Handbook of Geometry and Topology of Singularities III PDF
Preview Handbook of Geometry and Topology of Singularities III
José Luis Cisneros-Molina Lê Dũng Tráng José Seade Editors Handbook of Geometry and Topology of Singularities III Handbook of Geometry and Topology of Singularities III José Luis Cisneros-Molina (cid:129) Lê Du˜ng Tráng (cid:129) José Seade Editors Handbook of Geometry and Topology of Singularities III Editors JoséLuisCisneros-Molina LêDu˜ngTráng InstitutodeMatemáticas,Unidad CentredeMathématiquesetInformatique Cuernavaca,UniversidadNacional Universited’Aix-Marseille AutónomadeMéxicoand Marseille,France UnidadMixtaInternacionalCNRS LaboratorioSolomonLefschetz Cuernavaca,Mexico JoséSeade InstitutodeMatemáticas,Universidad NacionalAutónomadeMéxico MexicoCity,Mexico InternationalResearchLaboratoryCNRS LaboratorioSolomonLefschetz Cuernavaca,Mexico ISBN978-3-030-95759-9 ISBN978-3-030-95760-5 (eBook) https://doi.org/10.1007/978-3-030-95760-5 MathematicsSubjectClassification:14B05,32S15,14J17,14C17,32A27,32S35,32S60 ©SpringerNatureSwitzerlandAG2022 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface This is the third volume of the Handbook of the Geometry and Topology of Singularities,asubjectwhichisubiquitousinmathematics,appearingnaturallyina widerangeofdifferentareasofknowledge.Thescopeofsingularitytheoryisvast, itspurposeismultifold.Thisisameetingpointwheremanyareasofmathematics, andscienceingeneral,cometogether. Let us reminisce Bernard Teissier’s words in his foreword to Volume I of the Handbook: IclaimthatSingularityTheory sitsinsideMathematicsmuchasMathematicssitsinside thegeneralscientificculture.Thegeneralmathematicalcultureknowsabouttheexistence ofMorsetheory,parametrizationsofcurves,Bézout’stheoremforplaneprojectivecurves, zeroes of vector fields and the Poincaré-Hopf theorem, catastrophe theory, sometimes a version of resolution of singularities, the existence of an entire world of commutative algebra,etc.Butagain,forthesingularist,theseandmanyothersarelineamentsofasingle landscape and she or he is aware of its connectedness. Moreover, just as Mathematics does with science in general, singularity theory interacts energetically with the rest of Mathematics,ifonlybecausetheclosuresofnonsingularvarietiesinsomeambientspace or their projections to smaller dimensional spaces tend to present singularities, smooth functions onacompact manifoldmust have criticalpoints, etc. Butsingularity theory is also,againinaroleplayedbyMathematicsingeneralscience,acruciblewheredifferent typesofmathematicalproblemsinteractandsurprisingconnectionsareborn. TheHandbookhastheintentionofcoveringawidescopeofsingularitytheory, presenting articles on various aspects of the theory and its interactionswith other areas of mathematics in a reader-friendlyway. The authors are world experts;the variousarticlesdealwithbothclassicalmaterialandmoderndevelopments. ThefirstVolumeIofthiscollectiongatheredtenarticlesconcerningfoundational aspectsofthetheory.Thisincludes: – Thecombinatoricsandtopologyofplanecurvesandsurfacesingularities – An introductionto four of the classical methodsfor studying the topologyand geometry of singular spaces, namely resolution of singularities, deformation theory,Stratifications,andslicingthespacesàlaLefschetz – Milnorfibrationsandtheirmonodromy v vi Preface – Morsetheoryforstratifiedspacesandconstructiblesheaves – SimpleLiealgebrasandsimplesingularities Volume II also consists of ten articles. These coverfoundationalaspects of the theoryas well as some importantrelationswith other areas of mathematics. They include: – The analytic classification of plane curve singularities and the existence of complexandrealalgebraiccurvesintheplanewithprescribedsingularities – Anintroductiononthelimitsoftangentstoacomplexanalyticsurface,asubject thatoriginatesinWhitney’swork – IntroductionstoZariski’sequisingularityandintersectionhomology,whichare twoofthemaincurrentviewpointsforstudyingsingularities – AnoverviewofMilnor’sfibrationtheoremforrealandcomplexsingularities,as wellasanintroductiontoMassey’stheoryofLêcycles – Adiscussionofmixedsingularities,whicharerealanalyticsingularitieswith a richstructurethatallowstheirstudyviacomplexgeometry – The study of intersections of concentric ellipsoids in Rn and its relation with several areas of mathematics, from holomorphic vector fields to singularity theory,toricvarieties,andmoment-anglemanifolds – Areviewofthetopologyofquasi-projectivevarietiesandgeneralizationsabout thecomplementsofplanecurvesandhypersurfacesinprojectivespace ThisVolumeIIIalsoconsistsoftenchapters.Someofthesecomplementtopics explored previously in Volumes I and II, while other chapters bring in important new subjects. Let us say a few words about the content of this volume, though eachchapterhasitsownabstract,introductionandalargebibliographyforfurther reading.Thereisalsoaglobalindexoftermsattheend. Chapters 1 and 2 have as common thread the much celebrated Thom-Mather theory. In 1944, Whitney studied mappings Rn → R2n−1, the first pair of dimensions not covered by his immersion theorem, showing that in these setting singularitiescannotbeavoidedingeneral.Hethenintroducedtheconceptofstable mappingsandcharacterizedthestablemappingsfromRn toRp withp ≥ 2n−1, andalso thosefromthe planeinto itself, showingthatin all these cases the stable mappings form a dense set in the space of smooth proper mappings. Whitney conjectured that the density of stable mappings would hold for any pair (n,p). However,RenéThomshowedthatthisisnotthecasebygivingacounterexample. Thom then conjectured that the topologically stable maps are always dense and gaveanoutlineofthe proof.ThecompleteproofwasgivenbyJohnMather,who, from1965to1975,solvedalmostcompletelytheprogramdrawnbyThomforthe stabilityproblem.ThisisknownasThom-Mathertheory. Chapter3isaboutZariski’sequisingularity,previouslyenvisagedinParusin´ski’s chapter in Volume II. Among the various notions of equivalence of singularities, topological equisingularity is one of the oldest and easiest to define, but it is far from being well understood. Several challenging questions remain open. In this chapter,theauthorsurveysdevelopmentsintopologicalequisingularity,someofits Preface vii relationswithotherequisingularitynotions,andhintsonnewpossibleapproaches to old questions based in algebro-geometricmethods, Floer theory, and Lipschitz geometry.Topologicalequisingularityquestionswerecrucialmotivationsourcesfor thedevelopmentoftheComputerAlgebraprogramSINGULAR;thisisexplainedin anappendixbyG.-M.GreuelandG.Pfister. Chapter 4 somehow fits within the classical interplay between normal singu- larities in complex surfaces and 3-manifold theory, which has been studied for decades and was discussed from a topological perspective in F. Michel’s chapter inVolumeI.Nowtheauthorlooksatthesubjectfromanotherperspective,bringing insubtlestructures.Givenacomplexanalyticnormalsurfacesingularity(X,0)we know that its topology is fully determined by its link L , a 3-manifold which X is the intersection of X with a sufficiently small sphere in the ambient space, centeredat 0.The mainmotifof thischapteris studyingthe ties betweenanalytic and topological invariants of (X,0). Historically, this program was started by Artin and Laufer, which characterized topologically the rational and minimally elliptic singularities (respectively), and computed several analytic invariants from the resolution graph. This question brings us into the theory of the Casson and Casson-Walker invariant, the (refined) Turaev torsion, Seiberg-Witten invariants, lattice (co)homology,Heegaard-Floer theory, and other important invariants of 3- manifolds. This chapter starts from well-known elementary facts about surface singularitiesandbringsustothedepthsofthisrichandinterestingtheory. Chapters5–7discussdifferentaspectsofthetheoryofChernclassesforsingular varieties. For complex manifolds, their Chern classes are by definition those of its tangentbundle.These are importantinvariantsthat encodedeepgeometricand topologicalinformation.Whenweconsidersingularvarieties,thereisnotaunique wayofextendingthisconcept.ThissomehowdependsonwhichpropertiesofChern classesweareinterestedin,orhowweextendthenotionofthetangentbundleover the singular set. In these chapters, the authors introduce in elementary ways the variousnotionsofChernclassesforsingularvarietiesandtheirrelationswithother invariantsof singularvarieties. Chapter5 givesa thoroughaccountof the subject, fromthebirthofthetheoryofChernclassesuptothemoderntheoriesofmotivic, bivariant,and Hirzebruchcharacteristic classes. Chapter6 hasSegre classes as its core.TheseclassesareanimportantingredientinFulton-MacPhersonintersection theoryand providea powerfulmean for studying Chern classes of vector bundles in the algebraicsetting. Severalimportantinvariantsof algebraicvarietiesmay be expressed in terms of Segre classes. The goal of that chapter is to survey several invariantsspecificallyarisinginsingularitytheorywhichmaybedefinedorrecastin termsofSegreclasses.Chapter7looksatthesubjectfromatopologicalviewpoint, focusing on the relations between local and global invariants, particularly indices ofvectorfields,theMilnornumber,andLêcycles.Itincludesforcompletenessan introduction to the Hirzebruch-Riemann-Roch theorem and its generalizations to singularvarietiesthatgiverisetoseveraloftherecentdevelopmentsinthesubject. Chapter 8 studies the residuesin complexanalytic varietiesthat arise from the localization of characteristic classes via Alexander duality. A paradigm for this theoryistheclassicaltheoremofPoincaré-Hopfthatcanbeunderstoodasproviding viii Preface a localization of the top Chern class of a complex manifold at the singularities of a vector field. This was beautifully extended by Baum and Bott for singular holomorphic foliations on complex manifolds, providing expressions for certain Chern numbers in terms of residues localized at the singular set of the foliation. Thetheorythattheauthorpresentsinthischapterstartswiththestudyofresiduesof singularholomorphicfoliations,latertransferredtotheindextheoryofholomorphic self-maps. The philosophy behind is rather simple. Namely, once we have some kindofvanishingtheoremonthenon-singularpartofageometricobjectsuchasa foliation,certaincharacteristicclassesarelocalizedatthesetofsingularpoints,and thelocalizationgivesrisetoresiduesviatheAlexanderduality.Theauthorexplains how the relative Cˇech-de Rham theorem allows us to deal with the problemfrom both the topologicaland differentialgeometricviewpoints,and the comparisonof thetwoyieldsvariousinterestingexpressionsoftheresiduesandapplications. Chapter 9 surveys applications of mixed Hodge theory to the study of iso- lated singularities. Hodge theory deals with the cohomology of smooth complex projective varieties, or more generally, compact Kähler manifolds. A choice of a Riemannian metric enables one to define the Laplace operator (cid:2) on differential forms, and each de Rham cohomology class contains exactly one closed form ω with (cid:2)ω = 0, the harmonicrepresentative.One has the Hodge decompositionof cohomologyclassesviatheirharmonicrepresentatives: (cid:2) Hk(X,C)= Hp,q(X) p+q=k where Hp,q(X) is the subspace of Hk(X,C) consisting of classes of forms containing harmonic forms of type (p,q). Using Leray’s theory of sheaves and resolution of singularities, Grothendieck defined the de Rham cohomology of complex algebraic varieties in purely algebraic terms. A generalization of Hodge theory to arbitrary complex algebraic varieties was then developed by Deligne. He showed that the cohomologyof a complex algebraic variety carries a slightly more general structure, which presents Hk(X,C) as a successive extension of Hodgestructuresofdecreasingweights.ThisgeneralizationiscalledamixedHodge structure. We close this volume with Chap.10, a detailed introduction of the theory of constructible sheaf complexes in the complex algebraic and analytic settings. All concepts are illustrated by many interesting examples and relevant applications, while some important results are presented with complete proofs. This chapter is intendedasabroadlyaccessibleuser’sguidetothosetopics,providingthereaders not only with a presentation of the subject but also with concrete examples and applicationsthatmotivatethegeneraltheory.Theauthorsintroducethemainresults of stratified Morse theory in the framework of constructible sheaves, a subject discussed also in Goresky’s chapter in Volume I of this Handbook. Constructible sheafcomplexesandespeciallyperversesheaveshavebecomeindispensabletools for studying complexalgebraic and analytic varieties. They have seen spectacular Preface ix applications in geometry and topology, and several of these are discussed in this chapter. This handbookis addressed to graduatestudents and newcomersto the theory, as well as to specialists who can use it as a guidebook. It provides an accessible account of the state of the art in several aspects of the subject, its frontiers, and its interactionswith other areas of research. Thiswill continuewith a Volume IV, which will cover other aspects of singularity theory, and a Volume V, which will focusonholomorphicfoliations,aremarkablyimportantsubjectonitsownthathas closeconnectionswithsingularitytheory. WethankBernardTeissierforallowingustousehiswordsaboveandforvaluable andinspiringcomments. Cuernavaca,Mexico JoséLuisCisnerosMolina Marseille,France LêDu˜ngTráng MexicoCity,Mexico JoséSeade October2021 Contents 1 OldandNewResultsonDensityofStableMappings .................. 1 MariaAparecidaSoaresRuas 1.1 Introduction........................................................... 2 1.2 SettingtheProblem .................................................. 4 1.2.1 TheworkofHasslerWhitney:from1944to1958........ 6 1.2.2 RenéThomandJohnMather:from1958to1970........ 9 1.3 EquivalentNotionsofStability...................................... 12 1.3.1 TransverseStabilityandtheProofof2.⇔3.............. 16 1.3.2 Notes........................................................ 18 1.4 FiniteDeterminacyofMather’sGroups............................. 19 1.4.1 TheContactGroup ......................................... 19 1.4.2 FinitelyDeterminedGerms................................ 21 1.4.3 ClassificationofStableSingularities ...................... 30 1.4.4 MapsofFiniteSingularityType........................... 33 1.4.5 Notes........................................................ 34 1.5 TheNiceDimensions................................................ 35 1.5.1 Notes........................................................ 38 1.6 Thom’sExample ..................................................... 38 1.7 DensityofTopologicallyStableMappings ......................... 40 1.7.1 HowtoStratifyMappingsandJetSpaces................. 43 1.7.2 ProofthatTopologicallyStableMappingsare Dense(Mather,[66],§8) .................................. 46 1.7.3 TheGeometryofTopologicalStability.................... 47 1.7.4 Notes........................................................ 50 1.8 TheBoundaryoftheNiceDimensions ............................. 51 1.8.1 ACandidatefortheThom-MatherStratification inBND...................................................... 51 1.8.2 TheUnimodularStratainBND............................ 52 1.8.3 TopologicalTrivialityofUnimodularFamilies ........... 59 1.8.4 Notes........................................................ 65 xi
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