An Introduction to Intersection Homology Theory

Contents......Page 10
1 Introduction......Page 13
1.1 Poincaré duality......Page 15
1.2 Morse theory for singular spaces......Page 16
1.3 de Rham cohomology and L^2-cohomology......Page 19
1.4 The cohomology of projective varieties......Page 22
2.1 Simplicial homology......Page 26
2.2 Singular homology......Page 29
2.3 Homology with closed support......Page 31
2.4 Conclusion......Page 33
2.5 Further reading......Page 34
3.1 Sheaves......Page 35
3.2 Cech cohomology of sheaves......Page 39
3.3 Hypercohomology......Page 43
3.4 Functors and exactness......Page 44
3.5 Resolutions of sheaves and of complexes......Page 48
3.6 Cohomology and hypercohomology via derived functors......Page 52
3.7 Derived categories......Page 53
3.8 Right derived functors......Page 56
3.9 Further reading......Page 57
4.1 Stratified spaces and pseudomanifolds......Page 58
4.2 Simplicial intersection homology......Page 60
4.4 Simple examples of intersection homology......Page 63
4.5 Normalisations......Page 65
4.6 Relative groups and the Mayer-Vietoris sequence......Page 66
4.7 The intersection homology of a cone......Page 67
4.8 Functoriality of intersection homology......Page 70
4.9 Homology with local coefficients......Page 72
4.10 Quasi-projective complex varieties......Page 73
4.11 Further reading......Page 80
5.1 Generalised Poincare duality......Page 82
5.2 Witt spaces......Page 84
5.3 Signatures of Witt spaces......Page 87
5.4 The Witt-bordism groups......Page 88
5.5 Further reading......Page 92
6.1 L^2-cohomology and Hodge theory......Page 94
6.2 The L^2-cohomology of a punctured cone......Page 97
6.3 Varieties with isolated conical singularities......Page 102
6.4 Locally symmetric varieties......Page 105
6.5 Further reading......Page 110
7.1 Sheaves of singular chains......Page 111
7.2 Constructibility and an axiomatic characterisation......Page 115
7.3 The topological invariance of intersection homology......Page 118
7.4 Duality in the derived category......Page 121
7.5 Further reading......Page 124
8.1 Perverse sheaves......Page 125
8.2 Perverse sheaves on varieties......Page 129
8.3 Nearby and vanishing cycles......Page 130
8.4 The decomposition theorem......Page 133
8.5 Further reading......Page 138
9.1 Affine toric varieties......Page 140
9.2 Toric varieties from fans......Page 144
9.3 Maps and torus actions......Page 145
9.4 Projective toric varieties and convex polytopes......Page 147
9.5 Stratifications of toric varieties......Page 150
9.6 Subdivisions and desingularisations......Page 151
9.7 Equivariant intersection cohomology......Page 153
9.8 The intersection cohomology of fans......Page 157
9.9 Stanley's conjectures......Page 165
9.10 Further reading......Page 168
10.1 Statement of the Weil conjectures......Page 169
10.2 Basic properties of l-adic cohomology......Page 172
10.3 Etale topology and cohomology......Page 174
10.4 The Weil conjectures for singular varieties......Page 177
10.5 Further reading......Page 179
11.1 The Riemann-Hilbert problem......Page 181
11.2 Differential systems over C^n......Page 184
11.3 D_X-modules and intersection homology......Page 186
11.4 The characteristic variety of a D_X-module......Page 188
11.5 Holonomic differential systems......Page 191
11.6 Examples of characteristic varieties......Page 192
11.7 Left and right D_X-modules......Page 195
11.8 Restriction of D_X-modules......Page 196
11.9 Regular singularities......Page 199
11.10 The Riemann-Hilbert correspondence......Page 201
11.11 Further reading......Page 203
12.1 Verma modules......Page 205
12.2 D-modules over flag manifolds......Page 209
12.3 Characteristic p......Page 212
12.4 Hecke algebras and the Kazhdan-Lusztig polynomials......Page 213
12.5 Further reading......Page 215
Bibliography......Page 217
Index......Page 231