Tautological rings of stable map spaces

arXiv:math/0607432v2[math.AG]12Aug2006TAUTOLOGICALRINGSOFSTABLEMAPSPACESANCAM.MUSTAT¸ˇAANDANDREIMUSTAT¸ˇAAbstract.Wefindasetofgeneratorsandrelationsforthesystemofextendedtautologicalringsassociatedtothemodulispacesofstablemapsingenuszero,admittingasimplegeometricalinterpretation.Inparticular,whenthetargetisPn,thesegiveacompletepresentationforthecohomologyandChowringsinthecaseswith/withoutmarkedpoints.IntroductionLetd∈H2(X)beacurveclassonasmoothprojectivevarietyX.ThespaceM0,0(X,d)parametrizesmapsfromrationalsmoothornodalcurvesintoXwithimageclassd,suchthatanycontractedcomponentcontainsatleast3nodes.OverM0,0(X,d)thereexistsatowerofmodulispacesofsta-blemapswithmarkedpointsandmorphismsM0,m+1(X,d)→M0,m(X,d)forgettingonemarkedpoint,suchthatM0,m+1(X,d),togetherwithaneval-uationmapevm+1:M0,m+1(X,d)→Xandmnaturalsectionsσi,formtheuniversalfamilyoverM0,m(X,d).Inthispaperweinvestigatetherelationbetweenthestructureoftheco-homologyringofthevarietyXandthatofM0,0(X,d),whichislessthanobviousinparticularwhenm=0.Whenmarkedpointsexist,pullbackbythenaturalevaluationmapsevi:M0,m(X,d)→Xgenerateasetofclassesonthemodulispace.Asystemoftautologicalringsfor{M0,m(X,d)}misconstructedfromtheseclassesbyanalogywiththemodulispaceofcurves.WhenthetargetspaceisanSLn–flagvarity,thesecoincidewiththecoho-mologyringsasshownin([17]).Weexpectthatthesameresultholdsforalargerclassofvarieties.TheGromov-WitteninvariantsofXandtheirgravitationaldescendantsareintersectionnumbersintheserings.SpecialtautologicalclassesarethefirstChernclassesψiofthetautologicallinebundlesoverM0,m(X,d).Anotherspecialsetoftautologicalclassesarethek–classeska(α)definedin[10]bytheformulaka(α)=f∗(ψa+1iev∗iα),wherefistheforgetfulmapabove.Wedefineasetofextendedtautologicalrings,offeringamoreconvenientencodingoftheboundary,andforwhichtheoriginaltautologicalringsareinvariantsubrings.Theorems2.5and2.6inthetextidentifyasetofgenera-torsandrelationsfortheseringsintermsofboundarystrataandk–classesDate:February2,2008.12ANCAM.MUSTAT¸ˇAANDANDREIMUSTAT¸ˇAk−1(αi)ofthegenerators{αi}ifortheringH∗(X).RelationsinH∗(X)in-ducerelationsinthetautologicalringsviak–classdecomposition.Naturaluniversalrelationsexistontheboundary.Inparticular,whenX=Pn,theaboveformacompletepresentationoftheextendedcohomology(andChow)ringsofM0,0(Pn,d),with/withoutmarkedpoints,ofasimplegeometricalinterpretation.(Theorem3.3intext).Thuswerecoverthedegree3and2casesformulatedbyBehrendandO’Halloranin[3].1.DefinitionofTautologicalringsForanysmoothprojectivetargetX,themodulispacesM0,m(X,d)parametris-ingpointedstablemapsofclassdintoXareDeligne-Mumfordstackssharingacoupleofcommonfeatureswiththemodulispacesofstablecurves:(1)naturalforgetfulmorphismsf:M0,m+1(X,d)→M0,m(X,d);(2)naturalgluingmapsMτ→M0,m(X,d).HereMτisafiberproductofmodulispacesM0,mi(X,di)alongevaluationmorphismstoX,suchthatPidi=dandMτrepresentsmapsintoXfromstablereduciblecurvesofasplittypeτ.WesaythatMτisaboundarystratum.TheuniquefeaturesofM0,m(X,d)consistintheexistenceofmevaluationmapsintoX,andtheexistenceofvirtualfundamentalclasses[M0,m(X,d)]virinChowgroupsandhomology([1]).Theseclassesarecompatiblewiththenaturalmorphismsabove([2])andcompensateforthefactthatM0,m(X,d)ingeneralisnotasmoothstackoftheexpecteddimension.WiththesewecanconstructasystemofringsinsideH∗(M0,m(X,d)).Thesearecalledthetautologicalrings,andaredefinedbyanalogywiththemodulispaceofcurves.WewillworkinhomologyofM0,m(X,d)viathehomomorphismσ:H∗(M0,m(X,d))→H∗(M0,m(X,d))σ(β)=β∩[M0,m(X,d)]vir.Theclassesσ(ev∗iα)andtheirpushforwardsviaforgetfulandgluingmapsarethefirstexamplesoftautologicalclasses.Wenotethattheforgetfulmapsadmitnaturalflatpullbackinhomology.ThegluingmapsjoftheboundarystrataadmitaGysinmap,whichwillbedenotedbyj∗.IntersectionoftautologicalclassesinH∗(M0,m(X,d))canbedefinedinacanonicalway.ConsiderthreemodulispacesM1,M2,Mrelatedbynaturalmorphismsf1,f2,andthefibresquarediagramM1×MM2˜f2˜f1//M2f2M1f1//MTAUTOLOGICALRINGSOFSTABLEMAPSPACES3andletf:=f1◦˜f2=f2◦˜f1.Heref1andf2arestrataembeddings,forgetfulmapsorcompositionsofthese,ortheidentity.Similarlywedefineσ:H∗(Mi)→H∗(Mi)σ(β)=β∩[Mi]vir,fori=1,2.Definition1.1.Theintersectionproductoftwoclassesf1∗(σ(β1))andf2∗(σ(β2))isdefinedasf1∗(σ(β1))·f2∗(σ(β2)):=f∗(˜f∗1β2·˜f∗2β1∩˜f∗1([M2]vir))Notethatbytheaxiomsofthevirtualclass,˜f∗1([M2]vir)=˜f∗2([M1]vir)=f∗([M]vir),sotheproductiswelldefined.Theproductisclearlyassociative.WemaythusdefineDefinition1.2.ThetautologicalsystemofringsofX,disthesetofsmall-estQ–algebrasinsidethecohomologygroupsR(M0,m(X,d))⊂H∗(M0,m(X,d))satisfyingthefollowingproperties:(1)R(M0,m(X,d))containsallclassesσ(ev∗i(α)),whereα∈H∗(X),i=1,...,m.(2)Thesystemisclosedunderpush-forwardviaallforgettingmaps:f∗:R(M0,m+1(X,d))→R(M0,m(X,d)).(3)Thesystemisclosedunderpush-forwardviaallgluingmaps:j∗:R(M0,A1∪{•}(X,d1))⊗QR(M0,{•}∪A2(X,d2))→R(M0,m(X,d))whereA1SA2={1,...,m}andd1+d2=d.(4)Thesystemisclosedundertheproductdefinedabove.1.1.ConsiderasmoothprojectivevarietyXandanembeddingX֒→Qsi=1PnisuchthatthealgebraicpartofH2(X,Z)isZs,generatedbyfirstChernclassesoftheveryamplelinebundlesL1,...,Lsgivi