Instanton sheaves on complex projective spaces

arXiv:math/0412142v5[math.AG]15Sep2005InstantonsheavesoncomplexprojectivespacesMarcosJardimIMECC-UNICAMPDepartamentodeMatem´aticaCaixaPostal606513083-970Campinas-SP,BrazilFebruary1,2008AbstractWestudyaclassoftorsion-freesheavesoncomplexprojectivespaceswhichgeneralizethemuchstudiedmathematicalinstantonbun-dles.Instantonsheavescanbeobtainedascohomologiesoflinearmonadsandareshowntobesemistableifitsrankisnottoolarge,whilesemistabletorsion-freesheavessatisfyingcertaincohomologicalconditionsareinstanton.Wealsostudyafewexamplesofmodulispacesofinstantonsheaves.2000MSC:14J60;14F05Keywords:Monads,semistablesheavesContents1Monads42Basicpropertiesofinstantonsheaves113Semistabilityofinstantonsheaves154Simplicityoflinearsheaves285Modulispacesofinstantonsheaves291IntroductionThestudyofvectorbundlesandmoregeneralsheavesoncomplexprojectivespaceshasbeenatopicofgreatinteresttoalgebraicgeometersformanyyears,seeforinstancetheexcellentbookbyOkonek,SchneiderandSpindler[20]andHartshorne’sproblemlist[11].Inthispaperweconcentrateonaparticularclassofsheavesdefinedasfollows,generalizingtheconceptofadmissiblesheavesonP3duetoManin[17],seealso[10].Definition.AninstantonsheafonPn(n≥2)isatorsion-freecoherentsheafEonPnwithc1(E)=0satisfyingthefollowingcohomologicalconditions:1.forn≥2,H0(E(−1))=Hn(E(−n))=0;2.forn≥3,H1(E(−2))=Hn−1(E(1−n))=0;3.forn≥4,Hp(E(k))=0,2≤p≤n−2and∀k;Theintegerc=−χ(E(−1))iscalledthechargeofE.IfEisarank2mlocally-freeinstantonsheafonP2m+1oftrivialsplittingtype(i.e.thereexistalineℓ⊂P2m+1suchthatE|ℓ≃O2mℓ),thenEisamathematicalinstantonbundleasoriginallydefinedbyOkonekandSpindler[21].Thereisanextensiveliteratureonsuchobjects,seeforinstance[1,24].Thenomenclatureismotivatedbygaugetheory:mathematicalinstantonbundlesonP2n+1correspondtoquaternionicinstantonsonPHn[22].Thegoalofthispaperistoextendthediscussionintwodirections:theinclusionofeven-dimensionalprojectivespacesandtheanalysisofmoregen-eralsheaves,allowingnon-locally-freesheavesofarbitraryrank.Suchexten-sionismotivatedbytheconceptthatinordertobetterunderstandmodulispacesofstablevectorbundlesoveraprojectivevarietyonemustalsocon-sidersemistabletorsion-freesheaves[11].Itturnsoutthatmanyofthewell-knownresultsregardingmathematicalinstantonbundlesonP2n+1generalizeinsometimessurprisingwaystomoregeneralinstantonsheaves.2Thepaperisorganizedasfollows.WestartbystudyinglinearmonadsandtheircohomologiesinSection1,spellingoutcriteriatodecidewhetherthecohomologyofagivenmonadistorsion-free,reflexiveorlocally-free.Wethenshowthateveryinstantonsheafisthecohomologyofalinearmonad,andthatrankrinstantonsheavesonPnexistifandonlyifr≥n−1.FurtherpropertiesofinstantonsheavesarealsostudiedinSection2.ThebulkofthepaperliesinSection3,whereweanalyzethesemistabil-ity(inthesenseofMumford-Takemoto)ofinstantonsheaves.Itisshown,forinstance,thateveryrankr≤2n−1locally-freeinstantonsheafonPnissemistable,whileeveryrankr≤nreflexiveinstantonsheafonPnissemistable.Wealsodeterminewhenasemistabletorsion-freesheafonPnisaninstantonsheaf,showingforinstancethateverysemistabletorsion-freesheafonP2isaninstantonsheaf.InSection4itisshownthateveryrankn−1instantonsheafonPnissimple,generalizingaresultofAnconaandOttavianiformathematicalinstantonbundles[1,Proposition2.11].WethenconcludeinSection5withafewresultsconcerningthemodulispacesofinstantonsheaves.ItisalsoworthnotingthatBuchdahlhasstudiedmonadsoverarbitraryblow-upsofP2[3]whileCostaandMir´o-Roighaveinitiatedthestudyoflocally-freeinstantonsheavesoversmoothquadrichypersurfaceswithinPn[6],obtainingsomeresultssimilartoours.ManyoftheresultshereobtainedarealsovalidforinstantonsheavessuitablydefinedoverprojectivevarietieswithcyclicPicardgroup,see[16].Notation.WeworkoveranalgebraicallyclosedfieldFofcharacteristiczero.Itmightbeinterestingfromthealgebraicpointofviewtostudyhowtheresultshereobtainedgeneralizetofinitefields.Throughoutthispaper,U,VandWarefinitedimensionalvectorspacesoverthefixedfieldF,andweuse[x0:···:xn]todenotehomogeneouscoordinatesonPn.IfEisasheafonPn,thenE(k)=E⊗OPn(k),asusual;byHp(E)weactuallymean3Hp(Pn,E)andhp(E)denotesthedimensionofHp(Pn,E).Acknowledgment.TheauthorispartiallysupportedbytheFAEPEXgrantsnumber1433/04and1652/04,andtheCNPqgrantnumber300991/2004-5.WethankGiorgioOttavianiandRosaMariaMir´o-Roigfortheirvaluablecommentsonthefirstversionofthispaper.Thismaterialwasthetopicofatwo-weekcourseduringthe2005SummerProgramatIMPA;wethankEduardoEstevesfortheinvitationandIMPAforthefinancialsupport.1MonadsLetXbeasmoothprojectivevariety.AmonadonXisacomplexV•ofthefollowingform:V•:0→V−1α−→V0β−→V1→0(1)whichisexactonthefirstandlastterms.Here,VkarelocallyfreesheavesonX.ThesheafE=kerβ/ImαiscalledthecohomologyofthemonadV•.MonadswerefirstintroducedbyHorrocks,whohasshownthateveryrank2locallyfreesheafonP3canbeobtainedasthecohomologyofamonadwhereVkaresumsoflinebundles[15].Inthispaper,wewillfocusontheso-calledlinearmonadsonPn,whichareoftheform:0→V⊗OPn(−1)α−→W⊗OPnβ−→U⊗OPn(1)→0,(2)whereα∈Hom(V,W)⊗H0(OPn(1))isinjective(asasheafmap)andβ∈Hom(W,U)⊗H0(OPn(1))issurjective.ThedegenerationlocusΣofthemonad(2)consistsoft