Springer Monographs in Mathematics Takeo Ohsawa L² Approaches in Several Complex Variables Towards the Oka–Cartan Theory with Precise Bounds Second Edition Springer Monographs in Mathematics Editors-in-Chief IsabelleGallagher,Paris,France MinhyongKim,Oxford,UK SeriesEditors SheldonAxler,SanFrancisco,USA MarkBraverman,Princeton,USA MariaChudnovsky,Princeton,USA TadahisaFunaki,Tokyo,Japan SinanC.Güntürk,NewYork,USA ClaudeLeBris,MarnelaVallée,France PascalMassart,Orsay,France AlbertoPinto,Porto,Portugal GabriellaPinzari,Napoli,Italy KenRibet,Berkeley,USA RenéSchilling,Dresden,Germany PanagiotisSouganidis,Chicago,USA EndreSüli,Oxford,UK ShmuelWeinberger,Chicago,USA BorisZilber,Oxford,UK This series publishes advanced monographs giving well-written presentations of the “state-of-the-art” in fields of mathematical research that have acquired the maturity needed for such a treatment. 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Moreinformationaboutthisseriesathttp://www.springer.com/series/3733 Takeo Ohsawa 2 L Approaches in Several Complex Variables Towards the Oka–Cartan Theory with Precise Bounds Second Edition 123 TakeoOhsawa ProfessorEmeritus NagoyaUniversity Nagoya,Japan ISSN1439-7382 ISSN2196-9922 (electronic) SpringerMonographsinMathematics ISBN978-4-431-56851-3 ISBN978-4-431-56852-0 (eBook) https://doi.org/10.1007/978-4-431-56852-0 LibraryofCongressControlNumber:2018959147 ©SpringerJapanKK,partofSpringerNature2015,2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerJapanKKpartofSpringerNature. The registered company address is: Shiroyama Trust Tower, 4-3-1 Toranomon, Minato-ku, Tokyo 105-6005,Japan Preface As in the study of complex analysis of one variable, the general theory of several complex variables has manifold aspects. First, it provides a firm ground for systematic studies of special functions such as elliptic functions, theta functions, andmodularfunctions.Thegeneraltheoryplaysaroleofconfirmingtheexistence and uniqueness of functions with prescribed zeros and poles. Another aspect is to give an insight into the connection between two different fields of mathematics by understanding how the tools work. The theory of sheaves bridged analysis and topologyinsuchaway.Intheconstructionofthisbasictheoryofseveralcomplex variables, a particularly important contribution was made by two mathematicians, Kiyoshi Oka (1901–1978) and Henri Cartan (1904–2008). The theory of Oka and Cartan is condensed in a statement that the first cohomology of coherent analytic sheavesoverCn iszero.Ontheotherhand,themethodofPDE(partialdifferential equations)hadturnedouttobeessentialintheexistenceofconformalmappings.By thisapproach,thefunctiontheoryonRiemannsurfacesasone-dimensionalcomplex manifoldswasexploredbyH.Weyl.Weyl’smethodwasdevelopedonmanifoldsof higherdimensionbyK.KodairawhogeneralizedRiemann’sconditionforAbelian varieties by establishing a differential geometric characterization of nonsingular projective algebraic varieties. This PDE method, based on the L2 estimates for ¯ the ∂-operator, was generalized by J. Kohn, L. Hörmander, A. Andreotti, and E. Vesentini. As a result, it enabled us to see the results of Oka and Cartan in a much higher resolution. In particular, based on such a refinement, existence theoremsforholomorphicfunctionswithL2 growthconditionshavebeenobtained by Hörmander, H. Skoda, and others. The purpose of the present monograph is to report on some of the recent results in several complex variables obtained by the L2 method which can be regarded as a continuation of these works. Among varioustopicsincludingcomplexgeometry,theBergmankernel,andholomorphic foliations,aspecialemphasisisputontheextensiontheoremsanditsapplications. In this topic, highlighted are the recent developments after the solution of a long- standingopenquestionofN.Suita.ItisaninequalitybetweentheBergmankernel and the logarithmic capacity on Riemann surfaces, which was first proved by Z.Błockiforplanedomains.Q.GuanandX.-Y.Zhouprovedgeneralizedvariants v vi Preface andcharacterizedthosesurfacesonwhichtheinequalityisstrict.Theirworkgave the author a decisive impetus to start writing a survey to cover these remarkable achievements. As a result, he could find an alternate proof of the inequality, basedonhyperbolicgeometry,whichispresentedinChap.3.However,thereaders are recommended to have a glance at Chap.4 first, where the questions on the Bergman kernels are described more systematically. (The author started to write the monograph from Chap.4.) Since there have been a lot of subsequent progress concerningthematerialsinChaps.3and4duringthepreparationofthemanuscript, itsoonbecamebeyondtheauthor’sabilitytogiveasatisfactoryaccountofthewhole development.Sohewillbehappytohaveachanceinthefuturetoreviseandenlarge thisratherbriefmonograph. Nagoya,Japan TakeoOhsawa March2015 Preface to the Second Edition Thanks to the goodwill of the publisher, the revision and enlargement have been realized. What made this edition possible was the recent remarkable activity after Błocki’ssolutionofSuita’sconjectureforplanedomains.Amongmanycorrections, themostimportantoneisthereplacementofanerroneousproofofTheorem3.2by thepresentonewhichishopefullycorrect.TheauthorisverygratefultoShigeharu Takayama for pointing out the mistake. Additions have been made to focus on the results which appeared in the past 3 years. Some of them are in Sect.4.4.5 “Berndtsson–Lempert Theory and Beyond” and in the section “A History of Levi FlatHypersurfaces”in5.3.Besidesthese,eachchapterhasbeensupplemented by asectiontitled“NotesandRemarks,”inwhichtheauthoralsotriedtoenhancethe depth feeling of complex analysis and convey the atmosphere of several complex variablessimilartosearchingforextraterrestrialintelligencesinceHartogsandOka. Nagoya,Japan TakeoOhsawa April2018 vii Contents 1 BasicNotionsandClassicalResults........................................ 1 1.1 FunctionsandDomainsOverCn....................................... 2 1.1.1 HolomorphicFunctionsandCauchy’sFormula............... 2 1.1.2 WeierstrassPreparationTheorem.............................. 4 1.1.3 DomainsofHolomorphyandPlurisubharmonicFunctions .. 7 1.2 ComplexManifoldsandConvexityNotions........................... 9 1.2.1 ComplexManifolds,SteinManifoldsandHolomorphic Convexity ....................................................... 10 1.2.2 ComplexExteriorDerivativesandLeviForm................. 14 1.2.3 PseudoconvexManifoldsandOka–GrauertTheory .......... 16 1.3 Oka–CartanTheory..................................................... 19 1.3.1 SheavesandCohomology...................................... 19 1.3.2 CoherentSheaves,ComplexSpaces,andTheoremsA andB ............................................................ 25 1.3.3 Coherence of Direct Images and a Theorem of AndreottiandGrauert .......................................... 30 ¯ 1.4 ∂-EquationsonManifolds.............................................. 31 ¯ 1.4.1 HolomorphicVectorBundlesand∂-Cohomology............ 32 1.4.2 CohomologywithCompactSupport........................... 36 1.4.3 Serre’sDualityTheorem ....................................... 38 1.4.4 FiberMetricandL2Spaces.................................... 41 1.5 NotesandRemarks ..................................................... 42 References..................................................................... 45 2 AnalyzingtheL2∂¯-Cohomology........................................... 47 2.1 OrthogonalDecompositionsinHilbertSpaces........................ 47 2.1.1 BasicsonClosedOperators.................................... 47 2.1.2 Kodaira’sDecompositionTheoremandHörmander’s Lemma .......................................................... 48 2.1.3 RemarksontheClosedness .................................... 50 ix x Contents 2.2 VanishingTheorems .................................................... 51 2.2.1 MetricsandL2∂¯-Cohomology ................................ 51 2.2.2 CompleteMetricsandGaffney’sTheorem.................... 54 2.2.3 SomeCommutatorRelations................................... 55 2.2.4 PositivityandL2Estimates .................................... 58 2.2.5 L2VanishingTheoremsonCompleteKählerManifolds..... 60 2.2.6 PseudoconvexCases............................................ 66 2.2.7 SheafTheoreticInterpretation ................................. 67 2.2.8 ApplicationtotheCohomologyofComplexSpaces ......... 70 2.3 FinitenessTheorems.................................................... 77 2.3.1 L2FinitenessTheoremsonCompleteManifolds............. 77 2.3.2 ApproximationandIsomorphismTheorems.................. 79 2.4 NotesonMetricsandPseudoconvexity ............................... 88 2.4.1 PseudoconvexManifoldswithPositiveLineBundles ........ 89 2.4.2 GeometryoftheBoundariesofCompleteKählerDomains.. 91 2.4.3 CurvatureandPseudoconvexity................................ 94 2.4.4 MiscellaneaonLocallyPseudoconvexDomains.............. 95 2.5 NotesandRemarks ..................................................... 99 References..................................................................... 111 3 L2Oka–CartanTheory ..................................................... 115 3.1 L2ExtensionTheorems ................................................ 115 3.1.1 ExtensionbytheTwistedNakanoIdentity .................... 115 3.1.2 L2ExtensionTheoremsonComplexManifolds.............. 121 3.1.3 ApplicationtoEmbeddings .................................... 125 3.1.4 ApplicationtoAnalyticInvariants............................. 127 3.2 L2DivisionTheorems.................................................. 128 3.2.1 AGauss–Codazzi-TypeFormula .............................. 129 3.2.2 Skoda’sDivisionTheorem ..................................... 131 3.2.3 FromDivisiontoExtension.................................... 135 3.2.4 ProofofaPreciseL2DivisionTheorem ...................... 138 3.3 L2ApproachestoAnalyticIdeals...................................... 140 3.3.1 Briançon–SkodaTheorem...................................... 140 3.3.2 Nadel’sCoherenceTheorem................................... 142 3.3.3 MiscellaneaonMultiplierIdealSheaves...................... 142 3.4 NotesandRemarks ..................................................... 153 References..................................................................... 161 4 BergmanKernels............................................................. 165 4.1 BergmanKernelandMetric............................................ 165 4.1.1 BergmanKernels ............................................... 166 4.1.2 TheBergmanMetric............................................ 169 4.2 TheBoundaryBehavior ................................................ 170 4.2.1 LocalizationPrinciple .......................................... 171 4.2.2 Bergman’sConjectureandHörmander’sTheorem ........... 172