Kähler Immersions of Kähler Manifolds into Complex Space Forms PDF
Preview Kähler Immersions of Kähler Manifolds into Complex Space Forms
Lecture Notes of the Unione Matematica Italiana Andrea Loi · Michela Zedda Kähler Immersions of Kähler Manifolds into Complex Space Forms Lecture Notes of 23 the Unione Matematica Italiana Moreinformationaboutthisseriesathttp://www.springer.com/series/7172 Andrea Loi • Michela Zedda ¨ ¨ Kahler Immersions of Kahler Manifolds into Complex Space Forms 123 AndreaLoi MichelaZedda DepartmentofMathematics&Computer DepartmentofMathematical,Physical Science &ComputerSciences UniversityofCagliari UniversityofParma Cagliari,Italy Parma,Italy ISSN1862-9113 ISSN1862-9121 (electronic) LectureNotesoftheUnioneMatematicaItaliana ISBN978-3-319-99482-6 ISBN978-3-319-99483-3 (eBook) https://doi.org/10.1007/978-3-319-99483-3 LibraryofCongressControlNumber:2018954939 MathematicsSubjectClassification(2010):32Q15,32A10 ©SpringerNatureSwitzerlandAG2018 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. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland TheUnioneMatematicaItaliana(UMI)has establishedabi-annualprize,sponsored bySpringer-Verlag,tohonoranexcellent, originalmonographpresentingthelatest developmentsinanactiveresearcharea ofmathematics,towhichtheauthormade importantcontributionsintherecent years. Theprize-winningmonographsarepublishedinthisseries. Detailsabouttheprizecanbefoundat: http://umi.dm.unibo.it/en/unione-matematica-italiana-prizes/book-prize-unione- matematicaitaliana/ This book has been awarded the 2017 Book Prize of the Unione Matematica Italiana. Themembersofthescientificcommitteeofthe2017prizewere: FabrizioCatanese UniversityofBayreuth,Germany CiroCiliberto(PresidenteoftheUMI) UniversitàdegliStudidiRomaTorVergata,Italy VittorioCotiZelati UniversityofNaplesFedericoII,Italy SusannaTerracini UniversitàdegliStudidiTorino,Italy ValentinoTosatti NorthwesternUniversity,Evanston,USA Preface The study of Kähler immersions of a given real analytic Kähler manifold into a finite- or infinite-dimensionalcomplex space form originates from the pioneering work of Eugenio Calabi [10]. With a stroke of genius, Calabi defined a powerful tool, a special (local) potential called the diastasis function, which allowed him to obtain necessary and sufficient conditions for a neighbourhood of a point to be locally Kähler immersed into a finite- or infinite-dimensional complex space form. As an application of this criterion, he also provided a classification of (finite-dimensional) complex space forms admitting a Kähler immersion into another.However,acompleteclassificationofKählermanifoldsadmittingaKähler immersion into complex space forms is not known, not even when the Kähler manifoldsinvolvedareofgreatinterest,e.g.whentheyareEinsteinorhomogeneous spaces. In fact, the diastasis function is not always explicitly given, and most of the time Calabi’s criterion,althoughtheoreticallyimpeccable,is difficultto apply. Nevertheless, throughoutthe last 60 years many mathematicians have worked on thesubjectandmanyinterestingresultshavebeenobtained. TheaimofthisbookistodescribeCalabi’soriginalwork,toprovideadetailed accountofwhatisknowntodayonthesubjectandtopointoutsomeopenproblems. Eachchapterbeginswithabriefsummaryofthetopicsdiscussedandendswith alistofexercisestotestthereader’sunderstanding. ApartfromthetopicsdiscussedinSect.3.1ofChap.3,whichcouldbeskipped withoutcompromisingthe understandingof the rest of the book,the prerequisites forthisbookareabasicknowledgeofcomplexandKählergeometry(treated,e.g. inMoroianu’sbook[61]). The authors are grateful to Claudio Arezzo and Fabio Zuddas for their careful readingof the textand fortheir valuablecomments,whichhave greatlyimproved thebook’sexposition. Cagliari,Italy AndreaLoi Parma,Italy MichelaZedda June2018 vii Contents 1 TheDiastasisFunction ....................................................... 1 1.1 Calabi’sDiastasisFunction.............................................. 1 1.2 ComplexSpaceForms................................................... 5 1.3 TheIndefiniteHilbertSpace............................................. 7 Exercises........................................................................ 10 2 Calabi’sCriterion............................................................. 13 2.1 KählerImmersionsintotheComplexEuclideanSpace................ 13 2.2 KählerImmersionsintoNonflatComplexSpaceForms............... 19 2.3 KählerImmersionsofaComplexSpaceFormintoAnother .......... 24 Exercises........................................................................ 27 3 HomogeneousKählerManifolds............................................ 29 3.1 AResultAboutKählerImmersionsofHomogeneousBounded DomainsintoCP∞....................................................... 29 3.2 Kähler Immersionsof HomogeneousKähler Manifolds intoCN≤∞ andCHN≤∞ ................................................ 32 3.3 KählerImmersionsofHomogeneousKählerManifolds intoCPN≤∞.............................................................. 35 3.4 BergmanMetricandBoundedSymmetricDomains................... 37 3.5 KählerImmersionsofBoundedSymmetricDomainsintoCP∞ ...... 40 Exercises........................................................................ 44 4 Kähler–EinsteinManifolds .................................................. 47 4.1 KählerImmersionsofKähler–EinsteinManifolds intoCHN orCN.......................................................... 48 4.2 KählerImmersionsofKEManifoldsintoCPN:TheEinstein Constant .................................................................. 52 ix x Contents 4.3 KählerImmersionsofKEManifoldsintoCPN:Codimension1 and2...................................................................... 56 Exercises........................................................................ 60 5 HartogsTypeDomains....................................................... 63 5.1 Cartan–HartogsDomains................................................ 63 5.2 Bergman–HartogsDomains............................................. 69 5.3 RotationInvariantHartogsDomains.................................... 70 Exercises........................................................................ 74 6 Relatives........................................................................ 75 6.1 RelativesComplexSpaceForms........................................ 75 6.2 Homogeneous Kähler Manifolds Are Not Relative toProjectiveOnes........................................................ 78 6.3 Bergman–HartogsDomainsAreNotRelativetoaProjective KählerManifold.......................................................... 80 Exercises........................................................................ 82 7 FurtherExamplesandOpenProblems..................................... 83 7.1 TheCigarMetriconC................................................... 83 7.2 Calabi’sCompleteandNotLocallyHomogeneousMetric............ 89 7.3 TheTaub-NUTMetriconC2............................................ 92 Exercises........................................................................ 93 References.......................................................................... 95 Index................................................................................ 99 Chapter 1 The Diastasis Function Abstract Inthischapterwedescribethediastasisfunction,abasictoolintroduced by Calabi (Ann Math 58:1–23, 1953) which is fundamental to study Kähler immersionsof Kähler manifoldsinto complexspace forms.In Sect.1.1 we define the diastasis function and summarize its basic properties, while in Sect.1.2 we describethediastasisfunctionsofcomplexspaceforms,whichrepresentthebasic examplesofKählermanifolds.Finally,inSect.1.3wegivetheformaldefinitionof whataKählerimmersionisandprovethattheindefiniteHilbertspaceconstitutesa universalKählermanifold,inthesensethatitisaspaceinwhicheveryrealanalytic KählermanifoldcanbelocallyKählerimmersed. 1.1 Calabi’s Diastasis Function LetM beann-dimensionalcomplexmanifoldendowedwitharealanalyticKähler metricg.RecallthattheKählermetricg isrealanalyticiffixedalocalcoordinate system z = (z ,...,z ) on a neighbourhood U of any point p ∈ M, it can be 1 n described on U by a real analytic Kähler potential Φ : U → R. In that case the potentialΦ(z)canbeanalyticallycontinuedtoanopenneighbourhoodW ⊂U×U ofthediagonal.DenotethisextensionbyΦ(z,w¯). Definition1.1 ThediastasisfunctionD(z,w)onW isdefinedby: D(z,w)=Φ(z,z¯)+Φ(w,w¯)−Φ(z,w¯)−Φ(w,z¯). (1.1) ThefollowingpropositiondescribesthebasicpropertiesofD(z,w). Proposition1.1(Calabi [10]) The diastasis function D(z,w) given by (1.1) satisfiesthefollowingproperties: (i) itisuniquelydeterminedbytheKählermetricganditdoesnotdependonthe choiceofthelocalcoordinatesystem; (ii) itisrealvaluedinitsdomainof(real)analyticity; ©SpringerNatureSwitzerlandAG2018 1 A.Loi,M.Zedda,KählerImmersionsofKählerManifoldsintoComplex SpaceForms,LectureNotesoftheUnioneMatematicaItaliana23, https://doi.org/10.1007/978-3-319-99483-3_1
The list of books you might like
