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