Chern Classes of Tautological Sheaves on Hilbert Schemes of Points on Surfaces

ManfredLehn

Introduction

Hilbertschemesof-tuplesofpointsonacomplexprojectivemanifoldarenaturalcompactificationsoftheconfigurationspacesofunordereddistinct-tuplesofpointson.Theirgeometryisdeterminedbythegeometryofitselfandthegeometryofthe‘punctual’Hilbertschemesofallzero-dimensionalsubschemesinaffinespacethataresupportedattheorigin.Thusoneisnaturallyledtothefollowingproblem:

DetermineexplicitlythegeometricortopologicalinvariantsoftheHilbertschemessuchastheBettinumbers,theHodgenumbers,theChernnumbers,thecohomol-ogyring,fromthecorrespondingdataofthemanifolditself.

Thisproblemismostattractivewhenisasurface,sincethentheHilbertschemesarethemselvesirreducibleprojectivemanifolds,byaresultofFogarty[9],whereasforhigherdimensionalvarietiestheHilbertschemesareingeneralneitherirreduciblenorsmoothnorpureofexpecteddimension.

Inthesurfacecase,theanswertotheproblemabovefortheBettinumberswasfirstgivenbyG¨ottschein[11].Theanswerturnsouttobeparticularlybeautiful(cf.Theorem2.1below).TheproblemfortheHodgenumberswassolvedbyS¨orgelandG¨ottsche[12].Foradifferentapproachtobothresultssee[3].TheanswerfortheChernclasseswillbeimplicitlygiveninaforthcomingpaperbyEllingsrud,G¨ottscheandtheauthor[4].

Thequestionfortheringstructureofthecohomologyismoredifficult.Ingeneral,isthequotientoftheblow-upofalongthediagonalbythecanonical

,.involutionthatexchangesthefactors.Thusthecaseofinterestis

TheringstructurewasfoundforbyEllingsrudandStrømme[5],andfor,smoothprojectiveofarbitrarydimension,byFantechiandGo¨ttsche[8].Inanother

,arbitrary,direction,EllingsrudandStrømme[6]gavegeneratorsfor

andanimplicitdescriptionoftherelations.

VafaandWitten[27]remarkedthatG¨ottsche’sFormulafortheBettinumbersisidenticalwiththePoincar´eseriesofaFockspacemodelledonthecohomologyof.Nakajima[21]succeededingivingageometricconstructionofsuchaFockspacestructureonthecohomologyoftheHilbertschemes,leadingtoanatural‘explanation’ofG¨ottsche’sresult.SimilarresultshavebeenannouncedbyGrojnowski[13].

1

FollowingthepresentationofGrojnowski,thiscanbemademorepreciseasfol-ofsubschemesoflengthand,respectively,andoflows:sendingapair

disjointsupporttotheiruniondefinesarationalmap

Thismapinduceslinearmapsontherationalcohomology

and

Ifwelettiplication

,thenthesemapsdefineamultiplicationandacomul-whichmakeacommutativeandcocommutativebigradedHopfalgebra.Theresult

ofNakajimaandGrojnowskisaysthatthisHopfalgebraisisomorphictothegraded

.symmetricalgebraofthevectorspace

1Moreexplicitly,Nakajimaconstructedlinearmaps

andprovedthattheysatisfythe‘oscillator’or‘Heisenberg’relations

Herethecommutatoristobetakeninagradedsense.

Themultiplicationandthecomultiplicationofarenotobviouslyrelatedtothequitedifferentringstructureof,whichisgivenbytheusualcupproductoneach

.(Strictlyspeaking,containsacountablenumberofdirectsummand

idempotentsbutnotaunitunlesswepasstosomecompletion).

ThispaperattemptstorelatetheHopfalgebrastructureandthecupproductstruc-ture.Moreprecisely:

,i.e.Letbelocallyfreesheafofrankon.Attachingtoapoint

,the-vectorspacedefinesalocallyazero-dimensionalsubscheme

freesheafofrankon.TheChernclassesofallsheavesonofthis

.Wewilldescribeapurelyalgebraicalgorithmtotypegenerateasubalgebra

determinetheactionofonintermsofthe-basisofprovidedbyNakajima’s

foragivensheafintoresults.WecollecttheChernclassesofallsheaves

operators

andgeometricallycomputethecommutatorsoftheseoperatorswiththe‘standardop-erators’definedbyNakajima.

Acentralrˆoleisplayedbytheoperator,which—uptoafactor

—canalsobeinterpretedastheintersectionwiththe‘boundaries’ofthe

Hilbertschemes,i.e.thedivisorsofalltupleswhichhaveamultiple

isdefinedby.pointsomewhere.Thederivativeofanyoperator

Ourmaintechnicalresultthensaysthat

(1)

where

isthecanonicalclassofembeddingand

ofthisrelationis

isthemapinducedbythediagonal.Animmediatealgebraicconsequence

(2)

for.Byinductiononeconcludesthattheoperatorsandsufficeto

.generateall,

withthestandardoperatorThecommutatoroftheCherncharacteroperator

canbeexpressedintermsofhigherderivativesof:

Thispaperisorganisedasfollows:InSection1werecallthebasicgeometricnotionsusedinthelaterparts.Section2providesanintroductiontoNakajima’sresults.

inSection3containsthecoreofthispaper:wefirstdefineVirasorooperators

analogytothestandardconstructionandshowhowthesearisegeometrically.Wethenintroducetheoperatorandcomputethederivativeof.Finally,inSection4weapplytheseresultstocomputetheactionoftheChernclassesoftautologicalbundles.

DiscussionswithA.KingwereimportanttomeinclarifyingandunderstandingthepicturethatNakajimadrawsinhisveryinspiringarticle.IamverygratefultoG.EllingsrudforallthethingsIlearnedfromhistalksandconversationswithhim

3

aboutHilbertschemes.Tosomeextendtheresultsinthisarticleareareflectiononaninductionmethodentirelyduetohim.MostoftheresearchforthispaperwascarriedoutduringmystayattheSFB343oftheUniversityofBielefeld.OnvariousoccasionsIwasallowedtolectureonHilbertschemesandtheircohomologyintheseminarofthealgebraicgeometrygroupinBielefeld:itisapleasuretothankS.Bauer,R.BrusseeandT.Zinkfortheirwillingnesstolistenattentivelyandcriticallyeventonotyetfullycorrectpreliminaryresults.IowespecialthankstoS.Bauerforhiscontinuousencouragement,interestandsupport.Bielefeld,8October,1997.

ManfredLehn

Contents

1Preliminaries

1.1Symmetricproducts..............1.2HilbertschemesandHilbert-Chowmorphism1.3Hilbertschemesofsmoothsurfaces.....1.4Incidenceschemes..............

....................................................

55678111213151521243134343044

2Thestructureofthecohomology

2.1Correspondences............................2.2Nakajima’sMainTheorem.......................3Theboundaryoperator

3.1Virasorogenerators.............3.2TheboundaryoftheHilbertscheme....

.............3.3Thederivativeof

3.4Thevertexoperator,completionoftheproof

........................................................

4Towardstheringstructureof

4.1Tautologicalsheaves..........................4.2Thelinebundlecase..........................4.3TopSegreclasses............................References

4

1Preliminaries

Inthissectionweintroducethebasicnotationsthatwillbeusedthroughoutthepaperandcollectanumberofresultsfromtheliterature,mostlywithoutproof.

1.1Symmetricproducts

actsonbyLetbeaquasi-projectiveschemeover.Thesymmetricgroup

permutationofthefactors,andthereexistsageometricquotientfor

isagainquasi-projective,andifisirreducible(reduced,integralorthisaction.

.Moreover,thisconstructionisfunctorial:anynormal)thenthesameistruefor

morphisminducesamorphism.

.Itfollowsfromthetheoremonelementarysymmetricfunctionsthat

Consequently,thesymmetricproductsofsmoothcurvesareagainsmooth.Ontheotherhand,ifisasmoothvarietyofdimensiongreaterthanone,thenissingu-.larfor

ByaresultofGrothendieck[15],thenaturalmap

isanisomorphismontothesubringofinvariantelementsundertheactionof

thisMacdonaldcomputedthefollowingformulafortheBettinumbersofpurelyalgebraicargument:

.Frombya

Theorem1.1(Macdonald[20])—TheBettinumbersofthesymmetricproductsaregivenbytheformulaThereisanotherpropertyofthesymmetricproduct,whichisimportantforthedefi-nitionoftheHilbert-Chowmorphism.Considerthefollowingset-valuedcontravariantfunctoronthecategoryoflocallyNoetherian-schemes:

betheprojection.ThenLetbea-scheme,andlet

isthesetofallisomorphismclassesofcoherentsheaveson,whereis-flat,

isafinitemap,andislocallyfreeofrank.Ifis

mapstheclassoftoa-morphism,then

.Hereandinthefollowingwewriteinsteadof.Grothendieck[14]assertedthatthereisanaturaltransformation

sendinganyzero-dimensionalsheaftoitsweightedsupport.Thismeansthatforanythereisaclassifyingmorphismsuchthat

forall,whereforanycoherentsheafwelet

5

denotethelengthofthestalkasamoduleover.Moreover,forany.

ThiswasfirstprovedbyIversen[18]usingthetechniqueoflineardeterminants.Infact,ifisnormalthencorepresentsthefunctor(cf.[16,Ex.4.3.6]).

1.2HilbertschemesandHilbert-Chowmorphism

Throughoutthispaper,theterm‘Hilbertscheme’willalwaysrefertoHilbertschemesofzero-dimensionalsubschemes.

Letbeaquasi-projectiveschemeover.TheHilbertfunctoristhefollowingset-valuedfunctoronthecategoryoflocallyNoetherian-schemes:

bethesetofallclosedsubschemessuchthattheLet

projectionisflatandfiniteofdegree.Ifisa-morphism,

.theinducedmapisgivenbypull-back:

isrepresentedbyaquasi-projectiveGrothendieck[14]showedthat

scheme.Ifisprojective,isprojectiveaswell.

isan(open,closed)‘Functoriality’inislimitedtoafewcases:if

immersion,thenthereisanatural(open,closed)immersion

definedbytakingtheimageofsubschemesunder.Moreover,supposethatisane´tale(surjective)morphism.Letdenotetheopensubsetofall

suchthattheset-theoreticsupportofismappedinjectivelyto.subschemes

´tale(surjective)morphism.Thentakingimagesunderdefinesane

Forsmallvaluesofthereareexplicitdescriptionsof:Clearly,isa

,andisthequotientforthe-actionontheblow-upofreducedpoint,

alongthediagonal.Proceedingbyinduction,itisnotdifficulttoseethatall

areconnectedifisconnected.Hilbertschemes

Observethatthereisanaturaltransformationoffunctors

whichsendsasubscheme

isrespresentedby

toitsstructuresheaf.As

,thistransformationinducesamorphismofschemes

theHilbert-Chowmorphism.Onapointmorphismisgivenby

,i.e.asubscheme

,this

Forexample,if

isasmoothcurve,then

6

isanisomorphism.

1.3Hilbertschemesofsmoothsurfaces

denoteasmoothirreducibleprojectivesurface.Thebasicgeo-Fromnowon,let

metryoftheHilbertschemesofpointsonsurfacesisgovernedbytwotheoremsduetoFogarty[9]andBrianc¸on[1].

Theorem1.2(Fogarty)—variety.isa-dimensionalsmoothirreducibleprojectiveHereisashortsketchoftheproof:projectivityisduetoGrothendieck.Healso

atapointiscanonicallyisomorphictoshowedthattheZariskitangentspaceof

.Sincewealreadyknowthatisconnected,itthereforesufficesto

forall.ThiscanbedoneusingSerredualityshowthat

andtheHirzebruch-Riemann-RochTheoremappliedtothegroups.

Remark1.3—Wealreadymentionedthatissmoothforsmoothcurves.Com-issmoothforaputingthedimensionofthetangentspaceonecanshowthat

issingularifsmoothvarietyofanydimension.Ontheotherhand,

and.

andletdenotetheclosedsubsetofallsubschemesFixapoint

with(withthereducedinducedsubschemestructure).Thisis

indeedaclosedsubset,asitisthefibreoftheHilbert-Chowmorphismover

.thepoint

Letdenotethelocalringofat.Sinceanypointmaybe

,andsince,allconsideredasasubschemeof

schemes—forvaryingand—are(non-canonically)isomorphic.Clearly,

and,moreoveritisnottoodifficulttoseethatis

,thevertexoftheisomorphictotheprojectiveconeoverthetwistedcubic

.Itisnotaccidentalthatintheseconecorrespondingtothesubscheme

examplesthedimensionofincreasesbyoneineachstep:

Theorem1.4(Brianc¸on)—Forall.,isanirreduciblevarietyofdimensionForaproofsee[1].AnewproofwithamoregeometricandconceptualargumentwasrecentlygivenbyEllingsrudandStrømme[7].

Brianc¸on’sTheorememphasisestheimportanceofcurvilinearschemes:recallthat

iscalledcurvilinearat,ifiscontainedazero-dimensionalsubscheme

.Equivalently,onemightsaythatisisomorphictoinsomesmoothcurve

the-algebra,where.Henceiscurvilinearatifiseither

.Fromthiscriterionitisclear,thatinanyflatempty,areducedpoint,orif

familyofzero-dimensionalsubschemesthepointsinthebasespacewhichcorrespondtocurvilinearsubschemesformanopensubset.

7

Inparticular,wemayconsidertheopensubsetnicestructure:Lemma1.5—If.Thissethasavery

,thenthemorphismisabundlemorphismwithaffinefibressmoothvarietyofdimension..Inparticular,isanirreducibleProof.Let

sendingthe

-tuple

belocalcoordinatesandconsidertheopensubset

.Thenthereisanisomorphism

tothesubsheafcorrespondingtotheideal

.

AsaconsequenceofthislemmaweseethatBrianc¸on’sTheoremisequivalenttosayingthatisdensein.Thisisaveryimportantinformation:curvi-linearsubschemesarefareasiertohandlethananyoftheothers.Theycontainonlyonesubschemeforanygivensmallerlength,anysmalldeformationofacurvilinearsubschemeisagainlocallycurvilinearetc.

slightly,letdenotethediagonal,andGeneralisingthedefinitionof

let,endowedwiththereducedinducedsubschemestructure.Thus

consistsofallsubschemesoflengthwhicharesupportedatsomepointin.Thefibresofthesurjectivemorphismaretheschemes

consideredabove.Infact,achoiceofregularparametersnearapointleadstoatrivialisationofthemorphismnear,i.e.isafibrebundlefortheZariskitopology.

AsanimmediateconsequenceofBrianc¸on’sTheoremweget

Corollary1.6—isanirreduciblevarietyofdimension.Notethatandhavecomplementarydimensionsassubvarietiesin.Theirhomologicalintersectionisthereforezero-dimensional.However,theinclusion

complicatesthecomputationoftheintersectionproduct.Thefollowing

resultwasobtainedbyEllingsrudandStrømme[7]byaninductivegeometricargu-ment:

Theorem1.7(Ellingsrud,Strømme)—.1.4Incidenceschemes

Since

representsthefunctor

,thereisauniversalfamilyofsubschemes

8

Again,forsmallvaluesofthereareexplicitdescriptions:isempty,isthe

,andistheblow-upofthediagonalin.Thediagonalin

identificationisgivenbythequotientmapandanyofthetwoprojections.

Assumethat.Thenthereisauniquelydeterminedclosedsubscheme

withthepropertythatanymorphism

factorsthroughcorrespondtopairs

ifandonlyif

ofsubschemeswith

.Closedpointsin

.Let

denotethetwoprojections.Then

parametrisestwoflatfamilies

Considerthecorrespondingexactsequence

(4)

isacoherentsheafonwhichisflatoverTheidealsheaf

andfibrewisezero-dimensionaloflength.Itthereforeinducesaclassifyingmor-phismtothesymmetricproduct,analogouslytotheHilbert-Chowmorphism,whichwewillalsodenoteby

Asbeforelet

isatriplein

,where

withand

isthesmalldiagonal.Apoint

.

Wemaydecompose

intolocallyclosedsubsets

,

,with

Lemma1.8—tively,andof.andareirreducibleofdimensionforall.Moreover,and,respec-iscontainedintheclosureProof.If

or,themap

isanopenimmersion

ItfollowsfromBrianc¸on’sTheoremthat

isirreducibleand

For

considertheembedding

9

Infact,theimageofiscontainedinaproperclosedsubsetofthetargetvariety:For

iscurvilinear,inwhichcasethereisonlyauniquesubschemeofeither

length,orisnotcurvilinearandthereforecontainedinaproperclosedsubsetof

.Now,thevarietyontherighthandsidehasdimension

Finally,ageneralpointinisoftheformwhereisacurvilinearsubschemesupportedatanddisjointfrom.Nowitiseasytodeformtoasubschemewithsupportedatapoint.Henceageneralpointofdeformsinto.

Definition1.9—Foranypairofnonnegativeintegersdefinesubvarieties

asfollows:iftively.Moreover,

betheclosureofand,respec-,and,whereas

.Ontheotherhand,if,letandunderthetwist

let

and

Byconstructionandareemptyorirreduciblevarietiesofdimension

and,respectively.

,themostbasicofallincidenceLetusreturntotheparticularcase

situations:considertheprojectivisation.Itisaneasyexercise

suchthatthediagramtoseethatthereisanaturalisomorphism

commutes.

Theorem1.10(Ellingsrud,Strømme[7])—Theincidenceschemeirreduciblevariety.isanAnimmediatecorollaryisthefollowing:thereisanaturalclosedimmersion

;sincebothareirreduciblevarieties,thismustbeaniso-definedabove.morphism.Theexceptionaldivisorispreciselythevariety

Henceinthissituationwemaywritethesequence(4)as

(5)

Infact,theincidenceschemeissmooth.Thishasindependentlybeprovedby

Ellingsrud,TikhomirovandCheah.Theproofsareunpublished.

10

2Thestructureofthecohomology

Asbefore,letbeasmoothirreducibleprojectivesurface.ByFogarty’sTheoremtheHilbertschemesareprojectivemanifoldsofrealdimension.Themotivating

intermsoftheprobleminthisstudyistounderstandthecohomologyrings

cohomologyring.

Asfarasthevectorspacestructureofthecohomologyisconcerned,i.e.ifweonlyaskforthedimensionsofthegradedpiecesofthecohomology,thisproblemwassolvedbyG¨ottsche[11].TheanswerisgivenbythefollowingbeautifulformulafortheBettinumbers.

aredeterminedbytheBettiTheorem2.1(G¨ottsche)—TheBettinumbersnumbers.Moreprecisely,thefollowingformulaholds:G¨ottschesoriginalproofusestheWeilConjectures[11].Foradifferentapproachsee[3].

AmongotherthingsonelearnsfromthisformulathatitisagoodideatoconsiderallHilbertschemessimultaneously.ThiswillbecomeevenmorestrikingthroughNakajima’smethodwhichwewillreviewinthenextsections.Asapreparationwecollectafewdefinitions:

denotethedoublegradedvectorspacewithDefinition2.2—Let

components.Sinceisapoint,.Theunitin

iscalledthe‘vacuumvector’anddenotedby.

Alinearmapishomogeneousofbidegreeifforalland.Ifarehomogeneouslinearmapsofbidegree

,respectively,theircommutatorisdefinedby

and

Weusethenotation,etc.todenotethecohomologicaldegreeofhomogeneous

cohomologyclasses,homogeneouslinearmapsetc.

Setting

foranyonadjoint

definesanon-degenerate(anti)symmetricbilinearform

andhenceon.Foranyhomogeneouslinearmapitsischaracterisedbytherelation

Clearly,

.

11

2.1Correspondences

besmoothprojectivevarieties,andletbeaclassintheChowgroup.(Wetacitlyassumerationalcoefficients.Thiswillnotalwaysbeneces-sary.Ontheotherhand,wearenotinterestedinintegralityquestionsforthemoment,andhencewillnotpayattentiontothisproblem).Theimageofinwillbedenotedbythesamesymbol.inducesahomogeneouslinearmapLet

and

whereisthePoincar´edualitymap.

Assumethatisanothersmoothprojectivevariety,and.Letbetheprojectionfromtothefactors,andconsidertheelement

Then

See[10,Ch.16]fordetails.Supposeandand.Let

areclosedsubschemessuchthat

Thentheclassdefinedaboveisalreadydefinedin.

Thefollowingtypeofargumentswilloftenshowupinthesequel:oneshowsthat

issmallerthanthedegreeof,whichforcestobezero;orthedimensionof

ofofmaximaldimensionwiththatthereisatmostoneirreduciblecomponent

‘correct’dimension.Inthiscaseonemusthaveanditsufficestodeterminethemultiplicity.

exchangethefactors.ThenaChowcycleinducesLet

twomaps

and

whicharerelatedbytheformula

Thisfollowsdirectlyfromtheprojectionformula.Thus.

ThefollowingoperatorswereintroducedbyNakajima[21].Thestudyoftheirpropertiesisthemajorthemeofthisarticle.Wetakethelibertytochangethenotationsandsignconventions.

Recallthatwedefined(1.9)subvarities

12

ofdimension

.Theirfundamentalclassesarecycles

Lettheprojectionstothefactorsbedenotedby

,

and

.

Definition2.3(Nakajima)—Definelinearmaps

asfollows:assumefirstthat

.For

and

let

Theoperatorsfornegativeindicesthenaredeterminedbytherelation

Bydefinition,isahomogeneouslinearmapofbidegree.

,andif,theoperatorisinducedbythesubvarietiesMoreover,

,.

2.2Nakajima’sMainTheorem

Inthissectionwereviewthemainresultof[21]andsomeoftheimmediateconse-quences.SimilarresultshavebeenannouncedbyGrojnowski[13].

andcohomologyclassesTheorem2.4(Nakajima)—Foranyintegersandand,theoperatorsandsatisfythefollowing‘oscillatorrelations’:Hereandinthefollowingweadopttheconventionthatequalsifandiszeroelse,andthatanyintegraliszeroif.

In[21]Nakajimaonlyshowedthatthecommutatorrelationholdwithsomeuniver-salnonzeroconstantinsteadofthecoefficient.ThecorrectvaluewasfirstcomputeddirectlybyEllingsrudandStrømme[7]:uptoasignfactor,whichdependsonourcon-vention,thisnumberistheintersectionnumberofTheorem1.7.Brieflyafterwards,Nakajimagaveadifferentproofusing‘vertexoperators’[22].

Considerthevectorspaces

and13

Defineanon-degenerateskew-symmetricpairingonthevectorspaceby

Notethatwearetakingtheexpression‘skew-symmetric’inagradedsense:

TheHeisenbergalgebraisthequotientofthetensoralgebra

withidealgeneratedbytheexpressions

bythetwo-sided

:

isthe(restricted)tensorproductofcountablymanycopiesofCliffordalgebras

andcountablymanycopiesofWeylalgebrasarisingfromarisingfrom

.Asisisotropicwithrespecttotheskew-form,thesubalgebra

isthesymmetricalgebra(takenagaininagradedsense).ingeneratedby

asThisbecomesadoublegradedvectorspaceifwedefinethebidegreeof

.

Usingthesenotations,Nakajima’sTheoremcanberephrasedbysaying:Sendingtodefinesarepresentationofon.

ofmonomialsofnegativedegreeannihilatesthevacuumvectorThesubspace

forobviousdegreereasons.Hencethereisanembedding

ItisnotdifficulttocheckthatthePoincar´eseriesof

ofG¨ottsche’sformula.Thisimplies:

equalstherighthandside

Corollary2.5(Nakajima)—Theactionofoninducesamoduleisomorphism.Inparticular,isirreducibleandgeneratedbythevacuumvector.Infact,thiscanbestrengthenedasfollows:Considertherationalmap

withdisjointsupportbyopensubsetofallpairs

mapinduceshomomorphisms

whichisdefinedonthe

.Thisrational

and

andhence

and

Corollary2.6(Nakajima,Grojnowski)—ThehomomorphismandendowwiththestructureofaHopfalgebra.IfisgiventhecanonicalHopfalgebraisanisomorphismofHopfstructureofthesymmetricproduct,thenalgebras.14

3Theboundaryoperator

Thissectioncontainsthemaintechnicalresultsofthepaper.ThekeytooursolutionoftheChernclassproblemistheintroductionoftheboundaryoperator.Thisisdonein3.2.Webeginwiththediscussionofrelatedtopicsandingredientsforlaterproofs.

3.1Virasorogenerators

andthefundamentaloscillatorrelationswewillStartingfromthebasicgenerators

definethecorrespondingVirasorogeneratorsinanalogytotheprocedureincon-formalfieldtheory.Wewillthengiveconcretegeometricinterpretationsforthesegenerators.

bethepush-forwardLet

mapassociatedtothediagonalembedding.Equivalently,thisisthelinearmapad-,wewillwriteforjointtothecup-productmap.If

Definition3.1—Defineoperators

,

,asfollows:

Theorem3.3—Theoperatorsandsatisfythefollowingcommutatorrelations:1.2.Proof.Assumefirstthat

.Foranyclasses

and

with

wehave

Ifwesumupoverall

and,weget

with

Similarly,for

,

Thussummingupoverall

wefindagain

Thisprovesthefirstpartofthetheorem.

Asforthesecondpart,assumefirstthattionsletusagreethat

.Inordertoavoidcaseconsidera-

Bythefirstpartofthetheoremwehave

Inthefollowingcalculationwesuppress

,weget:overall

and

uptotheveryend.Summingup

16

Hence

Nowsplitoffthesummandscorrespondingtotheindicesfromthesums.Substitutingforinthesecondsumontherighthandside,weareleftwiththeexpression:

if

isodd,

and

if

iseven.

.

Aneasycomputationshowsthatinbothcases

Recallthedefinitionofthevarieties

equals

in(1.9).

Definition3.4—Letbeanonnegativeintegerandlet

bethelinearmap

for

and

.

Thefollowingtheoremgivesa‘finite’geometricinterpretationoftheinfinitesums

whichdefinetheVirasorooperators.

17

Theorem3.5—Letbeanonnegativeinteger.1.ifelseor2.Wehaveseenearlier(1.8)thatthissethasdimensiondisregarded.Ontheotherhand

andhencemaybe

Againusing1.8weseethatthissethasonlyonecomponentof(maximal)dimension

.Moreover,thiscomponentistheimageoftheembedding

Let

and

.Thenwehave

Thisshowsthat

forsomeinteger.Henceitremainstocomputethemultiplicityof

andinspecttheintersectionofendwepickageneralpoint

andalongthefibre.

Ageneralpointinisoftheform

in.Tothis

with

whereisacurvilinearsubschemeofoflength,supportedinasinglepointwhichisdisjointfrom.Sinceiscurvilinear,thereisauniquesubschemeoflength,andhenceconsistsofthesinglepoint

Nearthevarietiesandarelo-callyisomorphic;andsimilarlytoandto

.Thuswemaysplitoffthefactorsfromthegeometricpicture.Inthe

.endthisamountstosayingthatwemayassumewithoutlossofgeneralitythat

Moreover,thecalculationislocalin,sothatwemayassumethat

and,and.Thenhasan

inwithcoordinatefunctionsaffineneighbourhood

whichparametrisesquadrupels

ofsubschemesin

givenbytheideals

19

where

and

Now

belongsto

,i.e.

,ifandonlyif

and

(6)

And

satisfied:

belongsto,i.e.

ifandonlyifthefollowingthreeconditionsare

and

(7)

withpolynomialsandsupportedat,i.e.

ofdegree

and,respectively;theidealsheaf

is

and

(8)

andfinally,

mustbecontainedin,whichimposesthecondition

(9)

Oneeasilychecksthattheequations(6)-(8)cutoutasmoothsubvarietywhichprojectsisomorphicallytotheaffinespace.Moreover,inthesecoordinatesthelastcondition(9)simplyreads.Hencethemultiplicityequalstheexponent.

Next,weconsiderthecasewith.Thereisnothingtoprove

.Henceassumethat.Dimensionargumentssimilartotheonesaboveif

showthatthecyclewhichinducesthecommutatormustbesupportedontheclosedsubsets

hasdegree,sothatitsufficestoshowthat.ThisfollowsfromLemma1.8.

with.AdimensioncheckoftheItremainstoconsiderthecase

set-theoreticsupportoftheintersectioncycleshowsthatwemusthave

Thecycle

forsomeinteger,independentlyof

braicallyandtakethecommutatorwith

and

.Todetermine:

,weproceedalge-20

Ontheotherhand,combiningtheJacobiidentity,theoscillatorrelationsandthefirstpartoftheproofyields

Itfollowsthat

.

Ad2:Considerthedifference

Ifandarepartitionsof,then

ifandonlyifthereisasurjectiontainedintheclosureof

iscon-

suchthat

forall.Itfollowsthat

Proof.Considerthefollowingincidenceschemewiththenaturalprojections:

22

Wehaveseenearlierin1.4that

.Thisshows

andhencethat

Ontheotherhand,byLemma3.7,

Therefore,ifweput

forall

Foranyendomorphismitsderivativeis

forthehigherderivatives.

.Asusual,wewrite

Itfollowsdirectlyfromthedefinitionofthecommutatorthat

the‘Leibnizrule’holds:i.e.foranytwooperators

isaderivation,

and

Moreover,if

,sothatand

isahomogeneouslinearmap,then

havethesameparity.Furthermore,

Indeed,

Let

benonnegativeintegers,andconsidertheincidencevariety.Recallthedefinitionoftheidealsheafandtheexactsequence

Thenisalocallyfreesheafofrankon.

Inacertainsense,thefollowinglemmasimplyisareformulationofthedefinitionofthederivative.

23

Lemma3.10—Letassociatedtoaclass.ThenbetheinducedlinearmapProof.Let

.Then

with

,and

3.3Thederivativeof

Inordertounderstandtheintersectionbehaviouroftheboundaryweneedtoknowhowtheoperatorcommuteswiththebasicoperators,inotherwords:weneedtocomputethederivativeof.

Thefollowingtheoremisthemaintechnicaltheoremofthispaper.Itdescribesthederivativeoftheoperatorintwoways:Byitsactiononanyoftheotherbasicoperators,andasapolynomialexpressioninthebasicoperators.

Letdenotethecanonicalclassofthesurface.

Theorem3.11—Forallandthefollowingholds:1.Corollary3.12—Theoperatorsfromthevacuum.and,,sufficetogenerateProofofthetheorem.Thesecondassertionisanimmediateconsequenceofthefirst:byNakajima’srelations2.4andtherelations3.3weseethat

24

Hencethedifferenceofandtheexpressionontherighthandsideinthetheorem

,.Sinceisanirreducible-module,itcommuteswithalloperators

followsfromSchur’sLemmathatthisdifferenceisgivenbymultiplicationwithascalar(say,afterpassagetosomealgebraicclosureof).Butthisisimpossiblefordegreereasons:thebidegreeofis.(Thecasebeingtrivialanyhow.)

Theproofofthefirstassertionhastwopartsofquitedifferentnature:Weneedto

andanddealwiththemseparately.distinguishthecases

Proposition3.13—withandcohomologyclasses.foranytwointegersProof.Step1:Assumethatandarepositive.WeproceedasintheproofofTheorem3.5.Letbenonnegative,andconsiderthediagram

Let

AccordingtoLemma3.10,theoperator

isinducedbytheclass

anddenotetheopensubsetsofthosetuplesand

,respectively,whereeitherorbutiscurvilinear.Certainly,

,butinfactweevenhavethatisanisomorphism:

fortheconditionsimposedonimplythatisalreadydeterminedbytheremaining

.data

isirreducibleofdimension.Claim:

ForitfollowsfromBrianc¸on’sTheoremthattheopenpartisir-,andtuplesofthesecondkind,i.e.reducibleofdimension

withcurvilinear,areeasilyseentodeformintothisopensubset.

.Inparticular,thecomplementofinClaim:

cannotsupportanycontributionto.

hasastratificationIndeed,theset

,wherethestratumisthelocallyclosedsetofalltupleswith.Letbetheclosedsubsetthatconsistsoftupleswhereisnotcurvilinear.

.Nowisirreducibleofdimension,Then

andisaproperclosedsubsetandthereforehasstrictlysmallerdimension.TheassertionnowfollowsfromLemma1.8.Let

25

Claim:Theintersectionofandgeneralpointsof.

Infact,theintersectionistransversalatallpointswithWeconclude,thattheintersectioncycleequals

istransversalat

andcurvilinear.

Thehomomorphism

isanisomorphismoffthediagonal.Ontheotherhandtheclo-sureofequalstheimageofthe‘diagonal’embedding

.Itfollowsthat

whereisthelengthofcoker

proves

atthegenericpointofthevariety

.This

anditremainstoshowthat

Ageneralpointofisoftheformwhere

andisacurvilinearsubschemesupportedat.AsthecomputationwemayapplythesamereductionprocessasintheproofofTheoremislocalin

,that,and3.5:wemayassumethat

.Thenthereisanopenneighbourhoodofthispointinwhich

suchthatthefamiliesisomorphicto

andaregivenbytheideals

and

where

.Wefind

Thecokernelof

isisomorphictothe

foreachsothat

,whicharesupportedonthediagonallyembeddedvarieties

,

forcertainconstants.Inordertodeterminetheseconstantsweapplythecommu-.Thentheoscillatorrelationsyieldfortherighthandsidetator

Ontheotherhand

Now

whichbyStep1equals

.Hence

Chooseclasseswith.Itfollowsthat.

Step3:Thegeneralcasecannowbereducedformallytothecasesalreadytreated.Theassertioniscertainlytrivialifeitheror.Iftheassertionisknownto

,wemayapplytheoperationtobothsidesandfind:betrueforsomepair

Thisandtheidentity

allowustoreduceanythingtocasescheckedinStep1andStep2.

Inordertoprovepart1ofTheorem3.11,itremainstotreatthecase

.Thiswillbedoneintwosteps.First,weproveaqualitativestatementaboutthestructureofthe‘correctionterm’,andafterwardswedeterminetheprecisevalueofthe‘coefficient’:

28

Proposition3.14—Thereexistrationaldivisorsandandsuchthat,,with(10)

forall.Proof.Thereisnothingtoprovefor

.Moreover,

Itfollowsthatifthereisadivisorsothat(10)holdsfor,then(10)alsoholdsforwiththechoice.Henceitsufficestoprovethepropositionforpositiveintegers.

Letbeanonnegativeintegerandconsiderthediagram

Let

AccordingtoLemma3.10,theoperator

isinducedbytheclass

first.Itiscontainedin,where

.Theclosureofisthediagonal

andisthereforeirreducibleofdimension.Whereasfor,

embedsintotheirreduciblevarietyofdimensiontheset

.

Theoff-diagonalpartisemptyif.Ifithasprecisely

ofmaximaldimension:itcontainsasadenseoneirreduciblecomponent

subsettheimageoftheembedding

Considerthediagonalpart

and

arepairwisedisjoint

Sincethefunction

itfollowsthat

issemicontinuousandisatleast

on

,

dimension.Aswewanttocomputeacycleofdegree,wemayrestrictour

andmaydisregardthecomplementofinitsclosure.attentiontotheopenpart

isanisomorphism,whichweusetoidentifyandthe

parametrisesfourflatfamiliesofsubschemeson:off-diagonalpartof.Now

besidesthefamiliesandoffibrewiselength,thesearethefamiliesand

offibrewiselengthand.Thecontributionoftoistheclass

Reversingtheorderoftheoperatorsandshowsthatthepartofthecycleinducingthecommutator,thatissupportedon,istheclass

Sincetheidealsheavesandareisomorphic,thisclassiszero.

Thuswemayfullyconcentrateonthecontributionofthediagonalpart.(Also

anydiagonalpartsmustbecontainedinnotethatforthereversedorder

andarethereforetoosmallandirrelevant.)

inhascodimensionThecomplementoftheopensubset

.Locallynearthereareisomorphismsbetweenand,andsimilarlybetweenand.Henceifisthe

,thenthegeneralcycleissimplygivenbyintersectioncycleforthespecialcase

.Butthatwasallwehadtoprove:acycleofthis

forminducesthelinearmap

Corollary3.15—ForallpositiveintegersonehasProof.Usethesameargumentasinthefirstparagraphoftheproofofthemain

theoremafterCorollary3.12.

TofinishtheproofofTheorem3.11itremainstoshow:Proposition3.16—Forallpositiveintegerssition3.14isgivenbytherationaldivisordefinedbyPropo-3.4Thevertexoperator,completionoftheproof

beanelementwhichisofevendegreethoughnotDefinition3.17—Let

,necessarilyhomogeneous,andletbeaformalparameter.Defineoperators

,by

Proof.Assumefirstthatwith.Then

isanoperatorofevendegree,andthat

commutes

Next,letbeafamilyofcommutingoperatorsofevendegreesuchthatany

commuteswithevery.ThenitfollowsfromStep1and

31

that

resultsgets

with

and

anduseourprevious

.One

where

isthenumberofpairsofpositiveintegers.

and

thataddupto,i.e.,

Letbeasmoothprojectivecurve.Theboundaryintersectsgenericallytransverselyintheboundaryof,i.e.inthesetofalltupleswithmultiplepoints.Thesubvarietiesandhavecomplementarydimensions

andinandwemaycomputetheintersectionnumber

Wewilldothisfirstusingouralgorithmiclanguage,andafterwardsusingageometricargument.Thecomparisonofthetworesultswillleadtotheidentificationofthedivisors.

Lemma3.20—and.Proof.Thefirstassertionfollowsfromthedefinitionoftheoperators

istheclassofthesubmanifoldNakajima’sTheorem,

henceaccordingtoLemma3.8:

.By,and

Lemma3.21—32

Proof.Indeed,

sinceandwith

is

.Nowcommuteswithanyproductif,.Thustheonlysummandinthatcontributestothecommutator

.Hence

ThisfinishestheproofofTheorem3.11.

33

.

4Towardstheringstructureof

4.1Tautologicalsheaves

aseriesoftauto-Thereisanaturalwaytoassociatetoagivenvectorbundleon

,.TheChernclassesofthelogical’vectorbundlesontheHilbertschemes

tautologicalbundlesmaybegroupedtogethertoformoperatorson.

Considerthestandarddiagram

Let

on

bealocallyfreesheafonisdefindedas

.Foreach

theassociatedtautologicalbundle

Since

isaflatfinitemorphismofdegree,

islocallyfreewith

Notethatand.

isashortexactsequenceoflocallyfreeFurthermore,if

sheaveson,thecorrespondingsequenceisagainexact.Hencesendingtheclassofalocallyfreesheaftogivesagrouphomomorphism

Definition4.1—Let

beaclassin

.Defineoperators

and

asfollows:Foreach

thetotalChernclass

Let

,theactionon

andtheCherncharacter

isgivenbymultiplicationwith

,respectively.

and

bethedecompositionsintohomogeneouscomponentsofbidegree.Sincealloftheseoperatorsareofevendegreeandonlyact‘vertically’onbymultiplication,theycommutewitheachotherandinparticularwiththepreviouslydefinedboundaryoperator.

Moreover,wehave

and

forall

.

34

Theorem4.2—Letbeaclassinofrankandlet.Thenor,moreexplicitely,HereweusedLemma3.10whichsaysthatthecycleinducestheoperator

.ThisistheequationfortheCherncharacter.TheequationforthetotalChernclassisprovedanalogously.

Remark4.3—Thesequence(11)wasusedbyEllingsrudinarecursivemethodtodetermineChernclassesandSegreclassesoftautologicalbundles(unpublished,butsee[25],[4]).HeexpressestheclassesintermsoftheSegreclassesofthe

.Thusoneneedstocontrolthebehaviouroftheseuniversalfamily

Segreclassesundertheinductionprocedure.Thismethodyieldsqualitativeresultsonthestructureofcertainclassesandintegrals,butallattemptstogetnumbershaveendedsofarinunsurmountablecombinatoricaldifficulties.

Remark4.4—Theresultsofthepresentandtheprevioussectionprovideanalgo-rithmicdescriptionofthemultiplicativeactionofthesubalgebrawhichisgeneratedbytheChernclassesofalltautologicalbundles:Theelements

generateasa-vectorspace.ByCorollary3.12,eachsuchelement

,whereisawordinancanbewrittenasalinearcombinationofexpression

,.ByTheorem4.2thealphabetconsistingofandoperators

commutatorofwithanyoftheseisagainawordinthisalphabet.Andfinally,Theorem3.11showshowsuchawordcanbeexpressedintermsofthebasicoper-ators.Admittedly,withoutafurtherunderstandingofthealgebraicstructurethis

onlyforsmallvaluesoforifdescriptionisusefulforcomputationsin

oneimplementsitinsomecomputeralgebrasystem.

4.2Thelinebundlecase

TheresultsoftheprevioussectionsufficetocomputetheChernclassesofthetauto-logicalbundlesassociatedtoalinebundleintermsofthebasicoperators.

Theorem4.5—Letbealinebundleon.Then(12)

36

ThisisNakajima’sresult3.18:forsupposeisasmoothcurveand

,thenaturalhomomorphismvanishesifandonlyifIf

Hencethevanishinglocusoftheglobalvectorbundlehomomorphism

.

.

isthesubvariety.ThereforeNakajima’sformula3.18

.Insertingthisinto(12),werecover

Expandingtherighthandsideyieldssummandswhicharewordsinthetwosymbols

and.Movingallfactorswithinagivenwordasfartothe

rightaspossibleusingthecommutationrelationsofthemaintheoremwecanwrite

Let

denoteapartitionandlet

,and

.Weget

arisesfromawordinandoflength.Itisnotdifficulttoseethat

equalsthenumberofpossibilitiestopartitionasetofelementsintosubsetsinsuchawaythattherearesubsetsofcardinality.Hence

Insertingthisintoequation(13)aboveonegets

fromtherightandsumupoverall

:

Thismeansthattheseries

38

satisfiesthelineardifferentialequation

Wefind

Thisshows

Hence

satisfiesthesystem(14)and(15)aswellandthereforeequals.Thisprovesthetheorem.

39

4.3TopSegreclasses

ThefollowingproblemwasposedbyDonaldsoninconnectionwiththecomputationofinstantoninvariants:letbeaninteger,andconsideralinearsystemof

inducingamap.Azero-dimensionalsubschemedimension

doesnotimposeindependentconditionsonthelinearsystemifthe

naturalhomomorphism

failstobesurjective.Thesubschemeofallsuchhasvirtualdimensionzero,

anditsclassisgivenby

Thusthenumberofthose

,whereisthevirtualvectorbundlethatimposedependentconditionsisgivenby

and40

Aslongasnoexplicitgeneratingfunctionisavailablewemustbecontentwiththefollowingsemi-explicitsolutiontoDonaldson’sproblem:

Thisyields:

andtherefore

Obviously,forhigher,thepracticalcalculationofquicklybecomesratherdifficult.Alreadythecaseofsurpassedmypersonalcalculationskills.UsingMAPLE,Icomputedthefollowingexpressions:

ThesecalculationsverifyLeBarz’trisecantformulafor[19,Th´eor`eme8]and

byTikhomirovandTroshina[26].Theformulaeforandthecomputationof

42

seemtobenew.Iomitthepresentationof:theinformationiscontainedinthefollowinganalysisofthesenumericaldata.

Itisalwayspossibletoorganisethesedataintothefollowingform:

(16)

Thefactthattherighthandsidein(16)dependslinearlyoncanbeprovedbythemethodsintheforthcomingpaper[4].

WethankDonZagierforpointingouttoustheexistenceofSloane’s‘Encyclope-diaofIntegerSequences’[24].Wehadhadreasonstobelievethatthesequenceof

.Afterdividingbycoefficientsofbedivisiblebythebinomialcoefficients

.Asearchforthisreducedthese,weareleftwiththesequence

sequenceintheencyclopediawassuccessfulandledtotheabovegiven(conjectural)identificationofthecoefficientsof.Unfortunately,thecorresponding‘reduced’se-quenceofcoefficientsofremainsmysterious:

43

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ManfredLehn

MathematischesInstitutderGeorg-August-Universtit¨atBunsenstraße3-5,D-37073G¨ottingen,Germanye-mail:lehn@uni-math.gwdg.de

45