)^{−1}}) ≤ ⋯

dim IH^{l}(X, Ω_{X}^{•}(log (D + H)) ⊗ L^{(p^{γ})^{−1}}) ≤

∑_{a+b=l} H^{b}(X, Ω_{X}^{a}(log (D + H)) ⊗ L^{(p^{γ})^{−1}})

for all γ > 0. For γ = η we have p^{η} = N − 1 and

\begin{matrix}L^{(p^{η})} = L^{(N−1)} = L^{N−1}(−[\frac{(N−1)·(D+H)}{N}]) = \\ = L^{N−1}(−D − [\frac{−D}{N} + \frac{(N−1)·H}{N}]) = L^{N−1}(−D + D_{red}). \end{matrix}

Hence L^{(p^{η})} is ample and from (11.3) we obtain, for

l < dim X ≤ char k,

that

∑_{a+b=l} H^{b}(X, Ω_{X}^{a}(log (D + H)) ⊗ L^{(p^{η})^{−1}}) = 0.

Since L^{(1)} = L we obtain for a + b < dim X ≤ char k

H^{b}(X, Ω_{X}^{a}(log (D + H)) ⊗ L^{−1}) = 0.

Finally, since L^{N} ⊗ 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, Ω_{H}^{a−1}(log D)|_{H}) ⊗ L^{−1}) ⟶ H^{b}(X, Ω_{X}^{a}(log D) ⊗ L^{−1}) ⟶
⟶ H^{b}(X, Ω_{X}^{a}(log (D + H)) ⊗ L^{−1}).

allows to prove (11.5) by induction.

□

11.6. Remarks..

a) By (5.7) and (5.4,d) the assumption (∗) in (11.5) implies that L is numerically effective and of maximal Iitaka dimension.

b) If, on the other hand, L is numerically effective and of maximal Iitaka dimension, then there exists some effective divisor D such that the sheaf

L^{ν_{0}+ν} ⊗ O_{X}(−D)

is ample for some ν_{0} ∈ IN and all ν ≥ 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 11.7. Problem. Let k be a perfect field of characteristic p > 0, let X be a proper smooth k-variety and L an invertible sheaf. Assume that X and L admit liftings to W_{2}(k), that L is numerically effective and that κ(L) = dim X.

Does this imply that

H^{b}(X, L^{−1}) = 0 for b < dim X ≤ char k ?

11.8. 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, [12], Cor. 2.8, gives the vanishing of H^{b}(X, L^{−1}), for b < 2, without assuming that L lifts to W_{2}(k).

b) As mentioned in Lecture 1 and 8, even if we restrict ourselves to the case where L is semi-ample and of maximal Iitaka dimension, we do not know the answer to problem (11.7) for higherdimensional X.

§ 12 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 [26], deduce the deformation theory for cohomology groups.

12.1. Theorem (Mumford). Let g : Z ⟶ Y be a projective flat morphism of noetherian schemes, let Y_{0} ⊂ Y be an affine open subscheme,

Z_{0} = g^{−1}(Y_{0}) , g_{0} = g|_{Z_{0}}

and let B be a locally free sheaf on Z. Then there exists a bounded complex (E^{•}, δ_{•}) of locally free O_{Y_{0}} modules of finite rank such that

H^{b}(E^{•} ⊗ F) = R^{b}g_{0_{∗}}(B|_{Z_{0}} ⊗ g_{0}^{∗}F)

for all coherent sheaves F on Y_{0}.

In order to construct E^{•}, D. Mumford uses in [50], II, §5, the description of higher direct images by Čech complexes. The “ Coherence Theorem” of Grauert-Grothendieck allows to realize (E^{•}) as a complex of locally free sheaves of finite rank. The proof of (12.1) can be found as well in [30], III, §12.

§ 12 Deformation theory for cohomology groups 133

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.

12.2. Example. Let y ∈ Y_{0} be a point and F = k(y). For Z_{y} = g^{−1}(y) one obtains

H^{b}(E^{•} ⊗ k(y)) ⟶^{=} H^{b}(Z_{y}, B|_{Z_{y}})
τ↑  ^{η}↑
H^{b}(E^{•}) ⊗ k(y) ⟶^{=} R^{b}g_{∗}(B) ⊗ k(y),

where η is the base change morphism ([30], III, 9.3.1). In general, due to the fact that the images of

δ_{b−1} : E^{b−1} ⟶ E^{b} and δ_{b} : E^{b} ⟶ E^{b+1}

are not subbundles of E^{b} and E^{b+1}, τ and hence η will be neither injective nor surjective.

12.3. Example. Let X be a projective manifold, defined over an algebraically closed field k and let Y ⊂ Pic^{0}(X) be a closed subscheme,

Z = X × Y and g = pr_{2} : Z ⟶ Y.

Recall that on Z we have a Poincaré bundle P (see for example [50]), i.e. an invertible sheaf P such that

P|_{g^{−1}(y)} ≃ N_{y},

if N_{y} is the linebundle on X corresponding to

y ∈ Y ⊂ Pic^{0}(X).

In more fancy terms, the functor

T ⟼ Pic(X × T/T)

is represented by a locally noetherian group-scheme Pic(X), whose connected component containing zero is Pic^{0}(X). The invertible sheaf P is the restriction of the universal bundle on X × Pic(X) to

X × Y ⊂ X × Pic^{0}(X)

(see [28]). For y ∈ Y let T_{y,Y} = (m_{y,Y} /m_{y,Y}^{2} )^{∗} be the Zariski tangent space. We have an exact sequence

0 ⟶ T_{y,Y}^{∗} ⟶ O_{y,Y} /m_{y,Y}^{2} ⟶ k(y) ⟶ 0.

Since

g^{∗}(T_{y,Y}^{∗}) ⊗_{O_{Z}} P = g^{∗}(T_{y,Y}^{∗}) ⊗ N_{y}

one obtains the exact sequence

0 ⟶ g^{∗}(T_{y,Y}^{∗}) ⊗ N_{y} ⟶ g^{∗}(O_{y,Y} /m_{y,Y}^{2} ) ⊗ P ⟶ N_{y} ⟶ 0

on X ≃ g^{−1}(y). If, identifying X with g^{−1}(y),

ζ ∈ H^{1}(X, g^{∗}(T_{y,Y}^{1} )) = H^{1}(X, O_{X}) ⊗ T_{y,Y}^{∗}

is the extension class of this sequence, the induced edge morphism

H^{b}(X, N_{y}) ⟶ H^{b+1}(X, g^{∗}(T_{y,Y}^{∗}) ⊗ N_{y}) = H^{b+1}(X, N_{y}) ⊗_{k(y)} T_{y,Y}^{∗}

is the cup-product with ζ.

12.4.

Keeping the notations from (12.3), let M be a locally free sheaf on X and B = P ⊗ pr_{1}^{∗}M. Since the exact sequence

0 ⟶ g^{∗}(T_{y,Y}^{∗}) ⊗ N_{y} ⊗ M ⟶ g^{∗}(O_{y,Y} /m_{y,Y}^{2} ) ⊗ B ⟶ N_{y} ⊗ M ⟶ 0

is obtained from

0 ⟶ g^{∗}(T_{y,Y}^{∗}) ⊗ N_{y} ⟶ g^{∗}(O_{y,Y} /m_{y,Y}^{2} ) ⊗ P ⟶ N_{y} ⟶ 0

by tensorproduct with M, the induced edge morphism

H^{b}(X, M ⊗ N_{y}) ⟶ H^{b+1}(X, M ⊗ N_{y}) ⊗_{k(y)} T_{y,Y}^{∗}

is again the cup-product with ζ. Let us finally remark that ζ induces a morphism

T_{y,Y} ⟶^{ζ} H^{1}(X, O_{X})

which, due to the universal property of P is injective. In fact, if we represent τ ∈ T_{y,Y} by a morphism

ζ′ : D = Spec k[ϵ] ⟶ Y with ζ′(< ϵ >) = y ,

where k[ϵ] = k[t]/t^{2} is the ring of dual numbers, then for τ ≠ 0 the pullback of P to X × D is non trivial and hence the extension class ζ(τ) of

0 ⟶ N_{y} ⟶^{·ϵ} P|X × D ⟶ N_{y} ⟶ 0

is non zero. If Y = Pic^{0}(X) one has dim T_{y,Y} = dim H^{1}(X, O_{X}) and ζ is surjective as well.

12.5. Notations. For X, M as above let us write

S^{b}(X, M) = {y ∈ Pic^{0}(X); H^{b}(X, N_{y} ⊗ M) ≠ 0}.

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.

12.6. Lemma.

a) S^{b}(X, M) is a closed subvariety of Pic^{0}(X).

b) If Y ⊂ S^{b}(X, M) is an ireducible component and

m = Min{dim H^{b}(X, N_{y} ⊗ M); y ∈ Y },

then the set

U = {y ∈ Y; dim H^{b}(X, N_{y} ⊗ M) = m}

is open and dense in Y.

c) For y ∈ U and ζ : T_{y,Y} ↪ H^{1}(X, O_{X}) as in (12.4) the cup-products

ζ(T_{y,Y}) ⊗ H^{b−1}(X, N_{y} ⊗ M) ⟶ H^{b}(X, N_{y} ⊗ M)

and

ζ(T_{y,Y}) ⊗ H^{b}(X, N_{y} ⊗ M) ⟶ H^{b+1}(X, N_{y} ⊗ M)

are both zero.

Proof: For any open affine P_{0} ⊂ Pic^{0}(X) let E^{•} be the complex from (12.1) describing the higher direct images of

B|_{X×P_{o}} = pr_{1}^{∗}M ⊗ P|_{X×P_{o}}

and there base change. If we write

W_{b} = Coker(δ_{b−1} : E^{b−1} ⟶ E^{b}),

then for F coherent on P_{0} we have

W_{b} ⊗ F = Coker(δ_{b−1} : E^{b−1} ⊗ F ⟶ E^{b} ⊗ F)

and an exact sequence

0 ⟶ H^{b}(E^{•} ⊗ F) ⟶ W_{b} ⊗ F ⟶ E^{b+1} ⊗ F ⟶ W_{b+1} ⊗ F ⟶ 0.

If S^{b}(X, M) ≠ Pic^{0}(X), then S^{b}(X, M) is just the locus where W_{b} → E^{b+1} is not a subbundle. Obviously this condition defines a closed subscheme of P_{0}. For y ∈ Y ∩ P_{0} = Y_{0} we have

dim H^{b}(E^{•} ⊗ k(y)) = −rank(E^{b+1}) + dim(W_{b} ⊗ k(y)) + dim(W_{b+1} ⊗ k(y))

and the open set U in b) is nothing but the locus where both,

W_{b} ⊗ O_{Y_{0}} and W_{b+1} ⊗ O_{Y_{0}}

are locally free O_{Y_{0}} modules. On U the sequence

0 ⟶ H^{b}(E^{•}|_{U}) ⟶ W_{b}|_{U} ⟶ E^{b+1}|_{U} ⟶ W_{b+1}|_{U} ⟶ 0

is an exact sequence of vector bundles and

H^{b}(E^{•} ⊗ F) = H^{b}(E|_{U}) ⊗ F

for all coherent O_{U} modules F. In particular for y ∈ U the exact sequence

0 ⟶ E^{•} ⊗ T_{y,Y}^{∗} ⟶ E^{•} ⊗ O_{y,Y} /m_{y,Y}^{2} ⟶ E^{•} ⊗ k(y) ⟶ 0

induces

H^{b}(E^{•} ⊗ T_{y,Y}^{∗}) ⟶ H^{b}(E^{•} ⊗ O_{y,Y} /m_{y,Y}^{2} ) ⟶ H^{b}(E^{•} ⊗ k(y))
=↓  ^{=}↓  ^{=}↓
H^{b}(E^{•}) ⊗ T_{y,Y}^{∗} ⟶ H^{b}(E^{•}) ⊗ O_{y,Y} /m_{y,Y}^{2} ⟶ H^{b}(E^{•}) ⊗ k(y)

and the edge morphisms

H^{i}(E^{•} ⊗ k(y)) ⟶ H^{i+1}(E^{•} ⊗ 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, M ⊗ N_{y}) ⟶ H^{i+1}(X, M ⊗ N_{y}) ⊗_{k(y)} T_{y,Y}^{∗}

with the extension class ζ ∈ H^{1}(X, O_{X}) ⊗ T_{y,Y}^{∗} .

□

12.7. Corollary(Green, Lazarsfeld [26]). If Y ⊂ S^{b}(X, M) is an irreducible component and y ∈ Y is a point in general position then

codim_{Pic^{0}(X)}(Y) ≥ codim(Γ ⊂ H^{1}(X, O_{X}))

where

Γ = {φ ∈ H^{1}(X, O_{X}); α ∪ φ = 0 and β ∪ φ = 0 for all

α ∈ H^{b−1}(X, N_{y} ⊗ M) and β ∈ H^{b}(X, N_{y} ⊗ M)}.

12.8. Remark. Even if one seems to loose some information, in the applications of (12.7) in Lecture 13 we will replace Γ by the larger space

{φ ∈ H^{1}(X, O_{X}); β ∪ φ = 0 for all β ∈ H^{b}(X, N_{y} ⊗ M)}

in order to obtain lower bounds for codim_{Pic^{0}(X)}(S^{b}(X, M)) for certain invertible sheaves M.

§ 13 Generic vanishing theorems [26], [14] 137

§ 13 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, M))

for the subschemes S^{b}(X, M) introduced in §12. In particular, we lose a little bit the spirit of the previous lectures, where we tried to underline as much as possible the algebraic aspects of vanishing theorems. Everything contained in this lecture is either due Green-Lazarsfeld [26] or to H. Dunio [14]. 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 [56] whenever needed.

13.1. Notations. Let X be a projective complex manifold. The Picard group Pic(X) is H^{1}(X, O_{X}^{∗}) and, using the exponential sequence, Pic^{0}(X) is identified with

H^{1}(X, O_{X})/H^{1}(X, ZZ).

Let P be the Poincaré bundle on X × Pic^{0}(X), and g = pr_{2}. If

ζ : T_{y,Pic^{0}(X)} ⟶ H^{1}(X, O_{X})

is the extension class of

0 ⟶ g^{∗}T_{y,Pic^{0}(X)} ⊗ N_{y} ⟶ g^{∗}(O_{y,Pic^{0}}/m_{y,Pic^{0}}^{2}) ⊗ P ⟶ N_{y} ⟶ 0

then ζ is the identity. Let

Alb(X) = H^{0}(X, Ω_{X}^{1})^{∗}/H_{1}(X, ZZ)

be the Albanese variety of X and

α : X ⟶ Alb(X)

be the Albanese map. The morphism α induces an isomorphism

α^{∗} : H^{0}(Alb(X), Ω_{Alb(X)}^{1}) ⟶ H^{0}(X, Ω_{X}^{1}).

In particular,

dim α(X) = rank_{O_{X}} (im(H^{0}(X, Ω_{X}^{1}) ⊗_{C l} O_{X} ⟶ Ω_{X}^{1})) .

13.2. Theorem (Green-Lazarsfeld [26]). Let X be a complex projective manifold and

S^{b}(X) = {y ∈ Pic^{0}(X); H^{b}(X, N_{y}) ≠ 0}.

Then

codim_{Pic^{0}(X)}(S^{b}(X)) ≥ dim(α(X)) − b.

In particular, if N ∈ Pic^{0}(X) is a generic line bundle, then H^{b}(X, N) = 0 for b < dim α(X).

Proof: Using (12.7) or (12.8) one obtains (13.2) from

□

13.3. Claim. Assume that H^{b}(X, N_{y}) ≠ 0 then for

Γ = {φ ∈ H^{1}(X, O_{X}); β ∪ φ = 0 for all β ∈ H^{b}(X, N_{y})}

one has

codim(Γ ⊂ H^{1}(X, O_{X})) ≥ dim(α(X)) − b.

Proof: Step 1: 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 N_{y} gives a complex antilinear isomorphism, the so called Hodge duality,

ι : H^{b}(X, Ω_{X}^{a} ⊗ N_{y}) ⟶ H^{a}(X, Ω_{X}^{b} ⊗ N_{y}^{−1})

(see (13.5) and (13.6) for generalizations). Moreover, if φ ∈ H^{1}(X, O_{X}) and if ω = φ̄ ∈ H^{0}(X, Ω_{X}^{1}) is the Hodge-dual of φ, then for β ∈ H^{b}(X, Ω_{X}^{a} ⊗ N_{y}) one has

ι(β ∪ φ) = ι(β) ∧ ω ∈ H^{a}(X, Ω_{X}^{b+1} ⊗ N_{y}^{−1}).

Hence

ι(Γ) = Γ̄ ⊂ H^{0}(X, Ω_{X}^{1})

is the subspace of forms ω ∈ H^{0}(X, Ω_{X}^{1}) such that

β ∧ ω = 0 for all β ∈ H^{0}(X, Ω_{X}^{b} ⊗ N_{y}^{−1}).

Step 2: Consider the natural map

γ : H^{0}(X, Ω_{X}^{1}) ⊗ O_{X} ⟶ Ω_{X}^{1}.

Since all one-forms are pullback of one-forms on α(X) ⊂ Alb(X), the subsheaf im(γ) of Ω_{X}^{1} is of rank dim α(X) and

r = rankγ(\overline{Γ} ⊗ O_{X}) = dim \overline{Γ} − rank(ker(γ) ∩ \overline{Γ} ⊗ O_{X}) ≥
dim \overline{Γ} − dim ker(γ) = dim \overline{Γ} − (dim H^{0}(X, Ω_{X}^{1}) − dim α(X))
= dim α(X) − codim(Γ ⊂ H^{1}(X, O_{X})) .

We assumed that H^{0}(X, Ω_{X}^{b} ⊗ N_{y}^{−1}) ≠ 0. Hence we have at least one element

β ∈ H^{0}(X, Ω_{X}^{b} ⊗ N_{y}^{−1}) and β ∧ γ(\overline{Γ} ⊗ O_{X}) = 0.

Since

(∧^{b}Ω_{X}^{1}) ⊗ (∧^{n−b}Ω_{X}^{1}) ⟶ Ω_{X}^{n}

is a nondegenerate pairing, for n = dim X, we find some meromorphic differential form

δ ∈ Ω_{X}^{n−b} ⊗ C l (X) with δ ∧ β ≠ 0.

Hence δ lies in

Ω_{X}^{n−b} ⊗ C l (X) − \{γ(\overline{Γ} ⊗ O_{X}) ∧ Ω_{X}^{n−b−1} ⊗ C l (X)\}.

This however is only possible if n − b ≤ n − r or b ≥ r. Altogether we find

b ≥ dim α(X) − codim(Γ ⊂ H^{1}(X, O_{X})).
□

13.4.

If one tries to use the same methods for S^{b}(X, M) one has to make sure that H^{b}(X, M⊗N_{y}) is in Hodge duality with H^{0}(X, Ω_{X}^{b} ⊗M′ ⊗N_{y}^{∗}) for some sheaf M′. As shown in (3.23) this holds true for the sheaves L^{(i)} arising from cyclic coverings, at least if one considers Ω_{X}^{b}(log D) instead of Ω_{X}^{b}. More generally one has:

13.5. Theorem (K. Timmerscheidt [59]). Let D be a normal crossing divisor on X, let V be a locally free sheaf and

∇ : V ⟶ Ω_{X}^{1}(log D) ⊗ V

an integrable logarithmic connection. Assume that for all components D_{i} of D, the real part of all eigenvalues of res_{D_{i}}(∇) lies in (0, 1) (which implies that conditions (*) and (!) of (2.8) are satisfied, and that V is the canonical extension defined by Deligne [10]). Assume moreover that the local constant system

V = ker(∇ : V|_{U} ⟶ Ω_{U}^{1} ⊗ V|_{U})

is unitary for U = X − D. Then one has:

a) The Hodge to de Rham spectral sequence

E_{1}^{ab} = H^{b}(X, Ω_{X}^{a}(log D) ⊗ V) ⟶ IH^{a+b}(X, Ω_{X}^{•}(log D) ⊗ V)

degenerates at E_{1}.

b) There exists a C l - antilinear isomorphism

ι : H^{b}(X, Ω_{X}^{a}(log D) ⊗ V) ⟶ H^{a}(X, Ω_{X}^{b}(log D) ⊗ V^{∗}(−D_{red}))

such that for φ ∈ H^{1}(X, O_{X}) and ω = \overline{φ} ∈ H^{0}(X, Ω_{X}^{1}) the diagram

H^{b}(X, Ω_{X}^{a}(log D) ⊗ V) ⟶^{ι} H^{a}(X, Ω^{b}(log D) ⊗ V^{∗}(−D_{red}))
∪φ↓  ^{∧ω}↓
H^{b+1}(X, Ω_{X}^{a}(log D) ⊗ V) ⟶^{ι} H^{a}(X, Ω_{X}^{b+1}(log D) ⊗ V^{∗}(−D_{red}))

commutes.

13.6. Examples.

a) If D = 0 and if N_{y} is the invertible sheaf corresponding to y ∈ Pic^{0}(X), then N_{y} has an integrable connection ∇, 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 N_{y} ⊗ N_{y}^{∗} see [11].

b) If the sheaf V in (13.5) is of the form V = L^{(i)^{−1}} for

D = ∑_{j=1}^{r} α_{j}D_{j} and L^{N} = O_{X}(D),

then the assumptions made in (13.5) are satisfied whenever

\frac{i · α_{j}}{N} ∉ ZZ for j = 1, ..., r.

Hence, using the notations from (3.2), one has D^{(i)} = D^{(N−i)} = D_{red} and

L^{(i)}(−D_{red}) = L^{i}(−[\frac{i · D}{N}] − D_{red}) = L^{i}(−\{\frac{i · D}{N} \}) =

L^{i}([\frac{−i · D}{N}]) = L^{i−N}([\frac{(N − i) · D}{N}]) = L^{(N−i)^{−1}}

Hence (3.2) and (3.23) imply (13.5) for M = L^{(i)^{−1}}.

c) Finally, for M = L^{(i)^{−1}} ⊗ 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.

13.7. Corollary (H. Dunio [14]). 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, V)) ≥ dim(α(X)) − b.

Proof: Again, it is sufficient to give a lower bound for

codim(Γ ⊂ H^{1}(X, O_{X}))

where

Γ = {φ ∈ H^{1}(X, O_{X}); β ∪ φ = 0 for all β ∈ H^{b}(X, N_{y} ⊗ V)},

or using (13.5,b), for

codim(Γ̄ ⊂ H^{0}(X, Ω_{X}^{1}))

where

Γ̄ = {ω ∈ H^{0}(X, Ω_{X}^{1}); β ∧ ω = 0 for all
β ∈ H^{0}(X, Ω_{X}^{b}(log D) ⊗ N_{y}^{∗} ⊗ V^{∗}(−D_{red}))}.

As in (13.3), if

γ : H^{0}(X, Ω_{X}^{1}) ⊗ O_{X} ⟶ Ω_{X}^{1}

is the natural map, one has

r = rank(γ(Γ̄ ⊗ O_{X})) ≥ dim α(X) − codim(Γ̄ ⊂ H^{0}(X, Ω_{X}^{1})).

Assume that one has some

0 ≠ β ∈ H^{0}(X, Ω_{X}^{b}(log D) ⊗ N_{y}^{∗} ⊗ V^{∗}(−D_{red})).

Let v_{1}, ...., v_{s} be a basis of

N_{y}^{∗} ⊗ V^{∗}(−D_{red}) ⊗ C l (X),

then one has β = ∑_{i=1}^{s} β_{i}v_{i} for some β_{i} ∈ Ω_{X}^{b} ⊗ C l (X) and β_{i} ∧ ω = 0 for all i and all

ω ∈ γ(Γ̄ ⊗ O_{X}) ⊗ C l (X).

As in (13.3) this is only possible if b ≥ r.

□

13.8. Example. If L is an invertible sheaf on X and if D is a normal crossing divisor let N ∈ IN be larger than the multiplicities of the components of D. If L^{N} = O_{X}(D) then V = L^{−1} satisfies the assumptions made in (13.7) and

H^{b}(X, L^{−1} ⊗ N_{y}) = 0

for y ∈ Pic^{0}(X) in general position and b < dim α(X). On the other hand, (5.12,e) tells us, that

H^{b}(X, L^{−1} ⊗ N_{y}) = 0

for all y ∈ Pic^{0}(X) and b < κ(L). Hence (13.7) is only of interest if

dim α(X) − κ(L) > 0.

In this situation the bounds given in (13.7) can be improved. The generic vanishing theorem remains true for

b < dim α(X) + κ(L) − dim α′(Z)

where Z is a desingularization of the image of the rational map

Φ_{ν} : X ⟶ IP(H^{0}(X, L^{ν}))

for ν sufficiently large (see (5.3)) and where

α′ : Z ⟶ Alb(Z)

is the Albanese map of Z. To be more precise:

13.9. Assumptions and Notations. Let X be a complex projective manifold, let L be an invertible sheaf on X, let

D = ∑_{j=1}^{r} α_{j}D_{j}

be a normal crossing divisor and let N be a natural number with

0 < α_{j} < N for j = 1, ..., r.

Assume that either L^{N}(−D) is semi-ample or that (more generally) L^{N}(−D) is numerically effective and

κ(L^{N}(−D)) = ν(L^{N}(−D))

( see (5.9) and (5.11)). For some μ > 0 the rational map ( see (5.3))

Φ_{μ} : X ⟶ Φ_{μ}(X) ⊂ IP(H^{0}(X, L^{μ}))

has an irreducible general fibre and dim(Φ_{μ}(X)) = κ(L). For such a μ let Z be a desingularization of Φ_{μ}(X) and X′ a blowing up of X such that the induced rational map

Φ′ : X′ ⟶ Z

is a morphism of manifolds. Φ^{′∗} defines a morphism

Φ^{∗} : Pic^{0}(Z) ⟶ Pic^{0}(X′) = Pic^{0}(X)

and Φ^{∗}(Pic^{0}(Z)) is an abelian subvariety of Pic^{0}(X) independent of the desingularization choosen. Let

α : X ⟶ Alb(X) and α′ : Z ⟶ Alb(Z)

be the Albanese maps.

13.10. Theorem (H. Dunio [14]). Under the assumptions made in (13.9) one has:

a) S^{b}(X, L^{−1}) = 0 for b < κ(L).

b) S^{b}(X, L^{−1}) lies in the subgroup of Pic^{0}(X), which is generated by torsion elements and by Φ^{∗}(Pic^{0}(Z)), for b = κ(L).

c) codim_{Pic^{0}(X)}(S^{b}(X, L^{−1})) ≥ dim α(X) − dim α′(Z) + κ(L) − b.

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, L^{−1}) is compatible with blowing ups of X, we may assume that the rational map Φ : X → Z is a morphism. Moreover, as in the proof of (5.12), we can assume that L^{N}(−D) is semi-ample or even, replacing N and D by some common multiple, that

L^{N}(−D) = O_{X}(H)

where H is a non singular divisor and D + H a normal crossing divisor. Since

L = L^{(1,D)} = L^{(1,D+H)}

we can as well assume that L^{N} = O_{X}(D). By (13.5,b) or (13.6,c) the space

H^{b}(X, L^{−1} ⊗ N_{y})

is Hodge dual to

H^{0}(X, Ω_{X}^{b}(log D) ⊗ L^{(N−1)^{−1}} ⊗ N_{y}^{−1}).

Let G_{Φ}^{b} ↪ Ω_{X}^{b}(log D) be the largest subsheaf which over some open non empty subvariety of X coincides with

Φ^{∗}Ω_{Z}^{κ} ∧ Ω_{X}^{b−κ}(log D).

Of course G_{Φ}^{b} = 0 for b < κ = κ(L) and δ ∈ G_{Φ}^{b} ⊗ C l (X) if and only if δ is a meromorphic b-form with δ ∧ Φ^{∗}Ω_{Z}^{1} = 0.

Since L^{(N−1)} ⊆ L^{N−1} and since

(L^{(N−1)})^{N} = O_{X}(N · D_{red} − D),

we have κ(L^{(N−1)}) = κ(L) and (L^{(N−1)})^{μ} contains L^{N} for some μ > 0.

13.11. Claim. If M is an invertible sheaf such that M^{μ} contains L^{N} for some μ > 0, then

H^{0}(X, G_{Φ}^{b} ⊗ M^{−1} ⊗ N_{y}^{−1}) = H^{0}(X, Ω_{X}^{b}(log D) ⊗ M^{−1} ⊗ N_{y}^{−1}).

Proof: The methods used to prove (13.11) are due to F. Bogomolov [6]. A section β ∈ H^{0}(X, Ω_{X}^{b}(log D) ⊗ M^{−1} ⊗ N_{y}^{−1}) gives an inclusion

β : M ⟶ Ω_{X}^{b}(log D) ⊗ N_{y}^{−1}

and we have to show that Φ^{∗}Ω_{Z}^{1} ∧ β(M) = 0.

If τ : X′ ⟶ X is generically finite and D′ = τ^{∗}D a normal crossing divisor, then τ^{∗}Ω_{X}^{b}(log D) is a subsheaf of Ω_{X′}^{b}(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

τ^{∗}x = ∏_{j=1}^{r} x′_{j}^{ν_{j}} and τ^{∗} \frac{dx}{x} = ∑_{j=1}^{r} ν_{j} \frac{dx′_{j}}{x′_{j}} ∈ Ω_{X′}^{b}(log D′).

Hence β induces

β′ : τ^{∗}M ⟶ Ω_{X′}^{b}(log D′) ⊗ τ^{∗}(N_{y}^{−1}).

Since

β′(τ^{∗}M) ∧ τ^{∗}Φ^{∗}Ω_{Z}^{1} = 0

implies that

β(M) ∧ Φ^{∗}Ω_{Z}^{1} = 0,

we can replace X by X′ whenever we like. For example, if

s_{0}, ...., s_{κ} ∈ H^{0}(X, L^{ν}) ⊂ H^{0}(X, M^{μ})

are choosen such that the functions

\frac{s_{1}}{s_{0}} , ...., \frac{s_{κ}}{s_{0}}

are algebraic independent, we can take X′ as a desingularization of the covering obtained by taking the μ-th root out of s_{0}, s_{1}, ...., s_{κ}. Hence to prove (13.11) we may assume that M itself has sections

s_{0}, ...., s_{κ} with f_{1} = \frac{s_{1}}{s_{0}} , ...., f_{κ} = \frac{s_{κ}}{s_{0}}

algebraic independent. From (13.5) we know that d(β(s_{i})) = 0, which by the Leibniz rule implies

0 = d(β(s_{i})) = d(f_{i} · β(s_{0})) = d(f_{i}) ∧ β(s_{0}).

However, d(f_{1}), ...., d(f_{κ}) are generators of Φ^{∗}Ω_{Z}^{1} over some non empty open subset.

□

Part a) of (13.10) follows from (13.11) since for b < κ the sheaf G_{Φ}^{b} = 0.

If b = κ then

G_{Φ}^{b} = Φ^{∗}ω_{Z} ⊗ O_{X}(Δ)

for some effective divisor Δ on X, not meeting the general fibre F of Φ. Hence, for y ∈ S^{κ}(X, L^{−1}) (13.11) implies that L^{(N−1)^{−1}} ⊗ N_{y}^{−1}|_{F} has a non trivial section and therefore N_{y}^{−1}|_{F} = L^{(N−1)}|_{F}. The divisor D + H does not meet the general fibre F and, as we claimed in (13.10,b), N · y ∈ Φ^{∗}(Pic^{0}(Z)).

13.12. Remark. If L is semi-ample and b > κ, then a similar argument shows that L^{(N−1)^{−1}} ⊗ N_{y}^{−1}|_{F} ⊗ Ω_{F}^{b−κ} has a non trivial section. This implies, as we have seen in the proof of (13.2), that those N_{y}|_{F} are corresponding to points y in a subvariety of Pic^{0}(F) of codimension larger than or equal to

dim(α(F)) − b + κ = dim(α(X)) − dim(α(Z)) − b + κ,

which gives (13.10,c).

We instead generalise the argument used in step 2 of the proof of (13.3): If

Γ = {φ ∈ H^{1}(X, O_{X}); β ∪ φ = 0 for all β ∈ H^{b}(X, L^{−1} ⊗ N_{y})}

then the Hodge dual of Γ is

Γ̄ = {ω ∈ H^{0}(X, Ω_{X}^{1}); β ∪ ω = 0 for all β ∈ H^{0}(X, G_{Φ}^{b} ⊗ L^{(N−1)^{−1}} ⊗ N_{y}^{−1})}.

If γ is the composed map

H^{0}(X, Ω_{X}^{1}) ⊗ O_{X} ⟶ Ω_{X}^{1} ⟶ Ω_{X/Z}^{1}

then

H^{0}(X, G_{Φ}^{b} ⊗ L^{(N−1)^{−1}} ⊗ N_{y}^{−1}) ≠ 0

implies that

γ(Γ̄ ⊗ O_{X}) ∧ β̄ = 0 for some β̄ ∈ Ω_{X/Z}^{b−κ} ⊗ C l (X)

or, in other terms, that

Ω_{X/Z}^{n−b} ⊗ C l (X) ≠ γ(Γ̄ ⊗ O_{X}) ∧ Ω_{X/Z}^{n−b−1} ⊗ C l (X).

Again this is only possible for

n − b ≤ n − κ − rank(γ(Γ̄ ⊗ O_{X}))

or

b − κ ≥ rank(γ(Γ̄ ⊗ O_{X})).

However,

rank(γ(Γ̄ ⊗ O_{X})) ≥ dim(α(X)) − dim(α′(Z)) − codim(Γ̄ ⊂ H^{0}(X, Ω_{X}^{1}))

and (13.10,c) follows from (12.7)

□

13.13. Remarks.

a) If Z(φ) denotes the zero locus of a global one-form φ and

w(X) = Max{codim_{X}Z(φ); φ ∈ H^{0}(X, Ω_{X}^{1})},

then a second result of Green and Lazarsfeld [26] says that, for a generic line bundle

N ∈ Pic^{0}(X) and a + b < w(X),

one has

H^{b}(X, Ω_{X}^{a} ⊗ N) = 0.

b) In [27] 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 [5], who studied S^{1}(X) and showed the same result in this case.

c) Finally, C. Simpson recently gave in [58] 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.

APPENDIX: Hypercohomology and spectral sequences

1.

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 [60], [29], [7], [8], [33], [31]).

2.

Through this section X is a variety over a commutative ring k.

3.

We consider complexes F^{•} of sheaves of O-modules, where O is a sheaf of commutative rings. For example

O = ZZ , O = k or O = structure sheaf of X.

Any map of O-modules

σ : F^{•} ⟶ G^{•}

between two such complexes induces a map of cohomology sheaves:

H^{i}(σ) : H^{i}(F^{•}) ⟶ H^{i}(G^{•})

where H^{i}(F^{•}) is the sheaf associated to the presheaf

U ↦ \frac{ker Γ(U, F^{i}) → Γ(U, F^{i+1})}{im Γ(U, F^{i−1}) → Γ(U, F^{i})}

in the given topology. One says that σ is a quasi-isomorphism if H^{i}(σ) is an isomorphism for all i.

4.

We will only consider complexes F^{•} which are bounded below, that means F^{i} = 0 for i sufficiently negative.

5. Example: the analytic de Rham complex.

X is a complex manifold. Then the standard map

C l ⟶ ( O_{X} → Ω_{X}^{1} → Ω_{X}^{2} → Ω_{X}^{3} → ⋯)

from the constant sheaf C l to the analytic de Rham complex Ω_{X}^{•} is a quasiisomorphism as by the so called “Poincaré lemma”

H^{i}(Ω_{X}^{•}) = 0 for i > 0, and H^{0}(Ω_{X}^{•}) = C l .

6. Example: the Čech complex.

Let U = {U_{α}; α ∈ A}, for A ⊂ IN, be some open covering of the variety X defined over k. To a bounded below complex F^{•} one associates its Čech complex G^{•} defined as follows.

G^{i} := ⨁_{a≥0} C^{a}(U, F^{i−a})

where

C^{a}(U, F^{i−a}) = ∏_{α_{0}<α_{1}<...<α_{a}} ϱ_{∗}F^{i−a}|_{U_{α_{0}...αa}} .

Here, for any

U_{α_{0}...α_{a}} := U_{α_{0}} ∩ … ∩ U_{α_{a}}

ϱ denotes the open embedding

ϱ = ϱ_{α_{0}...α_{a}} : U_{α_{0}...α_{a}} ⟶ X,

and for any sheaf F, one writes ϱ_{∗}F for the sheaf associated to the presheaf

U ↦ Γ(U_{α_{0}...α_{a}} ∩ U, F).

As F^{i} = 0 for i << 0, the direct sum in the definition of G has finitely many summands.

The differential Δ of G^{•} is defined by

Δ(s) = (−1)^{i}δs + d_{F^{•}}s for s ∈ C^{a}(U, F^{i−a}),

where δ is the Čech differential defined by

(δs)_{α_{0}...α_{a+1}} = ∑_{0}^{a+1}(−1)^{l}s_{α_{0}...α̂_{l}...α_{a+1}}|_{U_{α_{0}...α_{a+1}}}

and d_{F^{•}} is the differential of F^{•}. Then the natural map

σ : F^{•} ⟶ G^{•}

defined by

F^{i} ⟶^{ϱ} ∏_{α∈A} ϱ_{∗}F^{i}|_{U_{α}} = C^{0}(U, F^{i})

is a quasi-isomorphism.

To show this one considers first a single sheaf F (see [30], III 4.2) and then one computes that, whenever one has a double complex

↑  ↑  ↑
⟶ K^{1,i−1} ⟶ K^{1,i} ⟶ K^{1,i+1} ⟶
↑  ↑  ↑  ↑^{d_{vert}}
⟶ K^{0,i−1} ⟶ K^{0,i} ⟶ K^{0,i+1} ⟶
↑  ↑  ↑
⟶ F^{i−1} ⟶ F^{i} ⟶ F^{i+1} ⟶ … ⟶_{d_{hor}}

such that F^{i} ⟶ K^{•,i} is a quasi-isomorphism for all i, then F^{•} ⟶ D^{•} is a quasi-isomorphism as well, where D^{•} is the associated double complex:

D^{i} := ⨁_{a} K^{a,i−a} ⟶ D^{i+1} := ⨁_{a} K^{a,i+1−a}

with differential (−1)^{i}d_{vert} + d_{hor}.

7.

If F^{•} and G^{•} are two complexes of O-modules bounded below one defines the tensor product F^{•} ⊗ G^{•} by

(F^{•} ⊗ G^{•})^{i} := ⨁_{a} F^{a} ⊗ G^{i−a}

with differential from

F^{a} ⊗ G^{i−a} to F^{a+1} ⊗ G^{i−a} ⊕ F^{a} ⊗ G^{i+1−a}. given by
d(f_{a} ⊗ g_{i−a}) = df_{a} ⊗ g_{i−a} + (−1)^{a}f_{a} ⊗ dg_{i−a}.

As both F^{•} and G^{•} are bounded below, a takes finitely many values, and (F^{•} ⊗ G^{•}) is a complex of O-modules bounded below.

8.

If in 7 we assume moreover that locally the O-modules F^{a} and G^{b} are free and dF^{a−1} as well as dG^{b−1} are subbundles for all a and b, then locally one has some decomposition

\begin{aligned}F^{a} & = & dF^{a−1} & ⊕ & H^{a}(F^{•}) & ⊕ & F^{′a} \\ G^{b} & = & dG^{b−1} & ⊕ & H^{b}(G^{•}) & ⊕ & G^{′b} \end{aligned}

where

\begin{matrix}d : & F^{′a} & ⟶ & dF^{a} \\ d : & G^{′b} & ⟶ & dG^{b} \end{matrix}

are isomorphisms. In particular,

F^{a} ⊗ G^{b} = d(F^{a−1} ⊗ dG^{b−1} + F^{a−1} ⊗ H^{b} + H^{a} ⊗ G^{b−1})
⊕ H^{a}(F^{•}) ⊗ H^{b}(G^{•})
⊕ (dF^{a−1} + H^{a}(F^{•})) ⊗ G^{′b}
⊕ F^{′a} ⊗ (dG^{b−1} + H^{b}(G^{•}))
⊕ F^{′a} ⊗ G^{′b}

and therefore one has the Künneth decomposition

H^{i}(F^{•} ⊗ G^{•}) = ⨁_{a} H^{a}(F^{•}) ⊗ H^{i−a}(G^{•}).

9.

The map σ : F^{•} ⟶ I^{•} is called an injective resolution of F^{•} if I^{•} is a complex of O-modules bounded below, σ is a quasi-isomorphism, and the sheaves I^{i} are injective for all i, i.e.:

Hom_{O}(B, I^{i}) ⟶ Hom_{O}(A, I^{i})

is surjective for any injective map A ⟶ B of sheaves of O-modules. It is an easy fact that if O is a constant commutative ring, for example O = ZZ, O = k, then every complex of O-modules which is bounded below admits an injective resolution (see [33], (6.1)).

10.

From now on we assume that O is a constant commutative ring. Let F^{•} be a complex of O-modules, bounded below. One defines the hypercohomology group IH^{a}(X, F^{•}), to be the O-module

IH^{a}(X, F^{•}) := \frac{ker Γ(X, I^{a}) → Γ(X, I^{a+1})}{im Γ(X, I^{a−1}) → Γ(X, I^{a})}.

One verifies that this definition does not depend on the injective resolution choosen (see [30] III, 1.0.8).

In particular, if σ : F^{•} ⟶ G^{•} is a quasi-isomorphism, then σ induces an isomorphism of the hypercohomology groups:

IH^{a}(X, F^{•}) ⟶^{∼} IH^{a}(X, G^{•})

(by taking an injective resolution I^{•} of G^{•} which is also an injective resolution of F^{•}).

11.

By definition, H^{a}(X, I) = 0 for a > 0 if I is an injective sheaf. We will verify in (A.28) that if σ : F^{•} ⟶ G^{•} is a quasi-isomorphism and if H^{a}(X, G^{i}) = 0

APPENDIX: Hypercohomology and spectral sequences 151

for all a > 0 and all i, then

IH^{a}(X, F^{•}) = \frac{ker Γ(X, G^{a}) → Γ(X, G^{a+1})}{im Γ(X, G^{a−1}) → Γ(X, G^{a})}.

We call (G^{•}, σ) an acyclic resolution of F^{•} in this case. 12. IH transforms short exact sequences

0 ⟶ A^{•} ⟶ B^{•} ⟶ C^{•} ⟶ 0

of complexes of O-modules which are bounded below into long exact sequences

⋯ ⟶ IH^{i}(A^{•}) ⟶ IH^{i}(B^{•}) ⟶ IH^{i}(C^{•}) ⟶ IH^{i+1}(A^{•}) ⟶ … of O-modules. 13. We assume now that the complex F^{•} of O-modules, bounded below, has subcomplexes

\begin{aligned}& … ⊂ F ilt_{i−1} ⊂ F ilt_{i} ⊂ … ⊂ F^{•} \\ (or & … ⊂ F ilt^{i} ⊂ F ilt^{i−1} ⊂ … ⊂ F^{•}) \end{aligned}

such that

\begin{aligned}& ⋃_{i} F ilt_{i} = F^{•} \\ (or & ⋃_{i} F ilt^{i} = F^{•}) \end{aligned}

and such that

F ilt_{i} = 0 for i << 0
(or F ilt^{i} = 0 for i ≫ 0).

One says that F^{•} is filtered by the subcomplexes F ilt_{i} (or F ilt^{i}). Via (A.12), this filtration defines a filtration on the hypercohomology groups:

F ilt_{i}IH^{a}(X, F^{•}) := im(IH^{a}(X, F ilt_{i}) ⟶ IH^{a}(X, F^{•}))
(or F ilt^{i}IH^{a}(X, F^{•}) := im(IH^{a}(X, F ilt^{i}) ⟶ IH^{a}(X, F^{•}))).

This just means that the group IH^{a}(X, F^{•}) has subgroups

… ⊂ F ilt_{i−1}IH^{a}(X, F^{•}) ⊂ F ilt_{i}IH^{a}(X, F^{•}) ⊂ … ⊂ IH^{a}(X, F^{•})

(or … ⊂ F ilt^{i}IH^{a}(X, F^{•}) ⊂ F ilt^{i−1}IH^{a}(X, F^{•}) ⊂ … ⊂ IH^{a}(X, F^{•})).

We pass from increasing to decreasing filtrations by setting

F ilt_{i} := F ilt^{−i}.

14. We define

Gr_{i} := F ilt_{i}/F ilt_{i−1}.

One has a diagram of exact sequences

0  0
↓  ↓
0 ⟶ F ilt_{i−1}  ⟶  F ilt_{i} ⟶ Gr_{i} ⟶ 0
↓  ↓
F^{•}  \underrightarrow{identity} F^{•}
↓ ↓
0 ⟶ Gr_{i} ⟶ F^{•}/F ilt_{i−1} ⟶ F^{•}/F ilt_{i} ⟶ 0
↓ ↓
0  0

which gives via (A.12):

Gr_{i}(F ilt_{•}IH^{a}(F^{•})) := F ilt_{i}IH^{a}(F^{•})/F ilt_{i−1}IH^{a}(F^{•})
= \frac{IH^{a}(F ilt_{i})}{IH^{a}(F ilt_{i−1})+IH^{a−1}(F^{•}/F ilt_{i})}
= \frac{ker(IH^{a}(Gr_{i})→IH^{a+1}(F ilt_{i−1}))}{ker(IH^{a}(Gr_{i})→IH^{a}(F^{•}/F ilt_{i−1}))}.

We define

E_{∞}^{a−i,i} := Gr_{i}(F ilt_{•}IH^{a}(F^{•})) and E_{2}^{a−i,i} := IH^{a}(Gr_{i}).

Obviously E_{∞}^{a−i,i} is a subquotient of E_{2}^{a−i,i}. 15. The formations of spectral sequences has the aim to compute E_{∞}^{a−i,i} only in terms of E_{2}^{s,t}, by filtering the terms IH^{a+1}(F ilt_{i−1}) and IH^{a}(F^{•}/F ilt_{i−1}) appearing in the description (A.14) by the induced filtrations:

F ilt_{l} IH^{a+1}(F ilt_{i−1}) := im(IH^{a+1}(F ilt_{l}) ⟶ IH^{a+1}(F ilt_{i−1})) for l ≤ i − 1 and

F ilt_{l} IH^{a}(F^{•}/F ilt_{i−1}) := im(IH^{a}(F ilt_{l}/F ilt_{i−1}) ⟶ IH^{a}(F^{•}/F ilt_{i−1}))

for l ≥ i − 1, and by computing the corresponding graded quotients.

16.

If we assume that for some a, the filtration F ilt_{i}IH^{a}(F^{•}) is exhausting, that is:

⋃_{i} F ilt_{i}IH^{a}(F^{•}) = IH^{a}(F^{•}),

then

Gr IH^{a}(F^{•}) = ⨁_{i} E_{∞}^{a−i,i}

is the corresponding graded group. In particular, assume that IH^{a}(F^{•}) and E_{∞}^{a−i,i} are free O-modules (where O is ZZ or k), and that IH^{a}(F^{•}) is of finite rank. Then

rank_{O}IH^{a}(F^{•}) = ∑_{i} rank_{O}E_{∞}^{a−i,i}.

If E_{2}^{a−i,i} is also free, then rank_{O}IH^{a}(F^{•}) ≤ ∑_{i} rank_{O}E_{2}^{a−i,i} and one has equality if and only if

E_{∞}^{a−i,i} = E_{2}^{a−i,i} for all i.

17. Example: One step filtration:

0 = F ilt_{s−1} ⊂ F ilt_{s} = F^{•}.

Then IH^{a}(F^{•}) = IH^{a}(Gr_{s}) = E_{∞}^{a−s,s} = E_{2}^{a−s,s}.

18. Example: Two steps filtration:

0 = F ilt_{s−2} ⊂ F ilt_{s−1} ⊂ F ilt_{s} = F^{•}

Then one has the exact sequence

0 ⟶ Gr_{s−1} ⟶ F^{•} ⟶ Gr_{s} ⟶ 0

and

0 ⟶ E_{∞}^{a−(s−1),(s−1)} ⟶ IH^{a}(F^{•}) ⟶ E_{∞}^{a−s,s} ⟶ 0

with

E_{∞}^{a−(s−1),(s−1)} = IH^{a}(Gr_{s−1})/IH^{a−1}(Gr_{s})
E_{∞}^{a−s,s}  = ker IH^{a}(Gr_{s}) ⟶ IH^{a+1}(Gr_{s−1})

19. 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

0  0
↓  ↓
0 ⟶ Gr_{i−1} ⟶ F ilt_{i}/F ilt_{i−2} ⟶  Gr_{i}  ⟶ 0
∥  ↓  ↓
0 ⟶ Gr_{i−1} ⟶ F ilt_{i+1}/F ilt_{i−2} ⟶ F ilt_{i+1}/F ilt_{i−1} ⟶ 0
↓  ↓
Gr_{i+1}  \underrightarrow{identity}  Gr_{i+1}
↓  ↓
0  0

and the 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 ε = 1, 2 and the ε-steps filtration one has

E_{∞}^{a−i,i} = E_{ε+1}^{a−i,i},

as we saw in (A.17) and (A.18), and the filtration on IH^{a}(F^{•}) is exhausting. 20. The right vertical exact sequence of the diagram (A.19) gives a surjection

IH^{a}(F ilt_{i+1}/F ilt_{i−1}) ⟶^{γ} ker(IH^{a}(Gr_{i+1}) → IH^{a+1}(Gr_{i}))

and the middle horizontal sequence induces a morphism

ker(IH^{a}(Gr_{i+1}) → IH^{a+1}(Gr_{i})) ⟶^{d′_{3}} IH^{a+1}(Gr_{i−1})/im IH^{a}(Gr_{i})

Replacing i by i − 1, one obtains maps

IH^{a}(F ilt_{i}/F ilt_{i−2}) ⟶^{γ} ker(IH^{a}(Gr_{i}) → IH^{a+1}(Gr_{i−1})) ⟶^{d′_{3}} \frac{IH^{a+1}(Gr_{i−2})}{im IH^{a}(Gr_{i−1})}

where γ is surjective. As the extension

0 ⟶ Gr_{i−2} ⟶ F ilt_{i}/F ilt_{i−3} ⟶ F ilt_{i}/F ilt_{i−2} ⟶ 0

lifts to an extension

0 ⟶ F ilt_{i−2}/F ilt_{i−4} ⟶ F ilt_{i}/F ilt_{i−4} ⟶ F ilt_{i}/F ilt_{i−2} ⟶ 0

the image of d′_{3} ∘ γ lies in fact in

\frac{ker IH^{a+1}(Gr_{i−2}) → IH^{a+2}(Gr_{i−3})}{im IH^{a}(Gr_{i−1}) → IH^{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

IH^{a}(F ilt_{i}/F ilt_{i−2})

and similarly a class in E_{4}^{a−i,i} may be represented by a class in

IH^{a}(F ilt_{i}/F ilt_{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 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 IH^{a}(F ilt_{i}/F ilt_{i−r}).

When F ilt_{σ−1} = 0, and F ilt_{σ} ≠ 0, one defines the induced filtration on F ilt_{σ+ρ} by

F ilt_{l} F ilt_{σ+ρ} = \{ \begin{matrix}F ilt_{l} & if l ≤ σ + ρ \\ F ilt_{σ+ρ} & if l ≥ σ + ρ . \end{matrix}

Then one has:

E_{∞}^{a−i,i}(IH^{a}(F ilt_{σ+ϱ})) = E_{ϱ+2}^{a−i,i}

and the filtration on IH^{a}(F ilt_{σ+ϱ}) is exhausting.

21.

This shows that in general, for going from IH^{a}(F ilt_{σ+ϱ}) to IH^{a}(F^{•}) one has to introduce infinitely many r and groups E_{r}^{a−i,i}, a reason for the notation E_{∞}^{a−i,i}.

One says that the 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})) degenerates in E_{r} if:

The filtration on IH^{a}(F^{•}) is exhausting and E_{∞}^{a−i,i} = E_{r}^{a−i,i} for all i, or equivalently d_{r+l} = 0 for all l ≥ 0.

With this terminology, we have seen in (A.20) that a (ϱ + 1)-steps filtration defines an E_{2} spectral sequence which degenerates in E_{ϱ+2}.

22.

Under the assumptions of (A.16), assume moreover that E_{r}^{a−i,i} is also a free O-module (for example if O is a field). Then one has:

rank_{O} IH^{a}(F^{•}) ≤ ∑_{i} rank_{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.

23.

By (A.19), to say that the spectral sequence degenerates in E_{2} means that for all i, one has an exact sequence

0 ⟶ IH^{a}(F ilt_{i−1}) ⟶ IH^{a}(F ilt_{i}) ⟶ IH^{a}(Gr_{i}) ⟶ 0,

and of course that the filtration is exhausting.

24.

One says that the spectral sequence (E_{2}, d_{2}) converges to IH^{a}(F^{•}) if it degenerates in E_{r} for some r. We sometimes write

(E_{2}^{a−i,i}, d_{2}) ⟹ IH^{a}(F^{•})

instead of “(E_{2}, d_{2}) converges to IH^{a}(F^{•})”.

25. The Hodge to de Rham spectral sequence.

On F^{•}, a complex of O-modules, bounded below, we define the Hodge filtration (often called the stupid filtration) by:

F ilt^{i}F^{•} = F^{≥i} = F ilt_{−i}F^{•}

where

(F^{≥i})^{l} = \{ \begin{matrix}0 & if l < i \\ F^{l} & if l ≥ i. \end{matrix}

Then Gr_{−i} = F ilt_{−i}/F ilt_{−i−1} = F^{i}[−i] where [α] means:

(F^{•}[α])^{l} = F^{l+a}.

This is the so called shift by α to the right. The E_{2} spectral sequence reads:

E_{2}^{a−i,i} = IH^{a}(Gr_{i}) = IH^{a}(F^{−i}[i]) = H^{a+i}(F^{−i})

where the differential d_{2} goes to

IH^{a+1}(Gr_{i−1}) = IH^{a+1}(F^{−(i−1)}[i − 1]) = H^{a+i}(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} or E_{1}^{α,β} = E_{2}^{β+2α,−α}

with differentials:

E_{2}^{β+2α,−α} ⟶^{d_{2}} E_{2}^{β+2α+2,−α−1}
∥  ∥
E_{1}^{α,β}  ⟶^{d_{1}}  E_{1}^{α+1,β}

The E_{1} spectral sequence obtained is called the Hodge to de Rham spectral sequence, at least when F^{•} is some de Rham complex on X, possibly with some poles, possibly with non-trivial coefficients …

For a given a, one has

IH^{a}(F^{•}) = IH^{a}(F^{≤a+1})

where

(F^{≤i})^{l} = \{ \begin{matrix}F^{l} & if l ≤ i \\ 0 & if l > i. \end{matrix}

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 ⟶ IH^{a}(F^{≥i+1}) ⟶ IH^{a}(F^{≥i}) ⟶ H^{a−i}(F^{i}) ⟶ 0.

Putting those sequences together, one obtains exact sequences

0 ⟶ IH^{a}(F^{≥i+j}) ⟶ IH^{a}(F^{≥i}) ⟶ IH^{a}(F^{[i,i+j)}) ⟶ 0

where

(F^{[i,i+j)})^{l} = \{ \begin{matrix}F^{l} & if l ∈ [i, i + j) \\ 0 & if not. \end{matrix}

If moreover, one knows that IH^{a}(F^{•}) is a free O-module of finite rank, as well as H^{a−i}(F^{i}), then the E_{1} Hodge to de Rham spectral sequence degenerates in E_{1} if and only if

rank_{O} IH^{a}(F^{•}) = ∑_{i} rank_{O} H^{a−i}(F^{i}).

26. The conjugate spectral sequence.

On F^{•}, a complex of O-modules bounded below, we defined the τ-filtration:

(τ_{≤i}F^{•})^{l} = \{ \begin{matrix}F^{l} & for l < i \\ ker d & for l = i \\ 0 & otherwise. \end{matrix}

Then Gr_{i} = H^{i}[−i] is the cohomology sheaf in degree i. The E_{2} spectral sequence reads

E_{2}^{a−i,i} = IH^{a}(Gr_{i}) = H^{a−i}(H^{i})

where the d_{2} differential goes to

IH^{a+1}(Gr_{i−1}) = H^{a−i+2}(H^{i−1}).

It is called the conjugate spectral sequence. For a given a, one has

IH^{a}(F^{•}) = IH^{a}(τ_{≤a+1}F^{•}).

However τ_{≤a+1}F^{•} is a finite complex on which the τ-filtration induces a finite step filtration. Therefore the E_{2}-conjugate spectral sequence always converges. Furthermore, if σ : F^{•} ⟶ G^{•} is a quasi-isomorphism then σ induces quasiisomorphisms

τ_{≤i}F^{•} ⟶ τ_{≤i}G^{•}

for all i, and therefore σ 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 ⟶ IH^{a}(τ_{≤i−1}) ⟶ IH^{a}(τ_{≤i}) ⟶ H^{a−i}(H^{i}) ⟶ 0

for all i. If IH^{a}(F^{•}) is a free O-module of finite rank, as well as IH^{a−i}(H^{i}), then the degeneration in E_{2} is equivalent to

rank_{O}IH^{a}(F^{•}) = ∑_{i} rank_{O} H^{a−i}(H^{i}).

27. The Leray spectral sequence.

Let f : X ⟶ Y be a morphism between two k-varieties, and let F^{•} be a complex of O-modules on X, bounded below.

Let F^{•} ⟶ I^{•} be an injective resolution. We consider the direct image functor (already used and defined but not named in 6):

(f_{∗}K)_{x} = lim ; ⟶_{x∈U} H^{0}(f^{−1}(U), K)

for any O-sheaf K on X and any open set U in Y. In particular, by definition H^{0}(X, K) = H^{0}(Y, f_{∗}K). Therefore one has

IH^{a}(X, F^{•}) = \frac{ker H^{0}(Y, f_{∗}I^{a}) ⟶ H^{0}(Y, f_{∗}I^{a+1})}{im H^{0}(Y, f_{∗}I^{a−1}) ⟶ H^{0}(Y, f_{∗}I^{a})}.

One verifies immediately that, by definition, f_{∗}I^{i} is an injective sheaf as well, which allows to write (A.10):

IH^{a}(X, F^{•}) = IH^{a}(Y, (f_{∗}I^{•})),

where f_{∗}I^{•} is the complex

(f_{∗}I^{•})^{l} = f_{∗}I^{l}.

One considers the conjugate spectral sequence for (f_{∗}I^{•}):

E_{2}^{a−i,i} = H^{a−i}(Y, H^{i}(f_{∗}I^{•})).

One defines

R^{i}f_{∗}F^{•} := H^{i}(f_{∗}I^{•}).

By definition

(R^{i}f_{∗}F^{•})_{x} : = lim ; ⟶_{x∈U} \frac{ker H^{0}(f^{−1}(U),I^{i})→H^{0}(f^{−1}(U),I^{i+1})}{im H^{0}(f^{−1}(U),I^{i−1})→H^{0}(f^{−1}(U),I^{i})}

= lim ; ⟶_{x∈U} IH^{i}(f^{−1}(U), F^{•})

In particular, R^{i}f_{∗}F^{•} 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_{∗}F^{•}),

with d_{2} differential to H^{a−i+2}(Y, R^{i−1}f_{∗}F^{•}), and is called the Leray spectral sequence for f.

As the conjugate spectral sequence for (f_{∗}I^{•}) converges to IH^{a}(f_{∗}I^{•}) (A.26), the Leray spectral sequence for f always converges to IH^{a}(F^{•}). In particular, if F is just a sheaf for which R^{i}f_{∗}F = 0 for i > 0, one has:

H^{a}(X, F) = H^{a}(Y, f_{∗}F) for all a.

28.

Let G^{•} be a complex of O-modules, bounded below, such that H^{i}(G^{j}) = 0 for i > 0 and all j. Then the E_{1} Hodge to de Rham spectral sequence E_{1}^{ij} = H^{j}(G^{i}) degenerates in E_{2} and one has

E_{∞}^{i,0} = E_{2}^{i,0} = \frac{ker H^{0}(G^{i})→H^{0}(G^{i+1})}{im H^{0}(G^{i−1})→H^{0}(G^{i})}
= IH^{i}(G^{•})
= IH^{i}(F^{•}) for any quasi-isomorphism F^{•} ⟶ G^{•}.

29.

Take for F^{•} a complex of quasi-coherent sheaves (for example some de Rham complex). We consider a collection of very ample Cartier divisors D_{α} with empty intersection, such that the open covering of X defined by U_{α} := X −D_{α} consists of affine varieties. Then one has:

H^{a}(X, ϱ_{∗}F^{j}|_{U_{α_{0}...α_{i}}}) = H^{a}(U_{α_{0}...α_{i}}, F^{j}|_{U_{α_{0}...α_{i}}})

for all a where ϱ : U_{α_{0}...α_{i}} ⟶ X is the natural embedding of the affine set U_{α_{0}...α_{i}}. In fact, one has

(R^{i}ϱ_{∗}F^{j})_{x} = lim ; ⟶_{x∈V} H^{i}(V ∩ U_{α_{0}···α_{i}}, F^{j})

= 0 for i > 0

and one applies (A.27). By (A.28) one obtains:

IH^{a}(X, F^{•}) = \frac{ker ⊕C^{i}(U, F^{a−i}) → ⊕ C^{i′}(U, F^{a+1−i′})}{im ⊕ C^{i′}(U, F^{a−1−i′}) → ⊕ C^{i}(U, F^{a−i})}

where

\begin{aligned}C^{i}(U, F^{j}) & = H^{0}(X, C^{i}(U, F^{j})) \\ & = ⨁_{α_{0}<...<α_{i}} H^{0}(U_{α_{0}...α_{i}}, F^{j}). \end{aligned}

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Hélène Esnault and Eckart Viehweg Fachbereich 6, Mathematik Universität GH Essen Universitätsstr. 3 D-W-4300 Essen1 Germany

Index

< Δ >, 21 C^{−1}, 101 E_{2}^{a−s,s}, 153 E_{∞}^{a−s,s}, 153 H_{DR}^{b}(X/k), 82 Res_{D}(∇), 14 S^{b}(X, M), 134 W_{2}(k), 85 [Δ], 19 Ω_{X}^{a}(log D), 11 Ω_{X}^{a}(∗D), 11

Ω_{_{X̃}/_{S̃}}^{a}(log D̃), 89

Q l -divisors, 19 IH^{a}(X, F^{•}), 150 κ(L), 44 ⌈Δ⌉, 21 ν(L), 47 ω_{X}\{ \frac{−D}{N} \}, 67 τ_{≤1} F_{∗}Ω_{X/S}^{•} (log D)., 105 cd(X, D), 38 e(D), 67 e(L), 67 f-numerically effective, 59 f-semi-ample, 59 l-ample, 56 r(U), 17, 40 r(g), 40 C_{X}(D, N), 67 L^{(i)}, 19 L^{(i,D)}, 19 Čech complex, 148

Absolute Frobenius, 93 Acyclic resolution, 151 Adjoint linear systems on surfaces, 80 AKNV, 83 Albanese variety, 137 Analytic de Rham complex, 147

Bounds for e(L), 69

Cartier operator, 101 Cohomological dimension - cd(X, D), 38

- r(U), 17

- r(g), 40

- coherent, 38 Condition (∗), 16 Condition (!), 16 Connection

- logarithmic, 14 Covering construction - Kawamata, 30, 31

Cyclic cover, 22 - n-th root out of D, 22

- induced connection, 28 their residues, 28

- via geometric vector bundles, 27 - with quotient singularities, 34

cyclic cover - ramification index, 27

- singularities, 27

de Rham cohomology, 82 de Rham complex - logarithmic, 14

E_{1} degeneration, 19 Deformation

- of cohomology groups, 132

- of quotient singularities, 75 Degeneration

- of spectral sequences, 156

- of the Hodge to de Rham spectral

sequence, 121 for unitary local systems, 139

Differential forms - logarithmic, 11

exact sequences, 13

Filtrations - on hypercohomology groups, 151

General vanishing theorem - for cohomology groups, 39

- for restriction maps, 36, 38

- with analytic methods, 41 Generic vanishing

- Green Lazarsfeld, 137 \underline{166 INDEX}

Generic vanishing theorems - for nef Q l -divisors, 140

Hurwitz’s formula, 28

- generalized, 33 Hypercohomology group, 150

Iitaka-dimension, 44

- numerical, 47 Injective resolution, 150 Integral part of a Q l -divisor, 19 Isomorphism of liftings, 90

Kodaira-dimension, 44 - numerical, 47

Liftings of a scheme, 84

Multiplier ideals, 67

Numerically effective (nef), 45

One step filtration, 153

Poincaré bundle, 137

Quasi-isomorphism, 147

Reider’s theorem, 80 Relative Frobenius, 94 Relative vanishing theorem

- for f-numerically effective Q l -divisors, 59

- for Q l -divisors, 49

- for log differentials, 33 Residue map, 14

Second Witt vectors, 85 Semi-ample, 45 Semipositivity theorem

- Fujita, 73 Spectral sequence, 152

- conjugate, 158

- Hodge to de Rham, 82, 157

- Leray, 159 Splitting cohomology class, 108 Splitting of τ_{≤1}F_{∗}Ω_{X/S}^{•}(log D), 106, Surfaces of general type

- semi-ampleness of the canonical sheaf, 65

Tensor product of complexes, 149 Torsion freeness - Kollár, 60

Two step filtration, 153

Two term de Rham complex, 105

Vanishing theorem - Akizuki Kodaira Nakano, 4, 56 - Bauer Kosarew, 62

- Bogomolov Sommese, 58

- Deligne Illusie Raynaud, 83, 129 - for differential forms with values in l-ample sheaves, 56

- for direct images, 63

- for local systems, 17

- for logarithmic differential forms with values in Kodaira integral parts of Q l -divisors, 54

- for multiplier ideals, 63, 71

- for restriction maps related to Q l - divisors, 42, 49

- Grauert Riemenschneider, 45 - in characteristic p > 0, 43

- Kawamata Viehweg, 49

- Kodaira, 4

- Kollár, 45

- Serre, 4

Zeros of polynomials, 72