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1、J.DIFFERENTIALGEOMETRY29(1989)231-244ANANALOGUEOFDEMAILLY'SINEQUALITYFORSTRICTLYPSEUDOCONVEXCRMANIFOLDSEZRAGETZLERRecently,BismuthasreformulatedDemailly'sresultsontheasymptoticdimensionofthed-cohomologyofalinebundleLm,wherem—>oo[2],[3].Hismainresultist
2、hefollowing:mlimj2τr/m)j/)IΩ^nTrIΩO^MLML™)^)j7™=jdetInthisformula,nisthecomplexdimensionofthecompactcomplexmanifoldM,FisthecurvatureoftheholomorphiclinebundleLonM,and∆istheLaplacian(d+d)2actingonthespaceΩ°'9(M;Lm).UsingthebounddimHq{M;Lm)=dimker∆Ωo,g(M
3、;L-)<Ή
4、n°.«(Λf;L")e"t∆/m,weobtainDemailly'soriginalinequalityfromBismut'sformula:dimH"(M;Lm)<(^"mi:£det^-JL—^jTrAo,qτ.Me~tF+o(mn)nTOX'det(F)+o(mn).hasqnegativeeigenvalues}Inparticular,ifFispositivesemideίinite,combiningthisinequalitywiththeHirzebruch-
5、Riemann-Rochtheorem,weseethato(mn)ifg>0.Fortheapplicationsofthisinequality,seeDemailly'soriginalpaper[3].Bismutprovedthisformula(andasimilaronefortheDiracoperator)usingmuchthesameproofashehadearliergivenoftheindextheoremforDiracoperators.Oneofthegoalso
6、fthispaperistoshowhowthesesortsofresultsmaybeprovedusingthesymbolcalculustechniqueofourearlierpaper[6].WewillapplythismethodtostudyananalogueofDemailly'sasymptoticresultfortheD&operatoronstrictlypseudoconvexCRmanifolds(whichweshallrefertoasHeisenbergma
7、nifolds,intheinterestofbrevityofterminology).ReceivedFebruary24,1987and,inrevisedform,September28,1987.232EZRAGETZLERThisproblemisasimpleversionofthesortofformulasthatoneshouldtrytoproveformoregeneralCRmanifolds,andalsoforthed-Neumannproblem.However,we
8、havenotbeenabletofindanysimplificationforthecoefficientoftheleadingorderofdimkerD6
9、no,10、bergmanifoldisamanifoldmodeledontheHeisenberggroup—recallthatthisisthe2n+1-dimensionalnilpotentgroupHn=CnxRwiththemultiplication(αo?So)(fli,5χ)=(αo+fti,5oH~5iH~4Imαofli).ThisgrouphastheLiealgebraί)nwithunderlyingvectorspaceCn0RandLiebra
