The Development of Intersection Homology Theory Steven L

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The Development of Intersection Homology Theory

Steven L. Kleiman

Contents Foreword

Received October 9, 2006.

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Preface

The Development of Intersection Homology Theory

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Discovery

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Other work and open problems

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Recommended publications

Perverse Sheaves

Perverse Sheaves Bhargav Bhatt Fall 2015 1 September 8, 2015 The goal of this class is to introduce perverse sheaves, and how to work with it; plus some applications. Background For more background, see Kleiman's paper entitled \The development/history of intersection homology theory". On manifolds, the idea is that you can intersect cycles via Poincaréduality|we want to be able to do this on singular spces, not just manifolds. Deligne figured out how to compute intersection homology via sheaf cohomology, and does not use anything about cycles|only pullbacks and truncations of complexes of sheaves. In any derived category you can do this|even in characteristic p. The basic summary is that we define an abelian subcategory that lives inside the derived category of constructible sheaves, which we call the category of perverse sheaves. We want to get to what is called the decomposition theorem. Outline of Course 1. Derived categories, t-structures 2. Six Functors 3. Perverse sheaves—definition, some properties 4. Statement of decomposition theorem|\yoga of weights" 5. Application 1: Beilinson, et al., \there are enough perverse sheaves", they generate the derived category of constructible sheaves 6. Application 2: Radon transforms. Use to understand monodromy of hyperplane sections. 7. Some geometric ideas to prove the decomposition theorem. If you want to understand everything in the course you need a lot of background. We will assume Hartshorne- level algebraic geometry. We also need constructible sheaves|look at Sheaves in Topology. Problem sets will be given, but not collected; will be on the webpage. There are more references than BBD; they will be online.
Intersection Cohomology Invariants of Complex Algebraic Varieties

Contemporary Mathematics Intersection cohomology invariants of complex algebraic varieties Sylvain E. Cappell, Laurentiu Maxim, and Julius L. Shaneson Dedicated to LêD˜ungTrángon His 60th Birthday Abstract. In this note we use the deep BBDG decomposition theorem in order to give a new proof of the so-called “stratified multiplicative property” for certain intersection cohomology invariants of complex algebraic varieties. 1. Introduction We study the behavior of intersection cohomology invariants under morphisms of complex algebraic varieties. The main result described here is classically referred to as the “stratified multiplicative property” (cf. [CS94, S94]), and it shows how to compute the invariant of the source of a proper algebraic map from its values on various varieties that arise from the singularities of the map. For simplicity, we consider in detail only the case of Euler characteristics, but we will also point out the additions needed in the arguments in order to make the proof work in the Hodge-theoretic setting. While the study of the classical Euler-Poincarécharacteristic in complex al- gebraic geometry relies entirely on its additivity property together with its mul- tiplicativity under fibrations, the intersection cohomology Euler characteristic is studied in this note with the aid of a deep theorem of Bernstein, Beilinson, Deligne and Gabber, namely the BBDG decomposition theorem for the pushforward of an intersection cohomology complex under a proper algebraic morphism [BBD, CM05]. By using certain Hodge-theoretic aspects of the decomposition theorem (cf. [CM05, CM07]), the same arguments extend, with minor additions, to the study of all Hodge-theoretic intersection cohomology genera (e.g., the Iχy-genus or, more generally, the intersection cohomology Hodge-Deligne E-polynomials).
Homology Stratifications and Intersection Homology 1 Introduction

ISSN 1464-8997 (on line) 1464-8989 (printed) 455 Geometry & Topology Monographs Volume 2: Proceedings of the Kirbyfest Pages 455–472 Homology stratifications and intersection homology Colin Rourke Brian Sanderson Abstract A homology stratification is a filtered space with local ho- mology groups constant on strata. Despite being used by Goresky and MacPherson [3] in their proof of topological invariance of intersection ho- mology, homology stratifications do not appear to have been studied in any detail and their properties remain obscure. Here we use them to present a simplified version of the Goresky–MacPherson proof valid for PL spaces, and we ask a number of questions. The proof uses a new technique, homology general position, which sheds light on the (open) problem of defining generalised intersection homology. AMS Classification 55N33, 57Q25, 57Q65; 18G35, 18G60, 54E20, 55N10, 57N80, 57P05 Keywords Permutation homology, intersection homology, homology stratification, homology general position Rob Kirby has been a great source of encouragement. His help in founding the new electronic journal Geometry & Topology has been invaluable. It is a great pleasure to dedicate this paper to him. 1 Introduction Homology stratifications are filtered spaces with local homology groups constant on strata; they include stratified sets as special cases. Despite being used by Goresky and MacPherson [3] in their proof of topological invariance of intersec- tion homology, they do not appear to have been studied in any detail and their properties remain obscure. It is the purpose of this paper is to publicise these neglected but powerful tools. The main result is that the intersection homology groups of a PL homology stratification are given by singular cycles meeting the strata with appropriate dimension restrictions.
The Decomposition Theorem, Perverse Sheaves and the Topology Of

The decomposition theorem, perverse sheaves and the topology of algebraic maps Mark Andrea A. de Cataldo and Luca Migliorini∗ Abstract We give a motivated introduction to the theory of perverse sheaves, culminating in the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of algebraic varieties. While most proofs are omitted, we discuss several approaches to the decomposition theorem, indicate some important applications and examples. Contents 1 Overview 3 1.1 The topology of complex projective manifolds: Lefschetz and Hodge theorems 4 1.2 Families of smooth projective varieties . ........ 5 1.3 Singular algebraic varieties . ..... 7 1.4 Decomposition and hard Lefschetz in intersection cohomology . 8 1.5 Crash course on sheaves and derived categories . ........ 9 1.6 Decomposition, semisimplicity and relative hard Lefschetz theorems . 13 1.7 InvariantCycletheorems . 15 1.8 Afewexamples.................................. 16 1.9 The decomposition theorem and mixed Hodge structures . ......... 17 1.10 Historicalandotherremarks . 18 arXiv:0712.0349v2 [math.AG] 16 Apr 2009 2 Perverse sheaves 20 2.1 Intersection cohomology . 21 2.2 Examples of intersection cohomology . ...... 22 2.3 Definition and first properties of perverse sheaves . .......... 24 2.4 Theperversefiltration . .. .. .. .. .. .. .. 28 2.5 Perversecohomology .............................. 28 2.6 t-exactness and the Lefschetz hyperplane theorem . ...... 30 2.7 Intermediateextensions . 31 ∗Partially supported by GNSAGA and PRIN 2007 project “Spazi di moduli e teoria di Lie” 1 3 Three approaches to the decomposition theorem 33 3.1 The proof of Beilinson, Bernstein, Deligne and Gabber .
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.
Characteristic Classes of Mixed Hodge Modules

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The Decomposition Theorem, Perverse Sheaves and the Topology of Algebraic Maps

BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 46, Number 4, October 2009, Pages 535–633 S 0273-0979(09)01260-9 Article electronically published on June 26, 2009 THE DECOMPOSITION THEOREM, PERVERSE SHEAVES AND THE TOPOLOGY OF ALGEBRAIC MAPS MARK ANDREA A. DE CATALDO AND LUCA MIGLIORINI Abstract. We give a motivated introduction to the theory of perverse sheaves, culminating in the decomposition theorem of Beilinson, Bernstein, Deligne and Gabber. A goal of this survey is to show how the theory develops naturally from classical constructions used in the study of topological properties of alge- braic varieties. While most proofs are omitted, we discuss several approaches to the decomposition theorem and indicate some important applications and examples. Contents 1. Overview 536 1.1. The topology of complex projective manifolds: Lefschetz and Hodge theorems 538 1.2. Families of smooth projective varieties 538 1.3. Singular algebraic varieties 540 1.4. Decomposition and hard Lefschetz in intersection cohomology 540 1.5. Crash course on sheaves and derived categories 541 1.6. Decomposition, semisimplicity and relative hard Lefschetz theorems 545 1.7. Invariant cycle theorems 547 1.8. A few examples 548 1.9. The decomposition theorem and mixed Hodge structures 549 1.10. Historical and other remarks 550 2. Perverse sheaves 551 2.1. Intersection cohomology 552 2.2. Examples of intersection cohomology 553 2.3. Definition and first properties of perverse sheaves 555 2.4. The perverse filtration 558 2.5. Perverse cohomology 559 2.6. t-exactness and the Lefschetz hyperplane theorem 560 2.7. Intermediate extensions 561 3.
Intersection Homology Theory

TO~O~ORY Vol. 19. pp. 135-162 0 Perpamon Press Ltd.. 1980. Printed in Great Bntain INTERSECTION HOMOLOGY THEORY MARK GORESKY and ROBERT MACPHERSON (Received 15 Seprember 1978) INTRODUCTION WE DEVELOP here a generalization to singular spaces of the Poincare-Lefschetz theory of intersections of homology cycles on manifolds, as announced in [6]. Poincart, in his 1895 paper which founded modern algebraic topology ([18], p. 218; corrected in [19]), studied the intersection of an i-cycle V and a j-cycle W in a compact oriented n-manifold X, in the case of complementary dimension (i + j = n). Lefschetz extended the theory to arbitrary i and j in 1926[10]. Their theory may be summarized in three fundamental propositions: 0. If V and W are in general position, then their intersection can be given canonically the structure of an i + j - n chain, denoted V f~ W. I(a). a(V 17 W) = 0, i.e. V n W is a cycle. l(b). The homology class of V n W depends only on the homology classes of V and W. Note that by 0 and 1 the operation of intersection defines a product H;(X) X Hi(X) A Hi+j-, (X). (2) Poincar& Duality. If i and j are complementary dimensions (i + j = n) then the pairing Hi(X) x Hj(X)& HO( Z is nondegenerate (or “perfect”) when tensored with the rational numbers. (Here, E is the “augmentation” which counts the points of a zero cycle according to their multiplicities). We will study intersections of cycles on an n dimensional oriented pseudomani- fold (or “n-circuit”, see § 1.1 for the definition).
$Arxiv:Math/0307349V1 [Math.RT] 26 Jul 2003 Naleagba Rashbr Ait,O H Ouisaeof Interested Space Be Moduli Therefore the Varieties$

Arxiv:Math/0307349V1 [Math.RT] 26 Jul 2003 Naleagba Rashbr Ait,O H Ouisaeof Interested Space Be Moduli Therefore the Varieties

AN INTRODUCTION TO PERVERSE SHEAVES KONSTANZE RIETSCH Introduction The aim of these notes is to give an introduction to perverse sheaves with appli- cations to representation theory or quantum groups in mind. The perverse sheaves that come up in these applications are in some sense extensions of actual sheaves on particular algebraic varieties arising in Lie theory (for example the nilpotent cone in a Lie algebra, or a Schubert variety, or the moduli space of representations of a Dynkin quiver). We will ultimately therefore be interested in perverse sheaves on algebraic varieties. The origin of the theory of perverse sheaves is M. Goresky and R. MacPherson’s theory of intersection homology [13, 14]. This is a purely topological theory, the original aim of which was to find a topological invariant similar to cohomology that would carry over some of the nice properties of homology or cohomology of smooth manifolds also to singular spaces (especially Poincaréduality). While the usual cohomology of a topological space can be defined sheaf theoretically as cohomology of the constant sheaf, the intersection homology turns out to be the cohomology of a certain complex of sheaves, constructed very elegantly by P. Deligne. This complex is the main example of a perverse sheaf. The greatest part of these notes will be taken up by explaining Goresky and MacPherson’s intersection homology and Deligne’s complex. And throughout all of that our setting will be purely topological. Also we will always stay over C. The deepest result in the theory of perverse sheaves on algebraic varieties, the Decomposition Theorem of Beilinson, Bernstein, Deligne and Gabber [1], will only be stated.
Intersection Homology and Poincaré Duality on Homotopically Stratified Spaces

Intersection homology and Poincaréduality on homotopically stratified spaces Greg Friedman Texas Christian University April 16, 2009 Abstract We show that intersection homology extends Poincaréduality to manifold homo- topically stratified spaces (satisfying mild restrictions). These spaces were introduced by Quinn to provide \a setting for the study of purely topological stratified phenomena, particularly group actions on manifolds." The main proof techniques involve blending the global algebraic machinery of sheaf theory with local homotopy computations. In particular, this includes showing that, on such spaces, the sheaf complex of singular intersection chains is quasi-isomorphic to the Deligne sheaf complex. 2000 Mathematics Subject Classification: 55N33, 57N80, 57P99 Keywords: intersection homology, Poincaréduality, homotopically stratified space Contents 1 Introduction 2 2 Sheaves vs. Singular Chains 5 3 Background and Basic Terminology 6 3.1 Intersection homology . 6 3.1.1 A note on coefficients . 8 3.2 Stratified homotopies and fibrations . 8 3.3 Manifold homotopically stratified spaces . 9 3.3.1 Forward tameness and homotopy links . 9 3.3.2 Manifold homotopically stratified spaces (MHSSs) . 9 3.4 Neighborhoods in stratified spaces . 10 3.4.1 Teardrops . 10 3.4.2 Approximate tubular neighborhoods . 11 1 4 Local approximate tubular neighborhoods 11 5 IS∗ is the Deligne sheaf 13 5.1 Superperversities and codimension one strata . 21 6 Constructibility 22 7 PoincaréDuality 25 8 Homotopy Witt spaces and PoincaréDuality Spaces 30 9 More general ground rings 30 A Naturality of dualization 32 1 Introduction The primary purpose of this paper is to extend Poincaréduality to manifold homotopically stratified spaces using intersection homology.
Intersection Homology and Poincaré Duality on Homotopically Stratified

Intersection homology and Poincaréduality on homotopically stratified spaces Greg Friedman Texas Christian University April 17, 2007 Abstract We show that intersection homology extends Poincaréduality to manifold homo- topically stratified spaces (satisfying mild restrictions). This includes showing that, on such spaces, the sheaf of singular intersection chains is quasi-isomorphic to the Deligne sheaf. 2000 Mathematics Subject Classification: 55N33, 57N80, 57P99 Keywords: intersection homology, Poincaréduality, homotopically stratified space Contents 1 Introduction 2 2 Sheaves vs. Singular Chains 4 3 Background and Basic Terminology 5 3.1 Intersectionhomology .............................. 5 3.1.1 Anoteoncoefficients........................... 7 arXiv:math/0702087v2 [math.GT] 23 May 2007 3.2 Stratifiedhomotopiesandfibrations . .. 7 3.3 Manifold homotopically stratified spaces . 8 3.3.1 Forwardtamenessandhomotopylinks . 8 3.3.2 Manifold homotopially stratified spaces (MHSSs) . 9 3.4 Neighborhoodsinstratifiedspaces. .. 9 3.4.1 Teardrops ................................. 9 3.4.2 Approximate tubular neighborhoods . 10 4 Local approximate tubular neighborhoods 11 5 IS∗ is the Deligne sheaf 12 5.1 Superperversities and codimension one strata . ..... 20 1 6 Constructibility 21 7 PoincaréDuality 24 8 Homotopy Witt spaces and PoincaréDuality Spaces 29 9 More general ground rings 30 A Naturality of dualization 32 1 Introduction The primary purpose of this paper is to extend Poincaréduality to manifold homotopically stratified spaces using intersection homology. Intersection homology was introduced by Goresky and MacPherson [16] in order to extend Poincaréduality to manifold stratified spaces – spaces that are not manifolds but that are composed of manifolds of various dimensions. Initially, this was done for piecewise-linear pseudomanifolds [16], which include algebraic and analytic varieties1, but the result was soon extended to topological pseudomanifolds (Goresky-MacPherson [17]) and locally conelike topological stratified spaces, also called cs-spaces (Habegger-Saper [19]).