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Intersection Homology & Perverse Sheaves with Applications To LAUREN¸TIU G. MAXIM INTERSECTIONHOMOLOGY & PERVERSESHEAVES WITHAPPLICATIONSTOSINGULARITIES i Contents Preface ix 1 Topology of singular spaces: motivation, overview 1 1.1 Poincaré Duality 1 1.2 Topology of projective manifolds: Kähler package 4 Hodge Decomposition 5 Lefschetz hyperplane section theorem 6 Hard Lefschetz theorem 7 2 Intersection Homology: definition, properties 11 2.1 Topological pseudomanifolds 11 2.2 Borel-Moore homology 13 2.3 Intersection homology via chains 15 2.4 Normalization 25 2.5 Intersection homology of an open cone 28 2.6 Poincaré duality for pseudomanifolds 30 2.7 Signature of pseudomanifolds 32 3 L-classes of stratified spaces 37 3.1 Multiplicative characteristic classes of vector bundles. Examples 38 3.2 Characteristic classes of manifolds: tangential approach 40 ii 3.3 L-classes of manifolds: Pontrjagin-Thom construction 44 Construction 45 Coincidence with Hirzebruch L-classes 47 Removing the dimension restriction 48 3.4 Goresky-MacPherson L-classes 49 4 Brief introduction to sheaf theory 53 4.1 Sheaves 53 4.2 Local systems 58 Homology with local coefficients 60 Intersection homology with local coefficients 62 4.3 Sheaf cohomology 62 4.4 Complexes of sheaves 66 4.5 Homotopy category 70 4.6 Derived category 72 4.7 Derived functors 74 5 Poincaré-Verdier Duality 79 5.1 Direct image with proper support 79 5.2 Inverse image with compact support 82 5.3 Dualizing functor 83 5.4 Verdier dual via the Universal Coefficients Theorem 85 5.5 Poincaré and Alexander Duality on manifolds 86 5.6 Attaching triangles. Hypercohomology long exact sequences of pairs 87 6 Intersection homology after Deligne 89 6.1 Introduction 89 6.2 Intersection cohomology complex 90 6.3 Deligne’s construction of intersection homology 94 6.4 Generalized Poincaré duality 98 6.5 Topological invariance of intersection homology 101 6.6 Rational homology manifolds 103 6.7 Intersection homology Betti numbers, I 106 iii 7 Constructibility in algebraic geometry 111 7.1 Definition. Properties 111 7.2 Local calculus 113 7.3 Euler characteristics of constructible complexes. Applications 116 8 Perverse sheaves 121 8.1 Definition, examples 122 8.2 Gluing of t-structures 126 8.3 Examples of perverse sheaves 129 8.4 Intermediate extension 132 8.5 A splitting criterion for perverse sheaves 135 8.6 Artin’s Vanishing Theorem 137 9 The Decomposition Package and Applications 141 9.1 Lefschetz hyperplane section theorem 141 9.2 Hard Lefschetz Theorem for intersection cohomology 143 9.3 The BBDG Decomposition Theorem. Applications 144 Stratifications of algebraic maps 145 Deligne’s decomposition theorem 145 Semi-small maps 147 The decomposition theorem for semi-small maps 151 The BBDG decomposition theorem for arbitrary maps 154 First applications of the Decomposition Theorem 155 Set of supports of an algebraic map 160 9.4 Applications of the decomposition package 161 Topology of Hilbert schemes of points on a smooth complex surface 162 Stanley’s proof of McMullen’s conjecture 165 Huh-Wang’s proof of Dowling-Wilson’s conjecture 168 iv 10 Hypersurface singularities. Nearby and vanishing cycles 171 10.1 Brief overview of complex hypersurface singularities 171 10.2 Global aspects of hypersurface singularities 179 10.3 Nearby and vanishing cycles 186 Construction 187 Relation with perverse sheaves 192 Thom-Sebastiani for vanishing cycles 193 On Euler characteristic computations 195 10.4 Euler characteristics of complex projective hypersurfaces 196 10.5 Generalized Riemann-Hurwitz-type formulae 201 10.6 Homological connectivity of Milnor fiber and link of hypersurface singularity germs 203 10.7 Canonical and variation morphisms. 205 11 Overview of Saito’s mixed Hodge modules, and immediate applications 209 11.1 Classical Hodge theory 209 11.2 Mixed Hodge modules 215 11.3 Hodge theory on intersection cohomology groups 222 11.4 Intersection homology Betti numbers, II 228 12 Epilogue 233 12.1 Applications to enumerative geometry 233 12.2 Characteristic classes of complex algebraic varieties and applications 234 12.3 Perverse sheaves on semi-abelian varieties: cohomology jump loci, propagation, generic van- ishing. 236 12.4 Generic vanishing theory via mixed Hodge modules 239 12.5 Applications to representation theory 241 12.6 Alexander-type invariants of complex hypersurface complements 241 Bibliography 243 Index 263 v List of Figures Figure 1.1: wedge product................... 3 Figure 1.2: normalization................... 4 Figure 1.3: pinched torus.................... 6 Figure 1.4: hyperplane section................. 7 Figure 2.1: open cone...................... 11 Figure 2.2: not a pseudomanifold............... 13 Figure 2.3: X = S(S1 t S1) ................... 20 Figure 2.4: dual block of a simplex.............. 22 Figure 2.5: pinched torus.................... 23 Figure 2.6: coning off the boundary............. 24 Figure 2.7: normalization of pinched torus......... 26 Figure 2.8: coning off the boundary............. 35 Figure 2.9: pinch bordism................... 35 Figure 10.1: Milnor fibration.................. 176 vii Dedicated to my family: Bridget, Juliana and Alex. ix Preface In recent years, intersection homology and perverse sheaves have be- come indispensable tools for studying the topology of singular spaces. This book provides a gentle introduction of these concepts, with an emphasis on geometric examples and applications. Part of the motivation for the development of intersection homology is that the main results and properties of manifolds (such as Poincaré du- ality, existence of multiplicative characteristic class theories, Lefschetz- type theorems and Hodge theory for complex algebraic varieties, Morse theory, etc.) fail to be true for singular spaces when considering usual homology. Intersection homology was introduced by M. Goresky and R. MacPherson in 1974 for the purpose of recovering such properties and results when dealing with singular spaces.1 1 In fact, according to [Kleiman, 2007], The first part of these notes provides an elementary introduction of Goresky and MacPherson were initially seeking a theory of characteristic num- intersection homology and some of its applications. We first recall the bers for complex analytic varieties and results and main properties in the manifold case, then show to what other singular spaces. extent the intersection homology allows to recover these properties in the singular case. The guiding principle of these notes is to provide an explicit and geometric introduction of the mathematical objects that are defined in this context, as well as to present some of the most significant examples. The basic idea of intersection homology is that, if one wants to recover classical properties of homology (e.g., Poincaré duality) in the case of singular spaces, one has to consider only cycles that meet the singularities with a controlled defect of transversality (encoded by a perversity function). This approach is explained in Chapter 2. As an application of Poincaré duality, in Chapter 3 we explain how the duality pairing on the middle-perversity intersection homology groups and its associated signature invariant can be used to construct characteristic L-classes in the singular setting. One of the most important (and intriguing) properties of intersection homology is the local calculus. In fact, this is what distinguishes intersection homology theory from classical homology theory, in the sense that intersection homology is not a homotopy invariant. The local calculus also provides the transition to sheaf theory, and motivates the x second part of these notes, which presents a sheaf-theoretic description of intersection homology. A second definition of intersection homology makes use of sheaf theory and homological algebra, and it was introduced by Goresky and MacPherson in [Goresky and MacPherson, 1983a], following a suggestion of Deligne. In Chapters 4 and 5, we develop the necessary background on sheaves needed to define Deligne’s intersection coho- mology complex, whose (hyper)cohomology computes the intersection homology groups. This complex of sheaves, introduced in Chapter 6, can be described axiomatically in a way that is independent of the stratification or any additional geometric structure (such as a piecewise linear structure), leading to a proof of the topological invariance of intersection homology groups. In complex algebraic geometry, the middle-perversity Deligne inter- section cohomology complex is a basic example of a perverse sheaf. Perverse sheaves are fundamental objects at the crossroads of topology, algebraic geometry, analysis and differential equations, with notable applications in number theory, algebra and representation theory. For instance, perverse sheaves have seen striking applications in represen- tation theory (proof of the Kazhdan-Lusztig conjecture, proof of the fundamental lemma in the Langlands program, etc.), and in geometry and topology (the BBDG Decomposition theorem). They also form the backbone of Saito’s mixed Hodge module theory. However, de- spite their fundamental importance, perverse sheaves remain rather mysterious objects. After a quick introduction of the theory of constructible sheaves in complex algebraic geometry (Chapter 7), we present a down-to-earth treatment of the deep theory of perverse sheaves (Chapter 8), with an emphasis on basic geometric examples. Of particular importance here is Artin’s vanishing theorem for perverse sheaves on complex affine varieties, which plays an essential role in proving the Lefschetz hy- perplane section theorem for intersection homology
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