The Derived Moduli Space of Stable Sheaves

arXiv:1004.1884v1[math.AG]12Apr2010TheDerivedModuliSpaceofStableSheavesK.Behrend,I.Ciocan-Fontanine,J.Hwang,M.RoseMarch31,2010AbstractWeconstructthederivedschemeofstablesheavesonasmoothpro-jectivevarietyviaderivedmodulioffinitegradedmodulesoveragradedring.WedothisbydividingthederivedschemeofactionsofCiocan-FontanineandKapranovbyasuitablealgebraicgaugegroup.WeshowthatthenaturalnotionofGIT-stabilityforgradedmodulesreproducesstabilityforsheaves.ContentsIntroduction2Derivedgeometry...........................5Glossary................................6NotationandConventions......................71Thederivedschemeofsimplegradedmodules81.1ThedifferentialgradedLiealgebraL................8Thegradedvectorspace.......................8Thegaugegroup...........................9Thedifferential............................9Thebracket..............................10TheMaurer-Cartanequation....................111.2ThemodulistackofL........................11BundlesofmarkeddifferentialgradedLiealgebras........11Associateddifferentialgradedschemeorstack...........12Theassociatedfunctorondgschemes...............13Thederivedschemeofactions....................14Thederivedstackofmodules....................141.3Thederivedspaceofequivalenceclassesofsimplemodules....15Equivalenceofsimplemodules...................15Coprimecase.............................16Thederivedspaceofsimplemodules................161.4Thetangentcomplex.........................18Deformationtheoryforsmallextensions..............18Deformationsofmodules.......................2012Stability213Moduliofsheaves233.1Theadjointofthetruncationfunctor................233.2Openimmersion...........................243.3Stablesheaves.............................263.4ModuliofSheaves..........................273.5Anamplification...........................284Derivedmoduliofsheaves30IntroductionForsomeyearsithasbeenatenantofgeometry,thatdeformationtheoryprob-lemsaregovernedbydifferentialgradedLiealgebras.Thisleadstoformalmodulibeinggivenbydifferentialgradedalgebras,andgivesrisethederivedgeometryprogramme.Usually,theexpectationis,thattosolveagivenglobalmoduliproblemwithadifferentialgradedLiealgebra,thisdifferentialgradedLiealgebrawouldhavetoinfinite-dimensional,andthereforeill-suitedforalge-braicgeometry.Forexample,gaugetheorycanbeusedtoconstructanalyticmodulispacesofholomorphicvectorbundlesonacompactcomplexmanifoldY.Inthiscase,thedifferentialgradedLiealgebramightbeA0,•(Y,Mn),thealgebraofC∞-formsoftype(0,∗)withvaluesinn×n-matrices.ThedifferentialistheDol-beaultdifferential,andthebracketiscombinedfromwedgeproductofformsandcommutatorbracketofmatrices.Almostcomplexstructuresareelementsofx∈A0,1(Y,Mn),andtheyareintegrable,ifandonlyiftheysatisfytheMaurer-Cartanequationdx+12[x,x]=0.DividingtheMaurer-CartanlocusbythegaugegroupG=A0,0(Y,GLn),weobtainthemodulispaceofholomorphicbundles.Onecentralobservationofthispaperisthatthereexistsafinitedimensionalanalogueofthisconstructionformoduliofcoherentsheavesonasmoothprojec-tivevarietyoverC.Derivedmoduliofsheaveshavebeenconstructedbefore(see[3]or[14]),butwebelieveitisanewobservationthatthereisafinitedimen-sionaldifferentialgradedLiealgebrawithanalgebraicgaugegroup,solvingthismoduliproblemglobally.SimplybyvirtueofbeingthespaceofMaurer-CartanelementsinadifferentialgradedLiealgebrauptogaugeequivalence,themodulispaceautomaticallycomeswithaderived,ordifferentialgraded,structure.ThisconstructionalsoleadsoneimmediatelytotheexaminationofGeo-metricInvariantTheorystabilityforthisalgebraicgaugegroupaction.Thus,anotherresultofthispaperitthatGITstabilityforouralgebraicgaugegroupactionreproducesthestandardnotionofstabilityforsheaves.LetYbeasmoothprojectivevarietywithhomogeneouscoordinateringA,andα(t)∈Q[t]anumericalpolynomial.2WepresentaconstructionofthederivedmodulischemeofstablesheavesonY,asaGeometricInvariantTheoryquotientofthederivedschemeofactions.Thederivedschemeofactions,RAct,wasintroducedbyCiocan-FontanineandKapranovin[3]asanauxiliarytoolintheirconstructionofthederivedschemeofquotients,RQuot.ThebasicideaistodescribeacoherentsheafFonYwithHilbertpolyno-mialα(t)intermsoftheassociatedfinite-dimensionalgradedA-moduleΓ[p,q]F=qMi=pΓY,F(i)
,withdimensionvectorα|[p,q]=α(p),...,α(q)
,forq≫p≫0.Infact,foranyopenboundedfamilyUofsheaveswithHilbertpolynomialα(t)onY,thereexistq≫p≫0,suchthatΓ[p,q]:U−→gradedA-modulesin[p,q]withdimensionvectorα|[p,q]isanopenembeddingofmodulifunctors(i.e.,ofstacks).Weconstructafinite-dimensionaldifferentialgradedLiealgebraL=q−pMn=0LntogetherwithanalgebraicgaugegroupG(theLiealgebraofGisL0),actinglinearlyonL,suchthatMC(L)/G,thequotientofthesolutionsetoftheMaurer-Cartanequationdx+12[x,x]=0,x∈L1(1)bythegaugegroup,isequaltotheset(orratherstack)ofgradedA-modulesconcentratedindegreesbetweenpandqwithdimensionvectorα|[p,q],uptoisomorphism.WedothisbyfixingafinitegradedvectorspaceVofdimensionα|[p,q].Thenthedegree1partofourdifferentialgradedLiealgebraisessentiallyL1=Homgr(A,EndCV),thespaceofdegreepreservingC-linearmapsfromAtoEndCV,andthesolutionstotheMaurer-Cartanequation(1)turnoutt