Moduli of pre-D-modules, perverse sheaves and the

Moduliofpre-D-modules,perversesheavesandtheRiemann-Hilbertmorphism-INitinNitsureClaudeSabbahyMarch25,1995AbstractWeconstructamodulischemeforsemistablepre-D-moduleswithpre-scribedsingularitiesandnumericaldataonasmoothprojectivevariety.Thesepre-D-modulesaretobeviewedasregularholonomicD-moduleswith‘levelstructure’.Wealsoconstructamodulischemeforperversesheavesonthevarietywithprescribedsingularitiesandothernumericaldata,andrepresentthedeRhamfunctor(whichgivestheRiemann-Hilbertcorrespondence)byananalyticmorphismbetweenthetwomodulischemes.Contents1Introduction:::::::::::::::::::::::::::::::::::22Pre-D-modules::::::::::::::::::::::::::::::::::33Frompre-D-modulestoD-modules::::::::::::::::::::::84Semistabilityandmoduliforpre-D-modules.:::::::::::::::::125Perversesheaves,Verdierobjectsandnitedescriptions:::::::::::196Moduliforperversesheaves:::::::::::::::::::::::::::217Riemann-Hilbertmorphism:::::::::::::::::::::::::::228SomepropertiesoftheRiemann-Hilbertmorphism::::::::::::::24References::::::::::::::::::::::::::::::::::::::28TataInstituteofFundamentalResearch,BombayyCNRS,URAD0169,EcolePolytechnique,Palaiseau11IntroductionThispaperisdevotedtothemoduliproblemforregularholonomicD-modulesandperversesheavesonacomplexprojectivevarietyX.IttreatsthecasewherethesingularlocusoftheD-moduleisasmoothdivisorSandthecharacteristicvarietyiscontainedintheunionofthezerosectionTXXofthecotangentbundleofXandtheconormalbundleNS;XofSinX(alsodenotedTSX).Thesequel(partII)willtreatthegeneralcaseofarbitrarysingularities.AmodulispaceforO-coherentD-modulesonasmoothprojectivevarietywascon-structedbySimpson[S].Thesearevectorbundleswithintegrableconnections,andtheyarethesimplestcaseofD-modules.Inthismoduliconstruction,therequire-mentofsemistabilityisautomaticallyfullledbyalltheobjects.Nextinorderofcomplexityarethesocalled‘regularmeromorphicconnections’.TheseD-modulescanbegeneratedbyvectorbundleswithconnectionswhichhavelogarithmicsingularitiesondivisorswithnormalcrossing.TheseD-modulesarenotO-coherent,butaretorsionfreeasO-modules.AmodulischemedoesnotexistfortheseD-modulesthemselves(seesection1of[N]),butitispossibletodeneanotionofstabilityandconstructamoduliforvectorbundleswithlogarithmicconnections.Thiswasdonein[N].Thoughmanylogarithmicconnectionsgiverisetothesamemeromorphicconnection,thechoiceofalogarithmicconnectionisinnitesimallyrigidifitsresidualeigenvaluesdonotdierbynonzerointegers(seesection5of[N]).Inthepresentpaperanditssequel,wedealwithgeneralregularholonomicD-modules.SuchmodulesareingeneralneitherO-coherent,norO-torsionfreeorpuredimensional.Wedeneobjectscalledpre-D-modules,whichplaythesameroleforregularholonomicD-modulesthatlogarithmicconnectionsplayedforregu-larmeromorphicconnections.Wedeneanotionof(semi-)stability,andconstructamodulischemefor(semi-)stablepre-D-moduleswithprescribedsingularitystrat-icationandothernumericaldata.Wealsoconstructamodulischemeforperversesheaveswithprescribedsingularitystraticationandothernumericaldataonanonsingularvariety,andshowthattheRiemann-Hilbertcorrespondencedenesananalyticmorphismbetween(anopensetof)themoduliofpre-D-modulesandthemoduliofperversesheaves.Thecontentsofthispaperareasfollows.LetXbeasmoothprojectivevariety,andletSbeasmoothhypersurfaceonX.Insection2,wedenepre-D-moduleson(X;S)whichmayberegardedasOX-coherentdescriptionsofthoseregularholo-nomicDX-moduleswhosecharacteristicvarietyiscontainedinTXX[TSX.Thepre-D-modulesformanalgebraicstackinthesenseofArtin,whichisapropertythatdoesnotholdforthecorrespondingD-modules.Insection3,wedeneafunctorfromthepre-D-modulestoD-modules(infactwemainlyusethepresentationofholonomicD-modulesgivenbyMalgrange[Mal],thatwecallMalgrangeobjects).Thisisasurjectivefunctor,andthoughnotinjective,ithasaninnitesimalrigidityproperty(seeproposition3.9)whichgeneralizesthecorrespondingresultformeromorphicconnections.2Insection4,weintroduceanotionof(semi-)stabilityforpre-D-modules,andshowthatsemistablepre-D-moduleswithxednumericaldataformamodulischeme.Next,weconsiderperversesheavesonXwhichareconstructiblewithrespecttothestratication(XS)[S.ThesehavenitedescriptionsthroughtheworkVerdier,recalledinsection5.WeobservethatthesenitedescriptionsareobjectswhichnaturallyformanArtinalgebraicstack.Moreover,weshowinsection6thatS-equivalenceclasses(Jordan-Holderclasses)ofnitedescriptionswithgivennumericaldataformacoarsemodulispacewhichisananescheme.Here,nohypothesisaboutstabilityisnecessary.Insection7,weconsidertheRiemann-Hilbertcorrespondence.Whenapre-D-modulehasunderlyinglogarithmicconnectionsforeachofwhicheigenvaluesdonotdierbynonzerointegers,wefunctoriallyassociatetoitanitedescription,whichisthenitedescriptionoftheperversesheafassociatedtothecorrespondingD-modulebytheRiemann-HilbertcorrespondencefromregularholonomicD-modulestoperversesheaves.Weshowthatthisgivesananalyticmorphismofstacksfromtheanalyticopensubsetofthestack(ormoduli)ofpre-D-moduleson(X;S)wherethe‘residualeigenvaluesaregood’,tothestack(ormoduli)ofnitedescriptionson(X;S).Insection8,weshowthattheabove