Holomorphic line bundles on the loop space of the

arXiv:math/0210017v1[math.CV]2Oct2002HOLOMORPHICLINEBUNDLESONTHELOOPSPACEOFTHERIEMANNSPHERENINGZHANGAbstract.TheloopspaceLP1oftheRiemannsphereconsistingofallCkorSobolevWk,pmapsS1→P1isaninfinitedimensionalcomplexmanifold.TheloopgroupLPGL(2,C)actsonLP1.WeprovethatthegroupofLPGL(2,C)invariantholomorphiclinebundlesonLP1isisomorphictoaninfinitedimensionalLiegroup.Further,weprovethatthespaceofholomorphicsectionsofthesebundlesisfinitedimensional,andcomputethedimensionforagenericbundle.1.IntroductionLetMbeafinitedimensionalcomplexmanifold.ItsloopspaceLMwithaspecifiedregularity,forexampleCk(1≤k≤∞)orWk,p(1≤k∞,1≤p∞),consistsofallmapsofthecircleS1intoMwiththegivenregularity.LMisaninfinitedimensionalcomplexmanifold.ThispaperstudiesholomorphiclinebundlesontheloopspaceLP1oftheRiemannsphere.Adirectmotivationcomesfrom[9],whereMillsonandZombroconjec-turethatthereexistsaPGL(2,C)equivariantembeddingofLP1intoaprojectivizedBanach/Fr´echetspace.TheconjecturearisesinconnectionwithextendingMumford’sgeometricinvarianttheorytoaninfinitedimen-sionalsetting.Anotherindirectmotivationcomesfrom[11],whereWittensuggeststostudythegeometryandanalysisofrealandcomplexmanifoldsthroughtheirloopspaces.Infinitedimensionsitisaproblemoffundamen-talimportancetoidentifythePicardgroupofholomorphiclinebundlesonacomplexmanifoldandthespaceofholomorphicsectionsofthesebundles.Hereweaddressthisproblemforaclassofholomorphiclinebundlesonthefirstinterestingloopspace:LP1,andinparticularmakesomeprogresstowardansweringtheconjecturebyMillsonandZombro.Thefollowingarethemainresultsofthispaper.IfagroupGactsonasetV,letVGdenotetheG-fixedsubsetofV.TheloopspaceofafinitedimensionalcomplexLiegroupisacomplexLiegroupunderpointwisegroupoperation(loopgroup).LetPic(LP1)bethePicardgroupofLP1.ThegroupPGL(2,C)actsonP1,sotheloopgroup1991MathematicsSubjectClassification.58B12,32Q99,58D15.Keywordsandphrases.Picardgroup,Dolbeaultcohomologygroup,Loopspace.ThisresearchwaspartiallysupportedbyanNSFgrant.12NINGZHANGLPGL(2,C)actsonLP1andonPic(LP1).LetLC∗betheloopgroupofC∗=C\{0},andHom(LC∗,C∗)bethegroupofholomorphichomomor-phismsfromLC∗toC∗.Theorem1.1.Pic(LP1)LPGL(2,C)∼=Hom(LC∗,C∗)asgroups.NotethatHom(LC∗,C∗)isaZ-moduleofinfiniterank,whilethegroupoftopologicalisomorphismclassesoflinebundlesonLP1isisomorphictoZ(cf.[8]).EvaluationofloopsinP1att∈S1givesrisetoaholomorphicmapEt:LP1→P1.LetΛt∈Pic(LP1)bethepullbackofthehyperplanebundleonP1byEt.Theorem1.2.LetΛ∈Pic(LP1)LPGL(2,C).(1)IfΛ∼=Λn1t1⊗···⊗Λnrtr,whereni≥0andti6=tjfori6=j,then(n1+1)···(nr+1)≤dimH0(LP1,Λ)∞.(2)OtherwiseH0(LP1,Λ)=0.ThereforethesectionsofnoLPGL(2,C)invariantholomorphiclinebun-dlewillgiverisetoaprojectiveembeddingofLP1.TheisomorphisminTheorem1.1isgottenbyanexplicitconstructioninSection2.InSection3weproveTheorem1.2(2).InSection4westudythespaceofholomorphicsectionsorthezeroorderDolbeaultcohomologygroupoflinebundlesdefinedinTheorem1.2(1),andinparticularproveTheorem1.2(1).Anysuchlinebundleobviouslyhasholomorphicsections:productsofpulledbacksectionsbytheevaluationmaps.Wewillshowthatforagenericbundleofthistypetheseareallsections.Yettherearebundleswhichhaveothersectionsaswell;interestingly,inthiscasedimH0(LP1,Λϕ)dependsontheregularityoftheloops.ItisnaturaltoaskwhetherPic(LP1)PGL(2,C)=Pic(LP1)LPGL(2,C).Ifso,thentheconjecturemadebyMillsonandZombroisansweredinthenegative.TheauthorwouldliketothankProfessorL´aszl´oLempertformanyhelp-fulconversationsandhiscontinuousencouragement.ThanksarealsoduetoProfessorJohnJ.MillsonandProfessorJamesMcClureforvaluablecom-municationsonsomeissuesrelatedtothispaper.2.IdentificationofPic(LP1)LPGL(2,C)WefixaregularityclassFamongCk(1≤k≤∞)respectivelyWk,p(1≤k∞,1≤p∞).InthispaperwewriteLM(LkMresp.Lk,pM)todenotetheF(Ckresp.Wk,p)loopspaceofamanifoldM.LetMandNbefinitedimensionalcomplexmanifoldsandφ:M→Nbeaholomorphicmap.DefineLφ:LM∋x7→φ◦x∈LN.ThenLMandLNareinfinitedimensionalcomplexmanifoldslocallybiholomorphictoopensubsetsofcomplexBanach(Fr´echetwhenF=C∞)spaces,andLφisholomorphic.ThusLisafunctorfromthecategoryoffinitedimensionalHOLOMORPHICLINEBUNDLESONTHELOOPSPACE3complexmanifoldstothecategoryofallcomplexmanifolds.Lett∈S1.TheevaluationmapEt=ELMt:LM∋x7→x(t)∈Misholomorphic.SeeSection2of[6].WecallconstantmapsS1→MpointloopsinM.Theyformasubman-ifoldofLM,whichweidentifywithM.NextwedefineamapL:Hom(LC∗,C∗)→Pic(LP1)LPGL(2,C).WewillshowthatLisanisomorphismofgroups,whichwillthenproveTheorem1.1.InSection6of[9]MillsonandZombroconstructaholomorphiclinebundleonLP1,andasimilarideainfactyieldsamapfromHom(LC∗,C∗)toPic(LP1)asfollows.Letp:Q→P1betheprincipalC∗-bundleassociatedwiththehyperplanebundleH→P1.ApplyingtheloopfunctorweobtainaprincipalLC∗-bundleLp:LQ→LP1.Nowahomomorphismϕ:LC∗→C∗determinesarepresentationofLC∗onC.Recallthat,ingeneral,withaprincipalG-bundleP→BandarepresentationρofGonavectorspaceV,onecanfunctoriallyassociateavectorbundleE→BwithtypicalfiberV(seeSection12.5of[4]).IfhabaretheG-valuedtransitionfunctionsofPwithrespecttosometrivialization,thecorrespondingtransitionfunctionsofEwillbeρ(hab).AccordinglyweassociatewithLpandϕalinebundleΛϕ.DefinethemapL: