LNM1314 chaptre 2 Kähler-Einstein metrics and extremal Kähler metrics

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1、ChapterII.Kahler-EinsteinmetricsandextremalKahlermetrics§2.1Kahler-EinsteinmetricsLetMbeacompactcomplexmanifold.A(1,1)-form~onMissaidtobepositive(resp.negative)ifitislocallyexpressedby"--F=-~a.~dziAdzJ1Jwith(a.=)apositive(resp.negative)definiteHermit:tanmatrix;1jhenceaisnecessarilya

2、realform.AdeRhamclassoftype(1,1)issaidtobepositive(resp.negative)ifitisrepresentedbyapositive(resp.negative)form.Let(M,g)beacompac:Kablermanifold.Thenby(1.1.12)itsgahlerform~ispost:ireandthusitsKahlerclassQisalsopositive.Wesaythatgisagahler-Einsteinmetricif0=kmforsomerealnumberkwher

3、eOistheRicciformofg,see(t.1.19).IfMadmitsaKM~ler-EinsteinmetricthenCl(M)ispositive,zeroornegativeaccordingaskispositive,zeroornegative.Nowonecanasktheconverse:IfCl(M)ispositive,zeroornegativedoesMadmitaKahler-Einsteinmetric?Thisproblemcanbereducedtotheso-calledcomplexMonge-Ampereequ

4、ation(2.1.1)below.Toderivethisequation,noticethattheconditionp=koimpliesthat,whenk~O,~=~Cl(M)whereQistheKihlerclass.Whenk=0wefixanarbitraryKahlerclass~.NowstartwithaKahlermetricgwithitsKilerclassequaltoQ.WeseekaK/hler-Einsteinmetricoftheformg=(gi~÷Oil)whereisasmoothfunctionand~i]=a2

5、~r.SincebothOgandk~g8zIazjrepresentsCl(M),thereexistsasmoothfunctionFsuchthatpg-k~g=2~~FbyProposition11""26"Considerthefollowingequationwith~theunknown:(2.1.1)det(gi~+~i~)/det(gi~)=e-k~+F(gi]+~i~)>032Then(2.1.1)impliesthatP~-Pg=2~Pg/:'T=k~g--k-'~=k~pgsothatp~=k~.ThisimpliesthatgisaK

6、ahler-Einsteinmetric.Whenk~O,thisequationwasalreadysolvedbyAubin[AT2]andYau[YS1].OntheotherhanditisknownthatifCl(M)ispositiveMdoesnotnecessarilyadmitaKahler-Einsteinmetric.InfacttherearetwoknownnecessaryconditionsduetoMatsushima[MYt]andtotheauthor[FAll,bothofwhichweshallexplaininlat

7、ersections.ThesetwoobstructionsautomaticallyvanishifMadmitsnononzeroholomorphicvectorfield.CalabiconjecturesthatifCl(M)ispositiveandifthereisnonozeroholomorphicvectorfieldMwilladmitaKahler-Einsteinmetric(see[YS3]).WeremarkthatYau'ssolutioninthecasewhenk=0alsoprovestheCalabiconjectur

8、e(thisconjectureisd