Intersection homology & perverse sheaves. With applications to singularities. (English) Zbl 1476.55001

Maxim, Laurenţiu G.

Graduate Texts in Mathematics 281. Cham: Springer (ISBN 978-3-030-27643-0/hbk; 978-3-030-27644-7/ebook). xv, 270 p. (2019).

This book is a welcome addition to the family of introductions to intersection cohomology and perverse sheaves. This is an active research area with a broad range of applications to topology, analytic and algebraic geometry, representation theory and combinatorics. Given this scope, there is plenty of room for introductory texts at the graduate level ranging from comprehensive technical treatises on aspects of the theory such as [A. Borel et al., Intersection cohomology. Boston-Basel-Stuttgart: Birkhäuser (1984; Zbl 0553.14002); M. Banagl, Topological invariants of stratified spaces. Berlin: Springer (2007; Zbl 1108.55001); G. Friedman, Singular intersection homology. Cambridge: Cambridge University Press (2020; Zbl 1472.55001)], to those which survey a broader area in less detail such as [F. Kirwan and J. Woolf, An introduction to intersection homology theory. Boca Raton, FL: Chapman & Hall/CRC (2006; Zbl 1106.55001)]. This book is pitched in the middle, judiciously omitting the details of more technical proofs but containing detailed treatments of other topics. In this respect it is comparable to books such as [A. Dimca, Sheaves in topology. Berlin: Springer (2004; Zbl 1043.14003)]. However the route taken through the background material, and the choice of topics and applications, are different.
The main thrust of the technical exposition parallels the historical development of the area in the 1970s and 80s, starting with simplicial intersection homology groups, moving on to their sheaf-theoretic definition and the notion of perverse sheaf, and ending with an introduction to Mixed Hodge Modules. The aim of introducing all this theory is to apply it to the topology of singular complex varieties, and the author takes care to introduce and motivate the main objects of study with geometric examples. There are also regular exercises which will help readers come to grips with the material.
In summary, this book will appeal to, and be a very useful resource for, graduate students and researchers with a modest background in algebraic topology and geometry who wish to obtain a sound working knowledge of the area with a view to applications in complex algebraic geometry.
In more detail the contents are as follows.

Chapter 1

‘Topology of singular spaces: motivation, overview’ is a lightning review of topological manifolds and Poincaré duality, and then of the Kähler package (Hodge decomposition, Lefschetz Hyperplane and Hard Lefschetz Theorems) for nonsingular complex projective varieties, illustrated by examples of singular spaces for which these results fail.

Chapter 2

‘Intersection homology: definition, properties’ introduces the intersection homology groups of (PL) pseudomanifolds via complexes of PL chains, very much in the spirit of the original paper [M. Goresky and R. MacPherson, Topology 19, 135–165 (1980; Zbl 0448.55004)]. It includes short sections on the intersection homology of normalizations and open cones, on (generalized) Poincaré Duality and on the signatures of Witt spaces.

Chapter 3

‘L-classes of stratified spaces’ extends the final topic of Chapter 2 by using the signature to construct L-classes of Witt spaces. It begins with a very rapid review of cohomological characteristic classes of vector bundles (on smooth manifolds) followed by a description of the Pontrjagin-Thom construction of L-classes in homology which are Poincaré dual to the Hirzebruch L-classes. This construction is then extended to sketch the construction of the Goresky-MacPherson L-class in intersection homology.

Chapter 4

‘Brief introduction to sheaf theory’ is a modest title for a chapter which in 27 pages goes from the definition of a sheaf to a description of the derived category of sheaves on a topological space! After the initial section defining sheaves of modules and the standard functors, it treats local systems and homology with local coefficients, sheaf cohomology (via injective resolutions), complexes of sheaves and hypercohomology, the homotopy and derived categories, and finally derived functors.

Chapter 5

‘Poincaré-Verdier duality’ revisits Poincaré Duality in Verdier’s sheaf-theoretic formulation. It explains how the inverse image with compact support and the dualizing functor are constructed, and how the Poincaré and Alexander Duality Theorems quickly follow from their properties. The final section introduces the dual pair of ‘attaching triangles’ arising from a decomposition of a space into complementary open and closed subspaces.

Chapter 6

‘Intersection homology after Deligne’ introduces the intersection cohomology complex (obtained by sheafifying the intersection chains) and Deligne’s complex. These are shown to be quasi-isomorphic using the axiomatic approach of [M. Goresky and R. MacPherson, Invent. Math. 72, 77–129 (1983; Zbl 0529.55007)]. This leads to short proofs of the Generalized Poincaré Duality Theorem and the topological invariance of intersection homology. The chapter ends with a discussion of the intersection homology of rational homology manifolds and a first look at the intersection Betti numbers and Euler characteristics of complex algebraic varieties.

Chapter 7

‘Constructibility in Algebraic Geometry’ introduces constructibility for sheaf complexes, collating some important properties such as its preservation under Grothendieck’s six functors. This is followed by a treatment of the local calculus, i.e.stalks, and Euler characteristics of constructible complexes on complex algebraic varieties.

Chapter 8

‘Perverse sheaves’ defines the abelian category of (middle perversity) perverse sheaves on a complex algebraic or analytic variety first via vanishing conditions and then via gluing of t-structures. After some examples including perverse sheaves on complete intersections, the intermediate extension functor is studied, an important splitting criterion for perverse sheaves proved, and Artin’s Vanishing Theorem for perverse sheaves on an affine variety deduced from the t-exactness properties of affine morphisms.

Chapter 9

‘The Decomposition Package and Applications’ is a nice and very well-motivated account of the Lefschetz Hyperplane and Hard Lefschetz theorems for intersection cohomology as well as Beilinson, Bernstein, Deligne and Gabber’s Decomposition Theorem. The presentation follows the geometric proof of the Decomposition Theorem in [M. de Cataldo and L. Migliorini, Ann. Sci. Éc. Norm. Supér. (4) 38, No. 5, 693–750 (2005; Zbl 1094.14005)] with an extensive discussion of semi-small maps. The last section of the chapter highlights some of the important applications in algebraic geometry, and also in combinatorics (McMullen’s Conjecture) and enumerative geometry (Dowling-Wilson’s Top-Heavy Conjecture).

Chapter 10

‘Hypersurface Singularities. Nearby and Vanishing Cycles’ is a substantial discussion of the role of perverse sheaves in singularity theory. It starts with the foundational notions of Milnor, and Milnor-Lê, fibrations and then relates local invariants at the singularities of the hypersurface to the topology of the complement. This leads naturally to a discussion of nearby and vanishing cycles and the associated functors between categories of perverse sheaves. These are then applied to Euler characteristic calculations for complex hypersurfaces and Riemann-Hurwitz type formulae. The chapter ends by introducing the canonical and variation morphisms.

Chapter 11

‘Overview of Saito’s Mixed Hodge Modules, and Immediate Applications’ is a rapid introduction to the theory of mixed Hodge modules. It skirts the formidable technicalities involved in the definition by giving clear statements of the formal structure of the theory, and motivating these with plenty of examples connecting this with the classical notions of Hodge theory and variations of Hodge structures. The formal properties are used to explain how to construct mixed Hodge structures on intersection cohomology groups, in particular treating Durfee-Saito’s semi-purity theorem for the intersection homology groups of the link of a complex subvariety. The chapter concludes with an application to the computation of intersection Betti numbers of complex varieties.

Chapter 12

‘Epilogue’ summarises several interesting and active areas of research which build on the material in the book. The choice of topics is guided by the author’s taste, but is quite broad, encompassing Donaldson-Thomas invariants, characteristic classes of complex varieties, perverse sheaves on semi-abelian varieties, cohomological jump loci for holomorphic forms, and some pointers towards the wide-ranging and deep applications to representation theory and singularity theory.

Reviewer: Jon Woolf (Liverpool)

MSC:

55-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology
14-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry
55N33	Intersection homology and cohomology in algebraic topology
14F43	Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)

Keywords:

intersection cohomology; stratified spaces; Poincaré duality; Witt spaces; L-classes; constructible sheaves; Poincaré-Verdier duality; perverse sheaves; singularities; vanishing theorems; hypersurface singularities; Hodge theory; mixed Hodge modules

Citations:

Zbl 0553.14002; Zbl 1108.55001; Zbl 1106.55001; Zbl 1043.14003; Zbl 0448.55004; Zbl 0529.55007; Zbl 1094.14005; Zbl 1472.55001