Scalar curvature and projective embeddings I

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1、j.differentialgeometry59(2001)479-522SCALARCURVATUREANDPROJECTIVEEMBEDDINGS,IS.K.DONALDSONAbstractWeprovethatametricofconstantscalarcurvatureonapolarisedK¨ahlermanifoldisthelimitofmetricsinducedfromaspecificsequenceofprojectiveembeddings;satisfyingaconditionintroduce

2、dbyH.Luo.Thisgives,asaCorollary,theuniquenessofconstantscalarcurvatureK¨ahlermetricsinagivenrationalcohomologyclass.TheproofusesresultsintheliteratureontheasymptoticsoftheBergmankernel.Theargumentsarepresentedinageneralframeworkinvolvingmomentmapsfortwodifferentgroup

3、actions.1.IntroductionTwoclassicalresultsare:(1)AcompactRiemannsurfacecanbeembeddedincomplexprojec-tivespace,asacomplexalgebraiccurve;(2)AcompactRiemannsurfaceadmitsametricofconstantcurvature,uniqueuptotheactionoftheholomorphicautomorphismsofthesurface.Itisnaturalto

4、askifthereareconnectionsbetweenthesetworenownedresults,linkingthealgebro-geometricanddifferential-geometricpointsofview.Specifically,acomplexalgebraiccurveinCPNhasametricin-ducedfromtheFubini-Studymetriconprojectivespaceandonecanaskhowmetricsofthiskindarerelatedtothec

5、onstant-curvaturemetric.Thisisthequestionwestudyhere,notjustforaRiemannsurfacebutforapolarisedmanifoldofanydimension,i.e.,acompactcomplexmani-foldXofcomplexdimensionnwithagivenpositivelinebundleL→X.ReceivedNovember13,2001.479480s.k.donaldsonThemainmotivationforthewo

6、rkcomesfromthequestionsabouttheexistenceanduniquenessofK¨ahlermetricsofconstantscalarcurvatureonX—generalisingtheconstantcurvaturemetricsintheRiemannsur-facecase.(InthecasewhenthelinebundleisapowerofthecanonicalbundleofthemanifoldthemetricsinvolvedareK¨ahler-Einstei

7、nandtheissuesweinvestigateherefallverymuchintothemouldofideaswhichhavebeenpropoundedbyS.-T.Yauovermanyyears[11].)ThethrustofourresultsistoestablishaprecisecorrespondencebetweenconstantscalarcurvatureK¨ahlermetricsandtheasymptoticsofsequencesofprojectiveembeddings,wh

8、ereonereplacesLbyLkandletsk→∞.AcentralroleinthispaperisaplayedbyanideaduetoLuo[9].Let[Z0,...,ZN]bestandardhomogeneouscoordinatesonCPNandde