The General Definition of the Complex Monge-Amp`er

arXiv:0705.2099v1[math.CV]15May2007TheGeneralDefinitionoftheComplexMonge-Amp`ereOperatoronCompactK¨ahlerManifoldsYangXingAbstract.WeintroduceawidesubclassF(X,ω)ofquasi-plurisubharmonicfunc-tionsinacompactK¨ahlermanifold,onwhichthecomplexMonge-Amp`ereoperatoriswell-definedandtheconvergencetheoremisvalid.WealsoprovethatF(X,ω)isaconvexconeandincludesallquasi-plurisubharmonicfunctionswhichareintheCegrellclass.1.IntroductionLetXbeacompactconnectedK¨ahlermanifoldofdimensionn,equippedwiththefundamentalformωgiveninlocalcoordinatesbyω=i2Pα,βgα¯βdzα∧d¯zβ,where(gα¯β)isapositivedefiniteHermitianmatrixanddω=0.ThesmoothvolumeformassociatedtothisK¨ahlermetricisthenthwedgeproductωn.DenotebyPSH(X,ω)thesetofuppersemi-continuousfunctionsu:X→R∪{−∞}suchthatuisintegrableinXwithrespecttothevolumeformωnandωu:=ω+ddcu≥0onX,whered=∂+¯∂anddc=i(¯∂−∂).Thesefunctionsarecalledquasi-plurisubharmonicfunctions(quasi-pshforshort)andplayanimportantroleinthestudyofpositiveclosedcurrentsinX,seeDemailly’spaper[D1].Aquasi-pshfunctionislocallythedifferenceofaplurisubhar-monicfunctionandasmoothfunction.Therefore,manypropertiesofplurisubharmonicfunctionsholdalsoforquasi-pshfunctions.FollowingBedfordandTaylor[BT2],thecom-plexMonge-Amp`ereoperator(ω+ddc)nislocallyandhencegloballywelldefinedforallboundedquasi-pshfunctionsinX.SomeimportantresultsofthecomplexMonge-Amp`ereoperatorforboundedquasi-pshfunctionshavebeenobtainedbyKolodziej[KO1-2]andBlocki[BL1].ItisalsoknownthatthecomplexMonge-Amp`ereoperatordoesnotworkwellforallunboundedquasi-pshfunctions.Otherwise,weshalllosesomeoftheessentialpropertiesthatthecomplexMonge-Amp`ereoperatorshouldhave,seeKiselman’spaperYangXingDepartmentofMathematics,UniversityofUme˚a,S-90187Ume˚a,SwedenE-mailaddress:[e-mail]2000MathematicsSubjectClassification.Primary32W20,32Q15Keywords.complexMonge-Amp`ereoperator,compactK¨ahlermanifold1[KI]orBedford’ssurvey[B].InaboundeddomainofCnoneusuallyneedscertainas-sumptionsonvaluesoffunctionsneartheboundaryofthedomaintodefinecomplexMonge-Amp`eremeasuresofunboundedplurisubahrmonicfunctions,seetheCegrellclass[C1-2]whereCegrellintroducedthelargestsubclassE(Ω)ofplurisuhharmonicfunctionsinaboundedhyperconvexdomainΩforwhichthecomplexMonge-Amp`ereoperatoriswell-definedandthemonotoneconvergencetheoremisvalid.However,suchatechniquedoesnotseemtoworkforquasi-pshfunctionsinacompactK¨ahlermanifoldbecauseweloseboundary.Ontheotherhand,itwasalreadyobservedbyBedfordandTaylor[BT1]thatforeachquasi-pshfunctionuthecomplexMonge-Amp`eremeasureωnu:=(ω+ddcu)niswelldefinedonitsnon-polarsubset{u−∞}.ThecomplexMonge-Amp`eremeasuresωnuconcentratingon{u−∞}werestudiedbyGuedjandZeriahi[GZ].In[X3]weob-tainedseveralconvergencetheoremsforcomplexMonge-Amp`eremeasureswithoutmassonpluripolarsets.InthispaperweintroduceaquitelargesubclassF(X,ω)ofquasi-pshfunctionsonwhichimagesofthecomplexMonge-Amp`ereoperatorarewell-definedpositivemeasuresandmayhavepositivemassesonpluripolarsets.WeprovethatthesetF(X,ω)isaconvexconeandincludesallquasi-pshfunctionswhichareintheCegrellclass.OurmainresultisthefollowingconvergencetheoremofthecomplexMonge-Amp`ereoperatorinF(X,ω).Theorem5.(ConvergenceTheorem)Let0≤p∞.Supposethatu0∈F(X,ω)andthatg∈PSH(X,ω)∩L∞(X)isnonpositive.Ifuj,u∈F(X,ω)aresuchthatuj→uinCapωonXanduj≥u0,then(−g)pωnuj→(−g)pωnuweaklyinX.AsadirectconsequencewehaveCorollary5.Let0≤p∞and0≥g∈PSH(X,ω)∩L∞(X).Ifuj,u∈F(X,ω)aresuchthatujցuorujրuinX,then(−g)pωnuj→(−g)pωnuweaklyinX.Forboundedquasi-pshfunctions,Corollary5isaslightlystrongerversionofthewell-knownmonotoneconvergencetheoremduetoBedfordandTaylor[BT2].AcknowledgmentsIwouldliketothankUrbanCegrellforinspiringdiscussionsonthesubject.2.TheclassF(X,ω)InthissectionwefirstintroducethesubclassF(X,ω)ofquasi-pshfunctions,onwhichimagesofthecomplexMonge-Amp`ereoperatorarefinitepositivemeasuresinX.WeobtainsomecharacterizationsoffunctionsinF(X,ω).Finally,weprovethatF(X,ω)isastar-shapedandconvexset.RecallthattheMonge-Amp`erecapacityCapωassociatedtotheK¨ahlerformωisdefinedbyCapω(E)=supZEωnu;u∈PSH(X,ω)and−1≤u≤0 ,2foranyBorelsetEinX.ThecapacityCapωisintroducedbyKolodziej[KO1]andiscomparabletotherelativeMonge-Amp`erecapacityofBedfordandTaylor[BT2],andhencevanishesexactlyonpluripolarsetsofX.RecallalsothatasequenceμjofpositiveBorelmeasuresissaidtobeuniformlyabsolutelycontinuouswithrespecttoCapωonX,orwewritethatμj≪CapωonXuniformlyforallj,ifforanyε0thereexistsδ0suchthatμj(E)εforalljandBorelsetsE⊂XwithCapω(E)δ.DenotebyPSH−1(X,ω)thesubsetoffunctionsuinPSH(X,ω)withmaxXu≤−1.Givenafunctionu∈PSH−1(X,ω),wedefinethemeasure(−u)ωn−1u∧ωinXwhichiszeroin{u=−∞}andZE(−u)ωn−1u∧ω=limj→∞ZE∩{u−j}−max(u,−j)
ωn−1max(u,−j)∧ωforallk≥1andE⊂{u−k}.Inacompletelysimilarway,wedefinethemeasureωn−1u∧ω:=χ{u−∞}ωn−1u∧ω,whereχ{u−∞}isthecharacteristicfunctionoftheset{u−∞}.Itisworthtopointoutthatingeneralneitherthemeasure(−u)ωn−1u∧ωnorωn−1u∧ωislocallyfiniteinX.However,wehavethefollowingresult.Proposition1.Letu∈PSH−1(X,ω).Supposethat−max(u,−j)ωn−1max(u,−j)∧ω≪CapωonXuniformlyforallj=1,2,....Thenthefollowingstatementshold.(1)(−u)ωn−1u∧ω