Characteristic Classes of Mixed Hodge Modules
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The Development of Intersection Homology Theory Steven L
Pure and Applied Mathematics Quarterly Volume 3, Number 1 (Special Issue: In honor of Robert MacPherson, Part 3 of 3 ) 225|282, 2007 The Development of Intersection Homology Theory Steven L. Kleiman Contents Foreword 225 1. Preface 226 2. Discovery 227 3. A fortuitous encounter 231 4. The Kazhdan{Lusztig conjecture 235 5. D-modules 238 6. Perverse sheaves 244 7. Purity and decomposition 248 8. Other work and open problems 254 References 260 9. Endnotes 265 References 279 Foreword The ¯rst part of this history is reprinted with permission from \A century of mathematics in America, Part II," Hist. Math., 2, Amer. Math. Soc., 1989, pp. 543{585. Virtually no change has been made to the original text. However, the text has been supplemented by a series of endnotes, collected in the new Received October 9, 2006. 226 Steven L. Kleiman Section 9 and followed by a list of additional references. If a subject in the reprint is elaborated on in an endnote, then the subject is flagged in the margin by the number of the corresponding endnote, and the endnote includes in its heading, between parentheses, the page number or numbers on which the subject appears in the reprint below. 1. Preface Intersection homology theory is a brilliant new tool: a theory of homology groups for a large class of singular spaces, which satis¯es Poincar¶eduality and the KÄunnethformula and, if the spaces are (possibly singular) projective algebraic varieties, then also the two Lefschetz theorems. The theory was discovered in 1974 by Mark Goresky and Robert MacPherson. -
Mixed Hodge Structure of Affine Hypersurfaces
R AN IE N R A U L E O S F D T E U L T I ’ I T N S ANNALES DE L’INSTITUT FOURIER Hossein MOVASATI Mixed Hodge structure of affine hypersurfaces Tome 57, no 3 (2007), p. 775-801. <http://aif.cedram.org/item?id=AIF_2007__57_3_775_0> © Association des Annales de l’institut Fourier, 2007, tous droits réservés. L’accès aux articles de la revue « Annales de l’institut Fourier » (http://aif.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://aif.cedram.org/legal/). Toute re- production en tout ou partie cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation à fin strictement per- sonnelle du copiste est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Ann. Inst. Fourier, Grenoble 57, 3 (2007) 775-801 MIXED HODGE STRUCTURE OF AFFINE HYPERSURFACES by Hossein MOVASATI Abstract. — In this article we give an algorithm which produces a basis of the n-th de Rham cohomology of the affine smooth hypersurface f −1(t) compatible with the mixed Hodge structure, where f is a polynomial in n + 1 variables and satisfies a certain regularity condition at infinity (and hence has isolated singulari- ties). As an application we show that the notion of a Hodge cycle in regular fibers of f is given in terms of the vanishing of integrals of certain polynomial n-forms n+1 in C over topological n-cycles on the fibers of f. -
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 Poincar´eduality|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 LˆeD˜ungTr´angon 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-Poincar´echaracteristic 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. -
Mixed Hodge Structures
Mixed Hodge structures 8.1 Definition of a mixed Hodge structure · Definition 1. A mixed Hodge structure V = (VZ;W·;F ) is a triple, where: (i) VZ is a free Z-module of finite rank. (ii) W· (the weight filtration) is a finite, increasing, saturated filtration of VZ (i.e. Wk=Wk−1 is torsion free for all k). · (iii) F (the Hodge filtration) is a finite decreasing filtration of VC = VZ ⊗Z C, such that the induced filtration on (Wk=Wk−1)C = (Wk=Wk−1)⊗ZC defines a Hodge structure of weight k on Wk=Wk−1. Here the induced filtration is p p F (Wk=Wk−1)C = (F + Wk−1 ⊗Z C) \ (Wk ⊗Z C): Mixed Hodge structures over Q or R (Q-mixed Hodge structures or R-mixed Hodge structures) are defined in the obvious way. A weight k Hodge structure V is in particular a mixed Hodge structure: we say that such mixed Hodge structures are pure of weight k. The main motivation is the following theorem: Theorem 2 (Deligne). If X is a quasiprojective variety over C, or more generally any separated scheme of finite type over Spec C, then there is a · mixed Hodge structure on H (X; Z) mod torsion, and it is functorial for k morphisms of schemes over Spec C. Finally, W`H (X; C) = 0 for k < 0 k k and W`H (X; C) = H (X; C) for ` ≥ 2k. More generally, a topological structure definable by algebraic geometry has a mixed Hodge structure: relative cohomology, punctured neighbor- hoods, the link of a singularity, . -
Weak Positivity Via Mixed Hodge Modules
Weak positivity via mixed Hodge modules Christian Schnell Dedicated to Herb Clemens Abstract. We prove that the lowest nonzero piece in the Hodge filtration of a mixed Hodge module is always weakly positive in the sense of Viehweg. Foreword \Once upon a time, there was a group of seven or eight beginning graduate students who decided they should learn algebraic geometry. They were going to do it the traditional way, by reading Hartshorne's book and solving some of the exercises there; but one of them, who was a bit more experienced than the others, said to his friends: `I heard that a famous professor in algebraic geometry is coming here soon; why don't we ask him for advice.' Well, the famous professor turned out to be a very nice person, and offered to help them with their reading course. In the end, four out of the seven became his graduate students . and they are very grateful for the time that the famous professor spent with them!" 1. Introduction 1.1. Weak positivity. The purpose of this article is to give a short proof of the Hodge-theoretic part of Viehweg's weak positivity theorem. Theorem 1.1 (Viehweg). Let f : X ! Y be an algebraic fiber space, meaning a surjective morphism with connected fibers between two smooth complex projective ν algebraic varieties. Then for any ν 2 N, the sheaf f∗!X=Y is weakly positive. The notion of weak positivity was introduced by Viehweg, as a kind of birational version of being nef. We begin by recalling the { somewhat cumbersome { definition. -
The Derived Category of Sheaves and the Poincare-Verdier Duality
THE DERIVED CATEGORY OF SHEAVES AND THE POINCARE-VERDIER´ DUALITY LIVIU I. NICOLAESCU ABSTRACT. I think these are powerful techniques. CONTENTS 1. Derived categories and derived functors: a short introduction2 2. Basic operations on sheaves 16 3. The derived functor Rf! 31 4. Limits 36 5. Cohomologically constructible sheaves 47 6. Duality with coefficients in a field. The absolute case 51 7. The general Poincare-Verdier´ duality 58 8. Some basic properties of f ! 63 9. Alternate descriptions of f ! 67 10. Duality and constructibility 70 References 72 Date: Last revised: April, 2005. Notes for myself and whoever else is reading this footnote. 1 2 LIVIU I. NICOLAESCU 1. DERIVED CATEGORIES AND DERIVED FUNCTORS: A SHORT INTRODUCTION For a detailed presentation of this subject we refer to [4,6,7,8]. Suppose A is an Abelian category. We can form the Abelian category CpAq consisting of complexes of objects in A. We denote the objects in CpAq by A or pA ; dq. The homology of such a complex will be denoted by H pA q. P p q A morphism s HomCpAq A ;B is called a quasi-isomorphism (qis brevity) if it induces an isomorphism in co-homology. We will indicate qis-s by using the notation A ùs B : Define a new additive category KpAq whose objects coincide with the objects of CpAq, i.e. are complexes, but the morphisms are the homotopy classes of morphisms in CpAq, i.e. p q p q{ HomKpAq A ;B : HomCpAq A ;B ; where denotes the homotopy relation. Often we will use the notation r s p q A ;B : HomKpAq A ;B : The derived category of A will be a category DpAq with the same objects as KpAq but with a q much larger class of morphisms. -
Hodge Polynomials of Singular Hypersurfaces
HODGE POLYNOMIALS OF SINGULAR HYPERSURFACES ANATOLY LIBGOBER AND LAURENTIU MAXIM Abstract. We express the difference between the Hodge polynomials of the singular and resp. generic member of a pencil of hypersurfaces in a projective manifold by using a stratification of the singular member described in terms of the data of the pencil. In particular, if we assume trivial monodromy along all strata in the singular member of the pencil, our formula computes this difference as a weighted sum of Hodge polynomials of singular strata, the weights depending only on the Hodge-type information in the normal direction to the strata. This extends previous results (cf. [19]) which related the Euler characteristics of the generic and singular members only of generic pencils, and yields explicit formulas for the Hodge χy-polynomials of singular hypersurfaces. 1. Introduction and Statements of Results Let X be an n-dimensional non-singular projective variety and L be a bundle on X. Let L ⊂ P(H0(X; L)) be a line in the projectivization of the space of sections of L, i.e., a pencil of hypersurfaces in X. Assume that the generic element Lt in L is non-singular and that L0 is a singular element of L. The purpose of this note is to relate the Hodge polynomials of the singular and resp. generic member of the pencil, i.e., to understand the difference χy(L0) − χy(Lt) in terms of invariants of the singularities of L0. A special case of this situation was considered by Parusi´nskiand Pragacz ([19]), where the Euler characteristic is studied for pencils satisfying the assumptions that the generic element Lt of the pencil L is transversal to the strata of a Whitney stratification of L0. -
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. -
University of Alberta
University of Alberta FILTRATIONS ON HIGHER CHOW GROUPS AND ARITHMETIC NORMAL FUNCTIONS by Jos´eJaime Hern´andezCastillo A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics Department of Mathematical and Statistical Sciences c Jos´eJaime Hern´andezCastillo Spring 2012 Edmonton, Alberta Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author's prior written permission. Library and Archives Bibliothèque et Canada Archives Canada Published Heritage Direction du Branch Patrimoine de l'édition 395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre référence ISBN: 978-0-494-78000-8 Our file Notre référence ISBN: 978-0-494-78000-8 NOTICE: AVIS: The author has granted a non- L'auteur a accordé une licence non exclusive exclusive license allowing Library and permettant à la Bibliothèque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par télécommunication ou par l'Internet, prêter, telecommunication or on the Internet, distribuer et vendre des thèses partout dans le loan, distrbute and sell theses monde, à des fins commerciales ou autres, sur worldwide, for commercial or non- support microforme, papier, électronique et/ou commercial purposes, in microform, autres formats.