Table Of ContentSpringer 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. They are sufficiently self-contained to be
accessibletomorethanjusttheintimatespecialistsofthesubject,andsufficiently
comprehensive to remain valuable references for many years. Besides the current
stateofknowledgeinitsfield,anSMMvolumeshouldideallydescribeitsrelevance
to and interaction with neighbouring fields of mathematics, and give pointers to
futuredirectionsofresearch.
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
