To appear in Journal of Algebraic Geometry CHOW QU

ToappearinJournalofAlgebraicGeometryCHOWQUOTIENTSANDPROJECTIVEBUNDLEFORMULASFOREULER-CHOWSERIESE.JAVIERELIZONDOANDP.LIMA-FILHOAbstract.GivenaprojectivealgebraicvarietyX,letp(X)denotethemonoidofeectivealge-braicequivalenceclassesofeectivealgebraiccyclesonX.Thep-thEuler-ChowseriesofXisanelementintheformalmonoid-ringZ[[p(X)]]denedintermsofEulercharacteristicsoftheChowvarietiesCp;(X)ofX,with2p(X).Weprovideasystematictreatmentofsuchseries,andgiveprojectivebundleformulaswhichgeneralizepreviousresultsby[LY87]and[Eli94].ThetechniquesusedinvolvetheChowquotientsintroducedin[KSZ91],andthisallowsthecomputationofvari-ousexamplesincludingsomeGrassmanniansandagvarieties.Therearerelationsbetweentheseexamplesandrepresentationtheory,andfurtherresultspointtointerestingconnectionsbetweenEuler-ChowseriesforcertainvarietiesandthetopologyofthemodulispacesM0;n+1.Contents1.Introduction22.Preliminaries63.TheEuler-Chowseriesofprojectivevarieties74.Projectivebundleformulas104.1.Projectiveclosureoflinebundles145.ChowquotientsandEuler-Chowseries165.1.Examples19AppendixA.AlgebraicConstructions27A.1.Monoidswithpropermultiplication27A.2.InvariantsforM-gradedalgebras30References321991MathematicsSubjectClassication.Primary:14C25;Secondary:14C05.Keywordsandphrases.Chowvarieties,eectivealgebraicequivalence,monoid-gradedalgebras,generatingfunc-tions,Chowquotients,invariantcycles,projectivebundles,Grassmannians,agvarieties.TherstauthorwassupportedinpartbygrantsUNAM-DGAPAIN101296andCONACYT3936-E.ThesecondauthorwaspartiallysupportedbyNSFgrant#DMS-9401533.12E.JAVIERELIZONDOANDP.LIMA-FILHO1.IntroductionTheuseoftopologicalinvariantsonmodulispaceshasplayedavitalroleinvariousbranchesofmathematicsandmathematicalphysicsinthelasttwodecades.Alightsamplingunderthisvastumbrellaincludesworksingaugetheory,thetheoryofinstantons,variousmodulispacesofvectorbundles,modulispacesofcurvesandtheircompactications,ChowvarietiesandHilbertschemes.InthisworkwestudyaclassofinvariantsforprojectivevarietiesarisingfromtheEulercharac-teristicsoftheirChowvarieties.TheseinvariantsrstappearedintheworkofH.B.LawsonandSteveS.T.Yau[LY87],whosetechniquesplayanimportantroleinthispaper,andtheypresent,invariousinstances,aquiteniceandelegantbehaviorwhichcanoftenbecodiedinsimplegeneratingfunctions.Asamotivation,westartwithsomeparticularcases,whicharewellstudiedintheliterature.LetXbeaconnectedprojectivevarietyandletSP(X)denotethedisjointunion‘d0SPd(X)ofallsymmetricproductsofX,withthedisjointuniontopology,whereSP0(X)isasinglepoint.OnecandeneafunctionE0(X):Z+=0(SP(X))!ZwhichsendsdtotheEulercharacteristic(SPd(X))ofthed-foldsymmetricproductofX.Thisiswhatwecallthe0-thEuler-ChowfunctionofX.ThesameinformationcanbecodiedasaformalpowerseriesE0(X)=Pd0(SPd(X))td,andaresultofMacdonald[Mac62]showsthatE0(X)isgivenbytherationalfunctionE0(X)=(1=(1t))(X).Anotherfamiliarinstancearisesinthecaseofdivisors.Givenann-dimensionalprojectivevarietyX,letDiv+(X)denotethespaceofeectivedivisorsonXandassumethatPic0(X)=f0g.ConsiderthefunctionE:Pic(X)!ZwhichsendsL2Pic(X)todimH0(X;O(L)).Observethat1.GivenL2Pic(X),thenE(L)6=0ifandonlyifL=O(D)forsomeeectivedivisorD;2.Underthegivenhypothesis,algebraicandlinearequivalencecoincide,andtwoeectivedivi-sorsDandD0arealgebraicallyequivalentifandonlyiftheyareinthesamelinearsystem.ThelastobservationimpliesthatDiv+(X)canbewrittenasDiv+(X)=‘2An1XDiv+(X),whereAn1Xisthemonoidofalgebraicequivalenceclassesofeectivedivisors(cf.Fulton[Ful84,x12]),andDiv+(X)isthelinearsystemassociatedto2An1X.TherstobservationshowsthattheonlyrelevantdatatoEisgivenbyAn1XPic(X).Therefore,wemightaswellrestrictEanddenethe(n1)-stEuler-ChowfunctionofXasthefunctionEn1X:An1X!Z+whichsends2An1XtotheEulercharacteristic(Div+(X))=dimH0(X;O(L)),whereListhelinebundleassociatedto.Example1.1.AnevenmorerestrictivecaseariseswhenPic(X)=Z,andAn1X=Z+isgeneratedbytheclassofaveryamplelinebundleL.Thenthe(n1)-stEuler-ChowfunctionEn1X=Pd0dimH0(X;O(Ln))tnisjusttheHilbertfunctionassociatedtotheprojectiveembeddingofXinducedbyL.Thisisonceagainarationalfunction.CHOWQUOTIENTSANDEULER-CHOWSERIES3Ingeneral,thesituationisnotsosimple,andweneedtointroduceadditionalnotionsinordertoapproachcyclesofarbitrarydimension.WestartwiththeChowmonoidCpXofeectivep-cyclesonX,whichcanbewrittenasadisjointunion‘2pXCp;(X)ofconnectedpro-jective(Chow)varietiesCp;(X);cf.Section3.Here,pX=0(CpX)denotesthemonoidofeectivealgebraicequivalenceclassesofeectivep-cycles.ThismonoidshouldbecontrastedwithApX,themonoidofalgebraicequivalenceclassesofeectivep-cycles;cf.[Ful84,x12].Infact,thereisanitesurjectivemonoidmorphismpX!ApX,andtheGrothendieckgroupassociatedtobothmonoidsisApX,thegroupofalgebraicequivalenceclassesofp-cyclesonX;cf.Friedlander[Fri91].ThepropertiesandrelationsamongthesemonoidsisdiscussedinSection3.Thep-thEuler-ChowfunctionoftheprojectivevarietyXisthendenedasthefunctionEpX:pX!Z(1)7!(Cp;(X))whichcanbecompletelyencodedasaformalpowerseriesEpX=P2pX