Intersection Homology & Perverse Sheaves: with Applications to Singularities

Preface......Page 7
Contents......Page 11
List of Figures......Page 14
1.1 Poincaré Duality......Page 15
1.2 Topology of Projective Manifolds: Kähler Package......Page 18
Hodge Decomposition......Page 19
Lefschetz Hyperplane Section Theorem......Page 20
Hard Lefschetz Theorem......Page 22
2.1 Topological Pseudomanifolds......Page 24
2.2 Borel–Moore Homology......Page 26
2.3 Intersection Homology via Chains......Page 29
2.4 Normalization......Page 39
2.5 Intersection Homology of an Open Cone......Page 42
2.6 Poincaré Duality for Pseudomanifolds......Page 44
2.7 Signature of Pseudomanifolds......Page 45
3 L-Classes of Stratified Spaces......Page 50
3.1 Multiplicative Characteristic Classes of Vector Bundles: Examples......Page 51
3.2 Characteristic Classes of Manifolds: Tangential Approach......Page 53
3.3 L-Classes of Manifolds: Pontrjagin–Thom Construction......Page 57
Construction......Page 58
Coincidence with Hirzebruch L-Classes......Page 60
Removing the Dimension Restriction......Page 61
3.4 Goresky–MacPherson L-Classes......Page 62
4.1 Sheaves......Page 65
4.2 Local Systems......Page 70
Homology with Local Coefficients......Page 72
Intersection Homology with Local Coefficients......Page 74
4.3 Sheaf Cohomology......Page 75
4.4 Complexes of Sheaves......Page 78
4.5 Homotopy Category......Page 83
4.6 Derived Category......Page 85
4.7 Derived Functors......Page 88
5.1 Direct Image with Proper Support......Page 93
5.2 Inverse Image with Compact Support......Page 97
5.3 Dualizing Functor......Page 98
5.4 Verdier Dual via the Universal Coefficient Theorem......Page 100
5.5 Poincaré and Alexander Duality on Manifolds......Page 101
5.6 Attaching Triangles. Hypercohomology Long Exact Sequences of Pairs......Page 103
6.1 Introduction......Page 105
6.2 Intersection Cohomology Complex......Page 106
6.3 Deligne's Construction of Intersection Homology......Page 110
6.4 Generalized Poincaré Duality......Page 114
6.5 Topological Invariance of Intersection Homology......Page 118
6.6 Rational Homology Manifolds......Page 120
6.7 Intersection Homology Betti Numbers, I......Page 123
7.1 Definition: Properties......Page 129
7.2 Local Calculus......Page 131
7.3 EulerCharacteristicsofConstructibleComplexes. Applications......Page 134
8 Perverse Sheaves......Page 141
8.1 Definition, Examples......Page 142
8.2 Gluing of t-Structures......Page 147
8.3 Examples of Perverse Sheaves......Page 149
8.4 Intermediate Extension......Page 152
8.5 A Splitting Criterion for Perverse Sheaves......Page 156
8.6 Artin's Vanishing Theorem......Page 158
9.1 Lefschetz Hyperplane Section Theorem......Page 161
9.2 Hard Lefschetz Theorem for Intersection Cohomology......Page 163
Stratifications of Algebraic Maps......Page 165
Deligne's Decomposition Theorem......Page 166
Semi-Small Maps......Page 168
The Decomposition Theorem for Semi-Small Maps......Page 172
The BBDG Decomposition Theorem for Arbitrary Maps......Page 174
First Applications of the Decomposition Theorem......Page 176
Set of Supports of an Algebraic Map......Page 181
9.4 Applications of the Decomposition Package......Page 182
Topology of Hilbert Schemes of Points on a Smooth Complex Surface......Page 183
Stanley's Proof of McMullen's Conjecture......Page 186
Huh–Wang's Proof of Dowling–Wilson's Conjecture......Page 189
10.1 Brief Overview of Complex Hypersurface Singularities......Page 192
10.2 Global Aspects of Hypersurface Singularities......Page 200
10.3 Nearby and Vanishing Cycles......Page 207
Construction......Page 208
Relation with Perverse Sheaves......Page 213
Thom–Sebastiani for Vanishing Cycles......Page 214
On Euler Characteristic Computations......Page 216
10.4 Euler Characteristics of Complex Projective Hypersurfaces......Page 218
10.5 Generalized Riemann–Hurwitz-Type Formulae......Page 222
10.6 Homological Connectivity of Milnor Fiber and Link of Hypersurface Singularity Germs......Page 225
10.7 Canonical and Variation Morphisms......Page 227
11.1 Classical Hodge Theory......Page 231
11.2 Mixed Hodge Modules......Page 237
11.3 Hodge Theory on Intersection Cohomology Groups......Page 245
11.4 Intersection Homology Betti Numbers, II......Page 251
12.1 Applications to Enumerative Geometry......Page 255
12.2 Characteristic Classes of Complex Algebraic Varieties and Applications......Page 257
12.3 Perverse Sheaves on Semi-Abelian Varieties: Cohomology Jump Loci, Propagation, Generic Vanishing......Page 258
12.4 Generic Vanishing Theory via Mixed Hodge Modules......Page 261
12.6 Alexander-Type Invariants of Complex Hypersurface Complements......Page 263
Bibliography......Page 264
Index......Page 275