Table Of ContentLecture 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
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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
