An Introduction to Intersection Homology Theory

Contents
1 Introduction
1.1 Poincaré duality
1.2 Morse theory for singular spaces
1.3 de Rham cohomology and L^2-cohomology
1.4 The cohomology of projective varieties
2 Review of homology and cohomology
2.1 Simplicial homology
2.2 Singular homology
2.3 Homology with closed support
2.4 Conclusion
2.5 Further reading
3 Review of sheaf cohomology and derived categories
3.1 Sheaves
3.2 Cech cohomology of sheaves
3.3 Hypercohomology
3.4 Functors and exactness
3.5 Resolutions of sheaves and of complexes
3.6 Cohomology and hypercohomology via derived functors
3.7 Derived categories
3.8 Right derived functors
3.9 Further reading
4 The definition of intersection homology
4.1 Stratified spaces and pseudomanifolds
4.2 Simplicial intersection homology
4.3 Singular intersection homology
4.4 Simple examples of intersection homology
4.5 Normalisations
4.6 Relative groups and the Mayer-Vietoris sequence
4.7 The intersection homology of a cone
4.8 Functoriality of intersection homology
4.9 Homology with local coefficients
4.10 Quasi-projective complex varieties
4.11 Further reading
5 Witt spaces and duality
5.1 Generalised Poincare duality
5.2 Witt spaces
5.3 Signatures of Witt spaces
5.4 The Witt-bordism groups
5.5 Further reading
6 L^2-cohomology and intersection cohomology
6.1 L^2-cohomology and Hodge theory
6.2 The L^2-cohomology of a punctured cone
6.3 Varieties with isolated conical singularities
6.4 Locally symmetric varieties
6.5 Further reading
7 Sheaf-theoretic intersection homology
7.1 Sheaves of singular chains
7.2 Constructibility and an axiomatic characterisation
7.3 The topological invariance of intersection homology
7.4 Duality in the derived category
7.5 Further reading
8 Perverse sheaves
8.1 Perverse sheaves
8.2 Perverse sheaves on varieties
8.3 Nearby and vanishing cycles
8.4 The decomposition theorem
8.5 Further reading
9 The intersection cohomology of fans
9.1 Affine toric varieties
9.2 Toric varieties from fans
9.3 Maps and torus actions
9.4 Projective toric varieties and convex polytopes
9.5 Stratifications of toric varieties
9.6 Subdivisions and desingularisations
9.7 Equivariant intersection cohomology
9.8 The intersection cohomology of fans
9.9 Stanley's conjectures
9.10 Further reading
10 Characteristic p and the Weil conjectures
10.1 Statement of the Weil conjectures
10.2 Basic properties of l-adic cohomology
10.3 Etale topology and cohomology
10.4 The Weil conjectures for singular varieties
10.5 Further reading
11 D-Modules and the Riemann-Hilbert correspondence
11.1 The Riemann-Hilbert problem
11.2 Differential systems over C^n
11.3 D_X-modules and intersection homology
11.4 The characteristic variety of a D_X-module
11.5 Holonomic differential systems
11.6 Examples of characteristic varieties
11.7 Left and right D_X-modules
11.8 Restriction of D_X-modules
11.9 Regular singularities
11.10 The Riemann-Hilbert correspondence
11.11 Further reading
12 The Kazhdan-Lusztig conjecture
12.1 Verma modules
12.2 D-modules over flag manifolds
12.3 Characteristic p
12.4 Hecke algebras and the Kazhdan-Lusztig polynomials
12.5 Further reading
Bibliography
Index