The Hodge Numbers of the Moduli Spaces of Vector B

arXiv:math/0012260v1[math.AG]29Dec2000TheHodgeNumbersoftheModuliSpacesofVectorBundlesoveraRiemannSurfaceRichardEarlandFrancesKirwanFebruary1,20080IntroductionLetM(n,d)denotethemodulispaceofstableholomorphicvectorbundlesofcoprimeranknanddegreedoverafixedRiemannsurfaceΣofgenusg≥2.LetΛbeafixedlinebundleoverΣofdegreedandletMΛ(n,d)⊂M(n,d)denotethespaceconsistingofthosebundleswithdeterminantΛ.ThespacesM(n,d)andMΛ(n,d)arenonsingu-larcomplexprojectivevarietieswhosegeometryhasbeenmuchstudied.InparticularHarder,Narasimhan,DesaleandRamananfirstdescribedin1975aninductivemethodtodeterminetheBettinumbersofM(n,d)usingnumbertheoreticmethodsandtheWeilconjectures[12,7].Subsequentlyin1982AtiyahandBott[1]obtainedthesameinductivemethodusinggaugetheory.InthisnotewegiveasimilarinductivemethodfordeterminingtheHodge-Poincar´epolynomialsofM(n,d)andMΛ(n,d),thatisHP(M(n,d))(x,y)=Xp≥0Xq≥0hp,q(M(n,d))xpyq,andHP(MΛ(n,d))(x,y)=Xp≥0Xq≥0hp,q(MΛ(n,d))xpyq,wherehp,qdenotetheHodgenumbers(seeTheorem1andLemma3below).Theχ(t)-characteristicofMΛ(n,d),thatisχ(t)=HP(MΛ(n,d))(t,−1),hasanespeciallysimpleform(seeCorollary6),whiletheχ(t)-characteristicofM(n,d)isidenticallyzero.In[1]AtiyahandBottidentifyM(n,d)withthequotientofCs,theinfinitedimen-sionalspaceofstableholomorphicstructuresonafixedC∞complexbundleEofranknanddegreedoverΣ,byGc,theinfinitedimensionalgroupofsmoothcomplexauto-morphisms.TheyintroduceastratificationforC,theinfinitedimensionalaffinespaceofallholomorphicstructuresonE,whichisequivariantlyperfectwithrespecttotheactionofGc(orequivalentlythegaugegroupG)andwhichhasCsasanopenstratum.TheresultingMorseequalitiesgivetheequivariantcohomologyofthestablestratumandthusthecohomologyofM(n,d),intermsoftheclassifyingspaceofGandtheequivariantcohomologyoftheunstablestrata,whichcanbecalculatedinductively.In[14]themethodsofAtiyahandBottareadaptedtofinitedimensionalquotientsinthesenseofMumford’sgeometricinvarianttheory(GIT)[17,19],consideringthelinear1actionofacomplexreductivegrouponanonsingularcomplexprojectivevarietywhereeverysemistablepointisstable.ThismethodgeneralisestogiveaninductiveapproachdeterminingtheHodgenumbers[14,§14].ToapplythistothecaseofM(n,d)werecallfromNewstead[19]howthismodulispacemaybeexpressedasafinitedimensionalquotientinthesenseofGIT.Itisshownin[15]thattheresultingfinitedimensionalstratificationcomingfrom[14]correspondsnatu-rallytothestratificationofAtiyahandBottoutsideasubsetwhosecodimensiontendstoinfinitywithd,andtheequivariantcohomologyofthecorrespondingstrataagreesuptoadegreetendingtoinfinitywithd.SinceM(n,d)dependsondonlythroughitsremaindermodulon,wemaythenrefineAtiyahandBott’sinductiveformulasfortheBettinumberstogivetheHodgenumbersaswell.Inthecasen=2ourformulafortheHodgenumbersofMΛ(n,d)wasrecentlyprovedbydelBa˜noRollin[3,§3]andhadbeenindependentlydiscoveredbyNewstead[21].Theformulafortheχ(t)-characteristicofMΛ(2,1)wasfirstprovedbyNarasimhanandRamanan[18]in1975.Thelayoutofthispaperisasfollows.InSection1wereviewtheargumentsusedin[1]and[15]toobtaininductiveformulasfortheBettinumbersofMΛ(n,d).InSection2weadaptthoseargumentstoHodgenumbersandproveourmainresultTheorem1.InSection3weexplicitlycalculateHP(MΛ(n,d))(x,y)forn=2,3.InSection4wedis-cusstheχ(t)-charactersticandexplainhowχ(t)containsinformationaboutintersectionpairingsinvolvingcertaingeneratorsofH∗(MΛ(n,d)).1TheTwoApproachesLetEbeafixedC∞complexbundleofranknanddegreedoverΣandletCdenotetheinfinitedimensionalaffinespaceofholomorphicstructuresonE.ForeachholomorphicbundleE∈Cthereisastrictlyascendingcanonicalfiltration[12,p.221]0=E0⊂E1⊂···⊂EP−1⊂EP=EsuchthatthequotientsQj=Ej/Ej−1aresemistableandsuchthatdeg(Qj)rk(Qj)=μ(Qj)μ(Qj+1)=deg(Qj+1)rk(Qj+1).WethensaythatEhastypeμ=(μ(Q1),...,μ(QP))∈Qn(1)whereμ(Qj)appearsrk(Qj)times.AtiyahandBottdefinedastratification{Cμ:μ∈M}bysettingCμ⊂CtobethesetofallholomorphicbundlesEoverΣoftypeμ.ThesetCssofsemistablebundlesispreciselythestratumoftypeμ0=dn,···,dn!.Thisstratification{Cμ:μ∈M}isequivariantlyperfectwithrespecttotheactionofthegaugegroupG.ThusifweletPG(X,t)=Xj≥0tjdimQHjG(X,Q)2denotetheequivariantPoincar´epolynomialofaspaceXactedonbyagroupG,thenwehaveequivariantMorseequalitiesPG(C)(t)=PG(Css)(t)+Xμ6=μ0t2dμPG(Cμ)(t)wheredμisthecomplexcodimensionofCμinCwhichisgivenbytheformula[1,7.16]dμ=X1≤ji≤Pnidj−njdi+ninj(g−1).(2)TheG-equivariantcohomologyofthestratamaybedeterminedinductivelyviatheiso-morphisms[1,7.12]H∗G(Cμ)∼=O1≤j≤PH∗G(nj,dj)(C(nj,dj)ss).TheG-equivariantPoincar´epolynomialoftheaffinespaceCisgivenby[1,Thm.2.15]PG(C)(t)=Qnl=1(1+t2l−1)2g(1−t2n)Qn−1l=1(1−t2l)2.(3)UsingthisAtiyahandBottobtainaninductiveformulaPG(Css)(t)=PG(C)(t)−Xμ6=μ0t2dμPG(nj,dj)(C(nj,dj)ss)(t)(4)(wherethesumisoverallunstabletypesμ=(d1/n1,...,dP/nP))fortheG-equivariantBettinumbersofCss.TheyshowthatwhennanddarecoprimethequotientG=G/S1ofGbyitscentralsubgroup,consistingofmultiplicationbyscalarsinS1,actsfreelyonCssandthat[1,9.3]P(M(n,d))(t)=PG(Css)(t)=(1−t2)PG(Css)(t).Finallytheyshowthat[1,Prop.9.7]P(M(n,d))(t)=(1+t)2gP(MΛ(n,d))(t).TheGITapproachdescribedin[19]involvestheactionofaprojectivegenerallineargrouponavariet