Table Of ContentThe occurrence of singularities is pervasive in many problems in topology,
differential geometry and algebraic geometry. This book concerns the study
of singular spaces using techniques from a variety of areas of geometry and
topology and interactions among them. It contains more than a dozen expos-
itory papers on topics ranging from intersection homology, L2 cohomology
and differential operators, to the topology of algebraic varieties, signatures
andcharacteristicclasses,mixedHodgetheory,andellipticgeneraofsingular
complex and real algebraic varieties. The book concludes with a list of open
problems.
Mathematical Sciences Research Institute
Publications
58
Topology of Stratified Spaces
Mathematical Sciences Research Institute Publications
1 Freed/Uhlenbeck:Instantons and Four-Manifolds,secondedition
2 Chern(ed.):Seminar on Nonlinear Partial Differential Equations
3 Lepowsky/Mandelstam/Singer(eds.):Vertex Operators in Mathematics and Physics
4 Kac(ed.):Infinite Dimensional Groups with Applications
5 Blackadar:K-Theory for Operator Algebras,secondedition
6 Moore(ed.):Group Representations, Ergodic Theory, Operator Algebras, and
Mathematical Physics
7 Chorin/Majda(eds.):Wave Motion: Theory, Modelling, and Computation
8 Gersten(ed.):Essays in Group Theory
9 Moore/Schochet:Global Analysis on Foliated Spaces,secondedition
10–11 Drasin/Earle/Gehring/Kra/Marden(eds.):Holomorphic Functions and Moduli
12–13 Ni/Peletier/Serrin (eds.):Nonlinear Diffusion Equations and Their Equilibrium States
14 Goodman/delaHarpe/Jones:Coxeter Graphs and Towers of Algebras
15 Hochster/Huneke/Sally(eds.):Commutative Algebra
16 Ihara/Ribet/Serre(eds.):Galois Groups over Q
17 Concus/Finn/Hoffman(eds.):Geometric Analysis and Computer Graphics
18 Bryant/Chern/Gardner/Goldschmidt/Griffiths: Exterior Differential Systems
19 Alperin(ed.):Arboreal Group Theory
20 Dazord/Weinstein(eds.):Symplectic Geometry, Groupoids, and Integrable Systems
21 Moschovakis(ed.):Logic from Computer Science
22 Ratiu(ed.):The Geometry of Hamiltonian Systems
23 Baumslag/Miller(eds.):Algorithms and Classification in Combinatorial GroupTheory
24 Montgomery/Small(eds.):Noncommutative Rings
25 Akbulut/King:Topology of Real Algebraic Sets
26 Judah/Just/Woodin(eds.):Set Theory of the Continuum
27 Carlsson/Cohen/Hsiang/Jones(eds.):Algebraic Topology and Its Applications
28 Clemens/Kolla´r(eds.):Current Topics in Complex Algebraic Geometry
29 Nowakowski(ed.):Games of No Chance
30 Grove/Petersen(eds.):Comparison Geometry
31 Levy(ed.):Flavors of Geometry
32 Cecil/Chern(eds.):Tight and Taut Submanifolds
33 Axler/McCarthy/Sarason(eds.):Holomorphic Spaces
34 Ball/Milman(eds.):Convex Geometric Analysis
35 Levy(ed.):The Eightfold Way
36 Gavosto/Krantz/McCallum(eds.):Contemporary Issues in Mathematics Education
37 Schneider/Siu(eds.):Several Complex Variables
38 Billera/Bjo¨rner/Green/Simion/Stanley (eds.):New Perspectives in Geometric
Combinatorics
39 Haskell/Pillay/Steinhorn(eds.):Model Theory, Algebra, and Geometry
40 Bleher/Its(eds.):Random Matrix Models and Their Applications
41 Schneps(ed.):Galois Groups and Fundamental Groups
42 Nowakowski(ed.):More Games of No Chance
43 Montgomery/Schneider(eds.):New Directions in Hopf Algebras
44 Buhler/Stevenhagen(eds.):Algorithmic Number Theory: Lattices, Number Fields,
Curves and Cryptography
45 Jensen/Ledet/Yui:Generic Polynomials: Constructive Aspects of the Inverse Galois
Problem
46 Rockmore/Healy(eds.):Modern Signal Processing
47 Uhlmann(ed.):Inside Out: Inverse Problems and Applications
48 Gross/Kotiuga:Electromagnetic Theory and Computation: A Topological Approach
49 Darmon/Zhang(eds.):Heegner Points and Rankin L-Series
50 Bao/Bryant/Chern/Shen(eds.):A Sampler of Riemann–Finsler Geometry
51 Avramov/Green/Huneke/Smith/Sturmfels(eds.):Trends in Commutative Algebra
52 Goodman/Pach/Welzl(eds.):Combinatorial and Computational Geometry
53 Schoenfeld(ed.):Assessing Mathematical Proficiency
54 Hasselblatt(ed.):Dynamics, Ergodic Theory, and Geometry
55 Pinsky/Birnir(eds.):Probability, Geometry and Integrable Systems
56 Albert/Nowakowski(eds.):Games of No Chance 3
57 Kirsten/Williams (eds.):A Window into Zeta and Modular Physics
58 Friedman/Hunsicker/Libgober/Maxim(eds.):Topology of Stratified Spaces
Volumes 1–4, 6–8, and 10–27 are published by Springer-Verlag
Topology of
Stratified Spaces
Edited by
Greg Friedman
Texas Christian University
Eug´enie Hunsicker
Loughborough University
Anatoly Libgober
University of Illinois at Chicago
Laurentiu Maxim
University of Wisconsin-Madison
Greg Friedman Eug´enie Hunsicker
Department of Mathematics Department of Mathematical Sciences
Texas Christian University Loughborough University
[email protected] [email protected]
Anatoly Libgober Laurentiu Maxim
Department of Mathematics Department of Mathematics
University of Illinois at Chicago University of Wisconsin-Madison
[email protected] [email protected]
Silvio Levy (Series Editor)
Mathematical Sciences Research Institute
Berkeley, CA 94720
[email protected]
The Mathematical Sciences Research Institute wishes to acknowledge support by
the National Science Foundation and the Pacific Journal of Mathematicsfor the
publication of this series.
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town,
Singapore, S˜ao Paulo, Delhi, Tokyo, Mexico City
Cambridge University Press
32 Avenue of the Americas, New York, NY 10013-2473, USA
www.cambridge.org
Information on this title: www.cambridge.org/9780521191678
(cid:13)c Mathematical Sciences Research Institute 2011
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
First published 2011
Printed in the United States of America
A catalog record for this publication is available from the British Library.
Topology of stratified spaces / edited by Greg Friedman ... [et al.].
p. cm. – (Mathematical Sciences Research Institute publications ; 58)
Includes bibliographical references.
ISBN 978-0-521-19167-8 (hardback)
1. Singularities (Mathematics) 2. Algebraic spaces. 3. Algebraic topology.
I. Friedman, Greg, 1973–
QA614.58.T67 2011
′
516.35—dc22 2010051108
ISBN 978-0-521-19167-8 hardback
Cambridge University Press has no responsibility for the persistence or accuracy of URLs
for external or third-party Internet Web sites referred to in this publication and does not
guarantee that any content on such Web sites is, or will remain, accurate or appropriate.
TopologyofStratifiedSpaces
MSRIPublications
Volume58,2011
Contents
Preface ix
AnintroductiontoL2 cohomology 1
XIANZHE DAI
Thealmostclosedrangecondition 13
GILLES CARRON
RigidityofdifferentialoperatorsandChernnumbersofsingularvarieties 31
ROBERT WAELDER
Hodgetheorymeetstheminimalmodelprogram:asurveyoflogcanonicaland
duBoissingularities 51
SA´NDOR J. KOVA´CS AND KARL SCHWEDE
Ellipticgenera,realalgebraicvarietiesandquasi-Jacobiforms 95
ANATOLY LIBGOBER
Theweightfiltrationforrealalgebraicvarieties 121
CLINT MCCRORY ANDADAM PARUSIN´SKI
OntheMilnorclassesofcomplexhypersurfaces 161
LAURENTIU MAXIM
Anintroductiontointersectionhomologywithgeneralperversityfunctions 177
GREG FRIEDMAN
Thesignatureofsingularspacesanditsrefinementstogeneralizedhomology
theories 223
MARKUS BANAGL
IntersectionhomologyWangsequence 251
FILIPP LEVIKOV
Anexponentialhistoryoffunctionswithlogarithmicgrowth 281
MATT KERR ANDGREGORY PEARLSTEIN
Motiviccharacteristicclasses 375
SHOJI YOKURA
CharacteristicclassesofmixedHodgemodules 419
JO¨RG SCHU¨RMANN
Openproblems 471
vii
TopologyofStratifiedSpaces
MSRIPublications
Volume58,2011
Preface
This volume is based on lectures given at the Workshop on the Topology of
Stratified Spaces thatwasheldattheMathematical Sciences ResearchInstitute
in Berkeley, CA, from September 8 to September 12, 2008. Stratified spaces
areusuallynotquitemanifolds—theymaypossesssingularities—buttheyare
composedofmanifoldlayers,thestrata. Examplesofsuchspacesincludealge-
braicvarieties,quotientsofmanifoldsandvarietiesbygroupactions,homotopy
stratified spaces, topological and piecewise linear pseudomanifolds, and even
manifolds,augmentedbyfiltrationsthatcanarise,forexample,viaembeddings
and their singularities. In recent years, there has been extensive interest and
success in expanding to stratified spaces the triumphs of algebraic topology in
manifold theory, including the vast progress in the mid-twentieth century on
signatures, characteristic classes, surgery theories, and the special homological
properties of nonsingular analytic and algebraic varieties, such as the Ka¨hler
package and Hodge theories. Such extensions from manifold theory to strati-
fiedspacetheoryarerarelystraightforward—theytendtoinvolvethediscovery
and study of subtle interactions between local and global behavior—but vast
progresshasbeenmade,particularlyusingsuchtopologicaltoolsasintersection
homology and the related analytic L2 cohomology. The goal of the workshop
was,andofthisproceedingsis,toprovideanoverviewofthisprogressaswellas
ofcurrentresearchresults,withaparticularemphasisoncommunicationacross
the boundaries of the different fields of mathematics that encompass stratified
space research. Thus there is an emphasis in this volume on expository papers
thatgiveintroductionstoandoverviewsoftopicsintheareaofstratifiedspaces.
Four main areas were featured in the MSRI workshop: L2 cohomology and
Hodge theorems, topology of algebraic varieties, signature theory on singular
spaces, and mixed Hodge theory and singularities. For the purpose of giving
someorganizationtothevolume,wehavegroupedthepapersroughlyintothese
topics, although somepapers overlap more than onearea.
There are three papers on analysis and topology. The paper by Dai is an
introductiontoL2 cohomology,whichdiscussessomeoftheanalyticconsider-
ations that arise inthestudyof L2 cohomology and L2 signatures onstratified
ix
x PREFACE
spaces as well as relationships between these analytic objects and topological
analogues. Building on these basics, the paper of Carron gives an example of
howhisidea oftheexterior derivative onaspacehaving “almostclosedrange”
can be used to calculate the L2 cohomology in the case of two QALE-type
spaces, the Hilbert schemes of two and of three points on CP2. The paper by
Waeldercomesfromadifferentangle. Itconsiders“rigid”differentialoperators
on a smooth manifold commuting with an S1-action, and infinite dimensional
analog oftheDiracoperator, whoseindex yields thecomplexelliptic genus. In
this survey, Waelder discusses how rigidity theorems for such operators are re-
latedtotheproblemofdefiningChernnumbersonsingularvarieties. Thepaper
of Waelder also fits under the second category: complex algebraic varieties.
Thenextsection,onalgebraicvarieties,startswiththesurveybyKova´csand
Schwede, which gives an introduction to the study of singularities that are un-
avoidableinclassificationproblemsofsmoothalgebraicvarieties,andespecially
in the minimal model program, including log-canonical and Du Bois singular-
ities. The paper of Libgober is an overview of the work of the author and his
collaboratorsonextensionsoftheellipticgenustosingularvarieties. Itcontains
a new description of the class of holomorphic functions (quasi-Jacobi forms)
that are elliptic genera of complex manifolds (possibly without theCalabi–Yau
condition) and includes a survey of recent developments such as the “higher
elliptic genera”, which contain information on non-simply connected spaces.
It also constructs the elliptic genus for singular real algebraic spaces. In their
contribution, McCrory and Parusin´ski make a thorough study of the weight fil-
trationontheBorel–Moorecohomologyofrealalgebraicvarieties. Theauthors
give several descriptions of this filtration and show some nice applications of
their construction to real algebraic and analytic geometry. Finally, the paper of
MaximgivesanewformulationofresultsaboutMilnorclasses,whichgeneral-
ize Parusin´ski’s Milnor numbers to non-isolated hypersurface singularities. He
considers these classes for singular hypersurfaces in complex manifolds and,
using his new formulation, gives comparisons to the topologically defined L-
classes of Cappell–Shaneson and Goresky–MacPherson.
Threepapersdiscussaspectsofintersectionhomologyandsignaturesonstrat-
ified spaces. The paper by Friedman is an expository survey of perversities in
intersection homology, starting with the classical perversities of Goresky and
MacPhersonanddescribingvariouswaysinwhichthesehavebeengeneralized
in the past three decades. It can also be read as a general introduction to in-
tersectionhomology. Banagl’s contribution isanoverview ofbordisminvariant
signatures on singular spaces, including the category of Witt space, where the
signature on intersection cohomology provides a Witt-bordism invariant sig-
nature, and Banagl’s new signature on a category of non-Witt spaces under a
