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 = Σ(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 in the subsequent chapter. Chapter 9 is devoted to what is usually referred to as the “decom- position package”, consisting of Lefschetz-type results for perverse sheaves and intersection homology (Section 9.1), as well as the BBDG Decomposition theorem (Section 9.3). The Beilinson-Bernstein-Deligne- Gabber Decomposition theorem is one of the most important results of the theory of perverse sheaves, and it contains as special cases some of the deepest homological and topological properties of algebraic maps. Since its proof in 1981, the Decomposition theorem has found spectac- ular applications in algebraic topology and geometry, number theory, representation theory and combinatorics. In Section 9.3, we give a brief overview of the motivation and the main ideas of its proof, and dis- xi

cuss some of its immediate consequences. Furthermore, in Section 9.4, we sample several of the numerous applications of the decomposition package. We begin with a computation of topological invariants of Hilbert schemes of points on a surface, then move to combinatorial applications and overview Stanley’s proof of McMullen’s conjecture (about a complete characterization of face vectors of simplicial poly- topes) as well as Huh-Wang’s recent resolution of the Dowling-Wilson top-heavy conjecture (on the enumeration of subspaces of a projective space generated by a finite set of points). In Chapter 10, we indicate several applications of perverse sheaves to the study of local and global topology of complex hypersurface singularities. In Section 10.1 we give a brief overview of the local topological structure of hypersurface singularities. Global topological aspects of complex hypersurfaces and of their complements are dis- cussed in Section 10.2 by means of Alexander-type invariants inspired by knot theory. The nearby and vanishing cycle functors, introduced in Section 10.3, are used to glue the local topological data around singular- ities into constructible complexes of sheaves. We also discuss here the interplay between nearby/vanishing cycles and perverse sheaves. Very concrete applications of the nearby and vanishing cycles are presented in Section 10.4 (to the computation of Euler characteristics of com- plex projective hypersurfaces), Section 10.5 (for obtaining generalized Riemann-Hurwitz-type formulae), and in Section 10.6 (for deriving homological connectivity statements for the local topology of complex singularities). Chapter 11 gives a quick introduction of Saito’s theory of mixed Hodge modules, with an emphasis on concrete applications to Hodge- theoretical aspects of intersection homology. Mixed Hodge modules are extensions in the singular context of variations of mixed Hodge structures, and can be regarded, informally, as sheaves of mixed Hodge structures. Section 11.1 reviews some of the main concepts and results from the classical mixed Hodge theory, due to Deligne. In Section 11.2, we discuss the basic calculus of mixed Hodge modules and discuss some basic examples. In Section 11.3, we explain how to use Saito’s mixed Hodge module theory to construct mixed Hodge structures on the intersection cohomology groups of complex algebraic varieties and, respectively, of links of closed subvarieties. We also show that the generalized Poincaré duality isomorphism in intersection homology is compatible with these mixed Hodge structures. Each of the main actors of these notes, namely, intersection homol- ogy, perverse sheaves and mixed Hodge modules, is at the center of a large and growing subject, touching on many aspects of modern mathematics. As a consequence, there is a vast research literature. In the Epilogue, we provide a succinct summary of (and references for) xii

some of the recent applications of these theories (other than those al- ready discussed in earlier chapters) in various research fields such as topology, algebraic geometry, representation theory and number theory. This list of applications is by no means exhaustive, but rather reflects the author’s own research interests and mathematical taste. While the discussion will be limited to a small fraction of the possible routes the interested reader might explore, it should nevertheless serve as a starting point for those interested in aspects of intersection homology and perverse sheaves in other areas than those already considered in the text. This book is intended as a broadly accessible first introduction to intersection homology and perverse sheaves, and it is far from compre- hensive. In order to keep the size of the material within a reasonable level, many important results are stated without proof (whenever this is the case, a reference is provided), while some of their applications are emphasized. The goal of these notes is not necessarily to introduce readers to the general abstract theory, but to provide them with a taste of the subject by presenting concrete examples and applications that motivate the theory. At the end of this journey, readers should feel comfortable enough to delve further into more specialized topics and to explore problems of current research. For more complete details and further reading, the interested reader is advised to consult standard references such as [Borel, 2008], [Ba- nagl, 2007], [Dimca, 2004], [Friedman], [Kashiwara and Schapira, 1994], [Schürmann, 2003]. For a nice account on the history of intersection homology and its connections with various problems in mathemat- ics, see [Kleiman, 2007]. For excellent overviews of perverse sheaves and their many applications, the two ICM addresses [Lusztig, 1991] and [MacPherson, 1984], as well as the more recent [de Cataldo and Migliorini, 2009], are highly recommended. While the text presented here has a sizable (and unavoidable) overlap with some of the above- mentioned references (especially on background material and classical aspects of the theory), it also complements them in terms of the range of applications and/or the level of detail. Throughout these notes, we assume the reader has a certain familiar- ity with basic concepts of Algebraic Topology and Algebraic Geometry. While many of the relevant notions are still defined in the text (often in the form of footnotes), the novice reader is expected to consult standard textbooks on these subjects, such as [Hatcher, 2002], [Griffiths and Harris, 1994] or [Hartshorne, 1977]. Acknowledgements. These notes grew out of lectures given by the author at the University of Illinois at Chicago, University of Wisconsin- Madison, University of Science and Technology of China (USTC) in xiii

Hefei, China, and the Chinese University of Hong Kong (CUHK). In particular, I would like to thank my hosts, Xiuxiong Chen at USTC and Conan Leung at CUHK, for excellent working conditions. I also thank my students, colleagues and collaborators for valuable feedback, and I thank Mr. Okan Akalin for helping me with drawing the pictures. Special thanks go to Jörg Schürmann, Julius Shaneson, Sylvain Cappell, Alex Dimca, Markus Banagl, Botong Wang, Yongqiang Liu, Greg Fried- mann, Shoji Yokura, Mark Andrea de Cataldo and David Massey for many enlightening discussions and for reading parts of earlier versions of the manuscript. During the writing of this book I was partially supported by the Simons Foundation Collaboration Grant #567077, and by the Romanian Ministry of National Education, CNCS-UEFISCDI, grant PN-III-P4-ID- PCE-2016-0030.

Madison, November 2018