Lecture Notes in Mathematics 2075 CIME Foundation Subseries Giorgio Patrizio Zbigniew Błocki François Berteloot Jean-Pierre Demailly Pluripotential Theory Cetraro, Italy 2011 Editors: Filippo Bracci, John Erik Fornæss Lecture Notes in Mathematics 2075 Editors: J.-M.Morel,Cachan B.Teissier,Paris Forfurthervolumes: http://www.springer.com/series/304 FondazioneC.I.M.E.,Firenze C.I.M.E.stands forCentroInternazionale Matematico Estivo,thatis,International Mathe- maticalSummerCentre.Conceivedintheearlyfifties,itwasbornin1954inFlorence,Italy, andwelcomedbytheworldmathematicalcommunity:itcontinuessuccessfully,yearforyear, tothisday. Many mathematicians from all over the world have been involved in a way oranother in C.I.M.E.’sactivitiesovertheyears.ThemainpurposeandmodeoffunctioningoftheCentre maybesummarisedasfollows:everyyear,duringthesummer,sessionsondifferentthemes from pure and applied mathematics are offered byapplication to mathematicians from all countries. 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C.I.M.E.Director C.I.M.E.Secretary PietroZECCA ElviraMASCOLO DipartimentodiEnergetica“S.Stecco” DipartimentodiMatematica“U.Dini” Universita`diFirenze Universita`diFirenze ViaS.Marta,3 vialeG.B.Morgagni67/A 50139Florence 50134Florence Italy Italy e-mail:zecca@unifi.it e-mail:[email protected]fi.it FormoreinformationseeCIME’shomepage:http://www.cime.unifi.it CIMEactivityiscarriedoutwiththecollaborationandfinancialsupportof: -INdAM(IstitutoNazionalediAltaMatematica) -MIUR(Ministerodell’Istruzione,dell’Universita`edellaRicerca) Giorgio Patrizio Zbigniew Błocki (cid:2) Franc¸ois Berteloot Jean-Pierre Demailly (cid:2) Pluripotential Theory Cetraro, Italy 2011 Editors: Filippo Bracci John Erik Fornæss 123 GiorgioPatrizio ZbigniewBłocki DipartimentodiMatematica“U.Dini” InstituteofMathematics Universita`diFirenze JagiellonianUniversity Firenze,Italy Krakow,Poland Franc¸oisBerteloot Jean-PierreDemailly InstitutdeMathe´matiquesdeToulouse InstitutFourier,Laboratoirede Universite´PaulSabatier Mathe´matiques Toulouse,France UniversityofGrenobleI Saint-Martind’He`res,France ISBN978-3-642-36420-4 ISBN978-3-642-36421-1(eBook) DOI10.1007/978-3-642-36421-1 SpringerHeidelbergNewYorkDordrechtLondon LectureNotesinMathematicsISSNprintedition:0075-8434 ISSNelectronicedition:1617-9692 LibraryofCongressControlNumber:2013936248 MathematicsSubjectClassification(2010):32U99,14-XX,32-XX,31-XX (cid:2)c Springer-VerlagBerlinHeidelberg2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Pluripotential theory is a very powerful tool in geometry, complex analysis and dynamics. The principal subjects of investigation in pluripotential theory are plurisubharmonic functions, namely, those functions which remain subharmonic underholomorphicchangesofcoordinates.Plurisubharmonicfunctionsareobjects rather easy to handle and to be constructed; therefore, they are very useful and important tools in complex analysis, geometry (such as geometry of Ka¨hler– Einstein manifolds, hyperbolicity, Green–Griffiths conjecture) and holomorphic dynamics.Amongthose,maximalplurisubharmonicfunctionsandtheirassociated Monge–Ampe`reequationsplay a fundamentalrolein modernmathematics.Many problems related to manifolds endowed with particular geometric structure, such assymplectic,Ka¨hlerian,iperka¨hlerian,quaternionial-Ka¨hlerian,algebraicspinorial and Calabi–Yau manifoldsand their generalizations,can be rephrasedin terms of severaltypes of complexMonge–Ampe`reequationsaboutthe existence of metric withconstantcurvatureonalgebraicmanifolds.Inparticular,pluripotentialtheory plays a very basic role in the study of the equationsassociated with the existence ofEinsteinmetricsofconstantscalarcurvatureandextremal“ala` Calabi”,bothin thestaticversionandintheparabolicone(Ricci’sflowsandCalabi)whichrecently allowedtosolvethePoincare´andThurston’sconjectures. Acompleteanddeeptheoryhasbeendevelopedinordertocharacterizemaximal plurisubhharmonic functions by Bedford, Taylor, Demailly, Kiselman, Siciak, Błocki and others. Indeed, maximal plurisubharmonic functions are essentially solutionsofhomogeneouscomplexMonge–Ampe`reequations.Specialsolutionsto suchequationsarethepluricomplexGreenfunctionandthepluri-complexPoisson measures,introducedandusedinreproducingformulasforplurisubharmonicfunc- tionsbyKlimek,Demailly,Lempertandothers.SuchapluricomplexGreenfunction turnedoutalsotobestrictlyrelatedtotheKobayashidistanceandtohyperconvexity andothergeometricalpropertiesofdomainsinCn.Pluripotentialtheoryhasalsoa numberof very importantapplicationsin algebraic geometry,in particular related to jets bundles and the solution of some of the leading conjectures in the area suchasGreen–GriffithsandKobayashisconjectures.Otherapplicationstocomplex dynamics in higher dimensions are also available, both in the realm of discrete dynamics and in that of holomorphic foliations. On another side, pluripotential v vi Preface theory and complex Monge–Ampe`re equations are used to characterize complex manifolds(theso-calledparaboliccomplexmanifoldsandGrauerttubes). The CIME session in Cetraro on Pluripotential theory was a great and unique occasionto presenta fewcoursesontopicsof highinterestinthe areaandtojoin both experts and young mathematicians in a nice environment. The school, from whichthesenotesaretaken,wasaimedtoprovidecoursesonpluripotentialtheory and Monge–Ampe`re equation and applications to algebraic geometry, complex dynamics and differential geometry. The program with its wide range of topics broughttogethermathematiciansandyoungresearcherswithdifferentbackground: complex analysis and geometry, differential geometry, dynamics, and differential equations. Thecoursesandthenotestakenfromthemwhichconstitutethechaptersofthis volumearebrieflydescribedhereafter. Inhislectures,Franc¸oisBertelootgivesasyntheticandself-containedexposition ofthetheoryofbifurcationcurrentsinholomorphicfamiliesofrationalmaps,giving applicationsofpluripotentialtheorytocomplexdynamics.HeconstructstheGreen measure (the maximal entropy measure) for a fixed rational map and discusses its dynamical properties and proves an approximation formula for its Lyapunov exponent.Hepresentssomeconcreteholomorphicfamiliesofexamplesandproves theBranner–Hubbardresultaboutthecompactnessoftheconnectednesslocusand introduces the hypersurfaces Per .w/ whose distribution turns out to shape the n bifurcationlocus.HealsodescribesthemodulispaceMod ofdegreetworational 2 maps.Next,hestudiesthebifurcationcurrentT andgivesaproofofDeMarco’s bif fundamental results which precisely relates the Lyapunov exponent to the Green function evaluated on the critical points. He then studies how the asymptotic distribution of dynamically defined hypersurfaces is governed by the bifurcation current. Finally, he examines the higher exterior powers Tk of the bifurcation bif current.Heconcentratesonthehighestpowerandshowsthatthesupportofsucha measureistheseatofthestrongestbifurcations. ZbigniewBłocki’s lectures present two situations where the complex Monge– Ampe`reequationappearsinKa¨hlergeometry:theCalabiconjectureandgeodesics in the space of Ka¨hler metrics. In the first case the problem is to construct, in a given Ka¨hler class, a metric with prescribed Ricci curvature. It turns out that this is equivalent to finding a metric with prescribed volume form and thus to solving nondegenerate complex Monge–Ampe`re equation on a manifold with no boundary.In the secondcase to find a geodesic in a Ka¨hler class one hasto solve a homogeneouscomplex Monge–Ampe`re equation on a manifold with boundary. In his self-contained lecture notes, Błocki discusses both the geometric aspects and the PDE part, mostly a priori estimates, starting from a very elementary introduction to Ka¨hler geometry. He introduces the Calabi conjecture and its equivalencetocomplexMonge–Ampe`reequation.Laterhegivesbasicpropertiesof theRiemannianstructureofthespaceofKa¨hlermetrics,theAubin–Yaufunctional andtheMabuchiK-energyaswellasrelationtoconstantscalarcurvaturemetrics. TheLempert–Vivasexampleisalsodescribed.Thenotescontainalsofundamental resultsoncomplexMonge–Ampe`reequationssuchasthebasicuniquenessresults as well as the comparison principle. Among other things, the continuity method, Preface vii used to prove existence of solutions, is described and Yau’s proof of the L1- estimateusingMoser’siterationispresented. Pluripotentialtheoryisapowerfulandstrongtoolalsoinalgebraicgeometryas showninJean–PierreDemailly’slecturesnote.Inhislectures,hedescribesthemain techniques involved in the proof of holomorphic Morse inequalities which relate certaincurvatureintegralstotheasymptoticcohomologyoflargetensorpowersof line or vector bundles bring a useful complement to the Riemann–Roch formula. HealsodescribestheirlinkwithMonge–Ampe`reoperatorsandintersectiontheory. Finally,heprovidesapplicationstothestudyofasymptoticcohomologyfunctionals and the Green–Griffiths–Lang conjecture. The latter conjecture asserts that every entire curve drawn on a projective variety of general type should satisfy a global algebraic equation; via a probabilistic curvature calculation, holomorphic Morse inequalities imply that entire curves must at least satisfy a global algebraic differentialequation. GiorgioPatrizio’slecturesintheCIMEsessionwerebasedonthelecturenotes by himself and Andrea Spiro, included in this volume. In these notes the authors discussthe link betweenpluripotentialtheoryandMonge–Ampe`refoliations.The latterturnedouttohavemanyapplicationsincomplexgeometry,andtheselection ofa goodcandidatefortheassociatedMonge–Ampe`refoliationisalwaysthefirst step in the construction of well-behaved solutions of the complex homogeneous Monge–Ampe`reequation.AfterreviewingsomebasicnotionsonMonge–Ampe`re foliations,theauthorsconcentrateontwomaintopics.Theydiscusstheconstruction of (complete) modular data for a large family of complex manifolds, which carry regular pluricomplex Green functions. This class of manifolds naturally includes allsmoothlybounded,strictlylinearlyconvexdomainsandallsmoothlybounded, strongly pseudoconvex circular domains of Cn. Then they report on the problem ofdefiningpluricomplexGreenfunctionsinthealmostcomplexsetting,providing sufficientconditionsonalmostcomplexstructures,whichensureexistenceofalmost complexGreenpluripotentialsandequalitybetweenthenotionsofstationarydisks andofKobayashiextremaldisks,andallowextensionsofknownresultstothecase ofnon-integrablecomplexstructures. Itisarealgreatpleasuretothankthespeakersfortheirveryinterestinglectures andallthe authorsforthe nicelecturesnotestheyhavecarefullypreparedforthis volume.WealsowanttowarmlythankalltheparticipantstotheCIMEsessionfor their enthusiasm and interest in the subjectand for having created a very friendly environmentwhichmadepossibletoexperiencesuchagreatscientificatmosphere. Last but not least, we want to thank the CIME organization for giving us the opportunitytoorganizeandfinancingthisschoolandtheGNSAGAofINDAMfor support.OurspecialgratitudealsotoPietroZeccaandElviraMascolo. Also, a special thanks to Mrs. Ute McCrory at Springer for her assistance in preparingthevolume. Rome,Italy FilippoBracci AnnArbor,MI JohnErikFornæss Contents BifurcationCurrentsinHolomorphicFamiliesofRationalMaps ......... 1 Franc¸oisBerteloot TheComplexMonge–Ampe`reEquationinKa¨hlerGeometry.............. 95 ZbigniewBłocki ApplicationsofPluripotentialTheorytoAlgebraicGeometry ............. 143 Jean-PierreDemailly PluripotentialTheoryandMonge–Ampe`reFoliations....................... 265 G.PatrizioandA.Spiro ix