Scalar curvature and projective embeddings, II

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1、Scalarcurvatureandprojectiveembeddings,IIS.K.DonaldsonFebruary1,20081IntroductionThisisasequeltothepreviouspaper[6],whichstudiedconnectionsbetweenthedifferentialgeometryofcomplexprojectivevarietiesandcertainspecific“balanced”embeddingsinprojectivespace.Th

2、eoriginalplanwasthatthissequelwouldbealengthypaper,discussingvariousextensionsandramifi-cationsoftheideassudiedin[6].Howeverthisplanhasbeenmodifiedinthelightofsubsequentdevelopments.Ontheonehand,Mabuchi[11],[12],[13]hasextendedtheresultsof[6]tothecasewher

3、ethevarietieshavein-finitesimalautomorphims.Ontheotherhand,PhongandSturm[14],[15]havesharpenedsomeoftheargumentsin[6].TheyalsoexplaintherelationoftheideastotheDelignepairingandtheChownorm,andtoearlierworkofZhang[18],whichtheauthorwasunfortunatelynotaware

4、ofwhenwrit-ing[6].Thesedevelopmentsmeanthatsomeoftheresultsplannedforthesequelarenowredundant,whileontheotherhandtheexpositionofallthedifferentpointsofviewhasgrownintoadauntingtask.Thus,instead,thissequelisashortpaperdevotedtotheproofofoneresultwhichisqu

5、iteaneasyconsequenceofthemaintheoremin[6].Tostateourresult,supposethatXisacompactKahlermanifoldandfixaKahlerclass[ω0]onX.RecallthattheMabuchifunctional[9]isafunctionalM,defineduptoanarbitraryadditiveconstant,ontheKahlerarXiv:math/0407534v2[math.DG]14Jan20

6、05metricsinthiscohomologyclasswhichischaracterisedbytheformulaZnωδM=(S−Sˆ)δφ(1)Xn!HeremetricsωintheclassarerepresentedbyKahlerpotentialsω=ω0+i∂∂φandδφisaninfinitesimalvariationinφ.ThesymbolSdenotesthescalarcurvatureofthemetricωandSˆistheaveragevalueofSwi

7、threspecttotheωnvolumeform,whichisatopologicalinvariantoftheKahlerclass.Whatn!1equation(1)reallydefinesisa1-formonthespaceofmetricsintheKahlerclassandoneshowsthatthisisclosed,soisthederivativeofafunctionM,uniqueuptoaconstant.NowsupposethatLisapositivelin

8、ebundleoverXandtheKahlerclassis2πc1(L).Asin[6],wewriteAut(X,L)forthegroupofautomorphimsof∗thepair(X,L)modulothetrivialautomorphismsC(actingbyconstantscalarmultiplicationonthefibres).Theorem1SupposethatAut(X,L)isdiscreteandthatther