Holomorphic Morse Inequalities (Demailly)

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1、HOLOMORPHICMORSEINEQUALITIESJean-PierreDEMAILLY,Universit´edeGrenobleISeriesofLecturesgivenattheAMSSummerInstituteheldinSantaCruz,California,July1989.1.IntroductionLetMbeacompactC1manifold,dimM=m,andhaMorsefunction,Ri.e.afunctionsuchthatallcriticalpointsarenondegenerate.Thestandar

2、dMorseqinequalitiesrelatetheBettinumbersbq=dimHDR(M;R)andthenumberssq=#criticalpointsofindexq;wheretheindexofacriticalpointisthenumberofnegativeeigenvaluesoftheHessianform(@2h=@x@x).Specifically,thefollowing“strongMorseinequalities”ijhold:(1:1)b¡b+¢¢¢+(¡1)qb6s¡s+¢¢¢+(¡1)qsqq¡10qq¡1

3、0foreachintegerq>0.Asaconsequence,onerecoversthe“weakMorseinequali-ties”bq6sqandtheexpressionoftheEuler-Poincar´echaracteristic(1:2)Â(M)=b¡b+¢¢¢+(¡1)mb=s¡s+¢¢¢+(¡1)ms:01m01mThepurposeoftheselecturesistoexplainwhatarethecomplexanaloguesoftheseinequalitiesfor@¡cohomologygroupswithva

4、luesinholomorphicline(orvector)bundles,andtopresentafewapplications.LetXbeacompactcomplexmanifold,n=dimCXandE;FholomorphicvectorbundlesoverXwithrankE=1;rankF=r:Wedenoteherehq(F)=dimHq(X;O(F)):1AssumethatEisendowedwithaC1hermitianmetricanddenotebyc(E)itscurvatureform.Thenic(E)isare

5、al(1;1)¡formonX(cf.x2).Finally,considertheq¡indexsets()x2X;ic(E)xhasqnegativeeigenvaluesX(q;E)=;n¡qpositiveeigenvalues[X(6q;E)=X(j;E):16j6qObservethatX(q;E)andX(6q;E)areopensubsetsofX.MainTheorem.

6、ThesequenceoftensorpowersEk•Fsatisfythefollowingasymptoticestimatesask!+1:(1.3)WeakM

7、orseinequalities:Zkn¡i¢nhq(Ek•F)6r(¡1)qc(E)+o(kn):n!X(q;E)2¼(1.4)StrongMorseinequalities:ZXkn¡i¢n(¡1)q¡jhj(Ek•F)6r(¡1)qc(E)+o(kn):n!X(6q;E)2¼06j6q(1.5)AsymptoticRiemann-Rochformula:Zkn¡i¢nÂ(Ek•F)=rc(E)+o(kn):n!X2¼Observethat(1.5)isinfactaweakconsequenceoftheHirzebruch-Riemann-Roch

8、formula.Theabovetheoremwasfirstprovedin[De2]in1985.ItwaslargelymotivatedbySiu’ssolutionoftheGrauert-Riemenschneiderconjecture([Siu2],1984),whichwewillreprovebelowasaspecialcaseofastrongerstatement.ThebasictoolisaspectraltheoremwhichdescribestheeigenvaluedistributionofthecomplexLapl

9、ace-Beltramioperators.Theoriginal