. SOME REMARKS ON HOMOGENEOUS KA¨HLER MANIFOLDS 5 ANDREA LOI, ROBERTO MOSSA 1 0 2 Dipartimento di Matematica e Informatica, Universita` di Cagliari, Via Ospedale n 72, 09124 Cagliari, Italy a J Abstract. In this paper we provide a positive answer to a conjecture due 0 3 to A. J. Di Scala, A. Loi, H. Hishi (see [3, Conjecture 1]) claiming that a simply-connected homogeneous K¨ahlermanifoldM endowedwithanintegral G] K¨ahler form µ0ω, admits a holomorphic isometric immersion in the complex projective space, for a suitable µ0 >0. This result has two corollaries which D extend tohomogeneous K¨ahlermanifoldstheresultsobtainedbytheauthors . h in[8]andin[12]forhomogeneous boundeddomains. t a m [ 1. Introduction and statement of the main results 1 v Thepaperconsistsofthreeresults: Theorem1.1,Theorem1.2andTheorem1.3. 1 Our first result answer positively to a conjecture due to A. J. Di Scala, A. Loi, H. 1 0 Hishi (see [3, Conjecture 1]), namely: 0 0 Theorem 1.1. Let (M,ω) be a simply-connected homogeneous K¨ahler manifold . 2 such that its associated K¨ahler form ω is integral. Then there exists a constant 0 µ >0 such that µ ω is projectively induced. 5 0 0 1 Before stating our second result (Theorem 1.2), we recall the definitions of a : v Berezin quantization and diastasis function. Let (M,ω) be a symplectic manifold i X and let {·,·} be the associated Poisson bracket. A Berezin quantization on M is r a given by a family of associative algebras A~ ⊂C∞(M) such that • ~∈E ⊂R+ and infE =0 • exists a subalgebra A ⊂ (⊕h∈EAh,∗), such that for an arbitrary element f =f(~)∈A, where f(~)∈A~, there exists a limit lim~→0f(~)=ϕ(f)∈ C∞(M), E-mail address: [email protected]; [email protected] . Date:February3,2015. 2000 Mathematics Subject Classification. 53D05;53C55;58F06. Keywords andphrases. K¨ahlermetrics;Balancedmetrics;Berezinquantization; boundedhomo- geneous domain;Calabi’sdiastasisfunction;DiastaticEntropy. The authors were supported by Prin 2010/11 – Varieta` reali e complesse: geometria, topologia e analisi armonica – Italy and also by INdAM. GNSAGA - Gruppo Nazionale per le Strutture Algebriche,GeometricheeleloroApplicazioni. 1 2 A.LOI,R.MOSSA • hold the following correspondence principle: for f,g ∈A ϕ(f ∗g)=ϕ(f)ϕ(g), ϕ ~−1(f ∗g−g∗f) =i{ϕ(f),ϕ(g)}, (cid:0) (cid:1) • for any pair of points x ,x ∈ Ω there exists f ∈ A such that ϕ(f)(x ) 6= 1 2 1 ϕ(f)(x ) 2 Let(M,ω)bearealanalyticK¨ahlermanifold. LetU ⊂M beaneighborhoodofa pointp∈M andletψ :U →RbeaK¨ahlerpotentialforω. Thepotentialψ canbe analytically extended to a sesquilinear function ψ(p,q), defined in a neighborhood of the diagonal of U × U, such that ψ(q,q) = ψ(q). Assume (by shrinking U) that the analytic extension ψ is defined on U ×U. The Calabi’s diastasis function D:U ×U →R is given by: D(p,q)=ψ(p,p)+ψ(q,q)+ψ(p,q)+ψ(q,p). Denoted by D (q) := D(p,q) the diastasis centered in a point p, one can see that p D :U →R is a K¨ahler potential around p. p Our second result extends to any homogeneous K¨ahler manifolds the results obtained by the authors [8] for homogeneous bounded domains. Theorem 1.2. Let (M,ω) be a homogeneous K¨ahler manifold. Then the following are equivalent: (a) M is contractible. (b) (M,ω) admits a global K¨ahler potential. (c) (M,ω) admits a global diastasis D:M ×M →R. (d) (M,ω) admits a Berezin quantization. Also to state the third result (Theorem 1.3) we need some definitions. The diastatic entropy has been defined by the second author in [12] (see also [13]) following the ideas developed in [9] and [10]. Assume that ω admits a globally defined diastasis function D : M →R (centered at p). The diastatic entropy at p p is defined as ωn Ent(M, ω)(p)=min c>0 | e−cDp <∞ . (1) (cid:26) Z n! (cid:27) M The definition does not depend on the point p chosen (see [13, Proposition 2.2]). Assume that M is simply-connected and assume that there exists a line bundle L → M such that c (L) = [ω] (i.e. ω is integral). Let h be an Hermitian metric 1 on L such that Ric(h) = ω and consider the Hilbert space of global holomorphic sections of Lλ =⊗λL given by ωn H = s∈Hol(L) | ksk2 = hλ(s,s) <∞ , (2) λ,h (cid:26) Z n! (cid:27) M SOME REMARKS ON HOMOGENEOUS KA¨HLER MANIFOLDS 3 withthe scalarproductinducedby k·k. Let{s } , N ≤∞, be anorthonor- j j=0,...,N mal basis for H . The ε-function is given by λ,h N ε (x)= hλ(s (x),s (x)). (3) λ j j Xj=0 This definition depends only on the K¨ahler form ω. Indeed since M is simply- connected,thereexists(uptoisomorphism)auniqueL→M suchthatc (L)=[ω], 1 and it is easy to see that the definition does not depend on the orthonormal basis chosen or on the Hermitian metric h (see e.g [8] or [14, 15] for details). Under the assumption that ε is a (strictly) positive function, one can consider the coherent λ states map f :M →CPN defined by f(x)=[s (x),...,s (x),...]. (4) 0 j The fundamental link between the coherent states map and the ε-function is ex- pressed by the following equation (see [16] for a proof) i f∗ω =λω+ ∂∂ε , (5) FS λ 2 where ω is the Fubini–Study form on CPN. FS Definition. We say that λω is a balancedmetric if and only if the ε is a positive λ constant. We can now state our third and last result which extends to any homogeneous K¨ahler manifold, the result obtained by the second author [12, Theorem 2], about thelinkbetweendiastaticentropy andbalancedcondition onhomogeneousbounded domains. Theorem 1.3. Let (M,ω) be a contractible homogeneous K¨ahler manifold. Then λω is balanced if and only if Ent(M,ω)<λ. (6) 2. Proof the main results We start with the following two lemmata, which provide a necessary and suffi- cient condition on the non-triviality of the Hilbert space H . λ,h Lemma 2.1 (Rosemberg–Vergne [17]). 1 Let (M,ω) be a simply-connected ho- mogeneous K¨ahler manifold with ω integral. Then there exists λ > 0 such that 0 H 6={0} if and only if λ>λ and λω is integral. λ,h 0 Lemma 2.2. Let (M,ω) be a simply-connected homogeneous K¨ahler manifold. Then H 6={0} if and only if λω is a balanced metric. λ,h 1Theauthorsthanks HishiHideyukiforreportingthisresult. 4 A.LOI,R.MOSSA Proof. LetF ∈Aut(M)∩Isom(M,ω)beanholomorphicisometryandletF itslift toL(whichexistssinceM issimply-connected). Notethat,if{s ,...,s },N ≤∞, 0 N e is an orthonormal basis for H , then {F−1(s (F (x))),...,F−1(s (F (x)))} is λ,h 0 N an orthonormalbasis for Hλ,Fe∗h. Therefoere e N ǫ (x)= F∗hλ F−1(s (F(x))),F−1(s (F(x))) λ j j Xj=0 (cid:16) (cid:17) e e e N = hλ(s (F(x)),s (F(x)))=ε (F(x)). j j λ Xj=0 SinceAut(M)∩Isom(M,ω)actstransitivelyonM, ε isforcedtobe constant. (cid:3) λ Proof of Theorem 1.1 By Lemma 2.1 there exists λ>λ , such that the Hilbert 0 space H 6={0}. By Lemma 2.2, ε is a positive constant, so the coherent states λ,h λ map f given by (4) is well defined. Moreover, by (5), we have that f∗ω = λω, FS i.e. λω is projectively induced for all λ > λ . The conclusion follows by setting 0 µ >λ . (cid:3) 0 0 ThemainingredientsfortheproofofTheorem1.2arethefollowingtwolemmata. ThefirstoneisthecelebratedtheoremofDorfmeisterandNakajimawhichprovides a positive answer to the so called Fundamental Conjecture formulated by Vinberg and Gindikin. The second one is a reformulation due to Engliˇs of the Berezin quantization result in terms of the ε-function and Calabi’s diastasis function. Lemma 2.3 (Dorfmeister–Nakajima[4]). A homogeneous K¨ahler manifold (M,ω) isthetotalspaceofaholomorphic fiberbundleover ahomogeneous boundeddomain Ω in which the fiber F = E ×C is (with the induced K¨ahler metric) is the K¨ahler product of a flat homogeneous K¨ahler manifold E and a compact simply-connected homogeneous K¨ahler manifold C. Lemma 2.4 (Engliˇs [5]). Let Ω ⊂ Cn be a complex domain equipped with a real analyticK¨ahler form ω. Then, (Ω,ω)admits aBerezin quantizationif thefollowing two conditions are satisfied: (1) the function ε (x) is a positive constant (i.e. λω is balanced) for all suffi- λ ciently large λ; (2) thefunctione−D(x,y)isgloballydefinedonΩ×Ω,e−D(x,y) ≤1ande−D(x,y) = 1, if and only if x=y. Proofof Theorem 1.2(a) ⇒ (b). ByLemma2.3,sinceahomogeneousbounded domain is contractible, M is a complex product Ω×F, where F = E ×C is (with the induced metric) a K¨ahler product of a flat K¨ahler manifold E and a compact simply-connected homogeneous K¨ahler manifold C. On the other hand E is K¨ahler SOME REMARKS ON HOMOGENEOUS KA¨HLER MANIFOLDS 5 flat and therefore E = Ck ×T where T is a product of flat complex tori. Hence M is a complex product Ω×Ck ×T ×C. Since we assumed M contractible, the compact factor T ×C has dimension zero and M = Ω×Ck. It is well-know that Ω is biholomorphic to a Siegel domain (see, [6] for a proof), therefore Ω × Ck is pseudoconvex and a classical result of Hormander (see [2]) asserting that the equation ∂u = f with f ∂-closed form has a global solution on pseudoconvex domains, assures us the existence of a global potential ψ for ω (see also [11, 12], and the proof of Theorem 4 in [3] for an explicit construction of the potential ψ). (b) ⇒ (c). Let ψ :M →R be a globalK¨ahlerpotential for (M,ω). By Lemma 2.3, M =Ω×Ck×T×C. ThecompactfactorT×C isa K¨ahlersubmanifoldofM, thereforetheexistenceofaglobalK¨ahlerpotentialonM impliesthatdim(T×C)= 0. So M =Ω×Ck. Consider the Hilbert space H defined in (2). Since Ω×Ck is contractible λ,h the line bundle L ∼= M ×C. So, the holomorphic section s ∈ Hλ,h can be viewed as a holomorphic function s : M → C. As Hermitian metric h over L we can take the one defined by h(s,s) = e−ψ|s|2. Hence H can be identified with the λ,h weightedHilbert spaceH (see [7])ofsquareintegrableholomorphicfunctions on λψ M =Ω×Ck, with weight e−λψ, namely ωn H = s∈Hol(M) | e−λψ|s|2 <∞ . (7) λψ (cid:26) Z n! (cid:27) M Assume λ > λ with λ given by Lemma 2.1, so that H 6= {0}. Let {s } be an 0 0 λψ j orthonormalbasis for H , then the reproducing kernel is given by λψ ∞ K (z,w¯)= s (z)s (w). λψ j j Xj=0 Then, the ε-function (defined in (3)) reads as: ε (z)=e−λψ(z)K (z,z¯). (8) λ λψ Letψ(z,w) be the analytic continuationof the K¨ahlerpotential ψ. By Lemma 2.2 there exists a constant C such that ε (z, w)=e−λψ(z,w)K (z, w)=C >0. (9) λ λψ Hence K (z, w) never vanish. Therefore, for any fixed point z , the function λψ 0 K (z, w)K (z , z ) Φ(z, w)= λψ λψ 0 0 (10) K (z, z )K (z , w) λψ 0 λψ 0 is well defined. Note that 1 D (z)= logΦ(z, z) (11) z0 λ 6 A.LOI,R.MOSSA is the diastasis centered in z associated to ω and that D is defined on whole M. 0 z0 Since we can repeat this argument for any z ∈M, we conclude that the diastasis 0 D:M ×M →R is globally defined. (c) ⇒ (d). Arguing as in “(b) ⇒ (c)”, the existence of a global diastasis implies that M is a complex productΩ×Ck. Therefore,asin (7)Hλ,h ∼=HλDz0 = s∈Hol(M) | Me−λDz0x|s|2ωnn! <∞ . Assume λ>λ0 withλ0 givenby Lemma 2(cid:8).1 and considerRthe coherent states m(cid:9)ap f given by (4). By Lemma 2.2 ε is a λ positive constant and by (5) we conclude that f∗ω =λω. FS By [8, Example 6], the Calabi’s diastasis function D associated to ω is FS FS such that e−DFS is globally defined on CPN × CPN. Since the diastasis D is globally defined on M, by the hereditary property of the diastasis function (see [1, Proposition 6]) we get that, for all x,y ∈M, λ e−DFS(f(x),f(y)) =e−λD(x,y) = e−D(x,y) (12) (cid:16) (cid:17) is globally defined on M ×M. Since, by [8, Example 6], e−DFS(p,q) ≤ 1 for all p,q∈CPN it followsthat e−D(x,y) ≤1 for all x,y ∈M. By Lemma 2.4, itremains to show that e−D(x,y) = 1 iff x = y. By (12) and by the fact that e−DFS(p,q) = 1 iff p = q (again by [8, Example 6]) this is equivalent to the injectivity of the coherentstatesmapf. This followsby [3, Theorem3],whichassertsthata K¨ahler immersion of a homogeneous K¨ahler manifold into a finite or infinite dimensional complex projective space is one to one. (d) ⇒ (a). By the very definition of Berezin quantization there exists a global potential for (M,ω). By Lemma 2.3 we deduce, as above, that M is a complex productΩ×Ck,whereΩisaboundedhomogeneousdomainwhichiscontractible.(cid:3) Proof of Theorem 1.3 By (c) in Theorem 1.2, the diastasis D : M ×M →R is globally defined. Assume that λω is balanced i.e. that ε is a positive constant. λ Since the ε does not depend on the K¨ahler potential, by (9) we have λ ελ(z, w)=e−λDz0(z,w)KλDz0 (z, w)=C (13) whereDz0(z, w)denotetheanalyticcontinuationofDz0(z)andKλDz0 istherepro- ducingkernelforHλDz0. By(10),withw=z0,wegetDz0(z, z0)= λ1 logΦ(z, z0)= 0. Hence C =e−λDz0(z,z0)KλDz0 (z, z0)=KλDz0 (z, z0)∈HλDz0. Thus HλDz0 contains the constant functions and ωn e−λDz0 <∞. Z n! M SOME REMARKS ON HOMOGENEOUS KA¨HLER MANIFOLDS 7 Therefore, by definition of diastatic entropy, Ent(M,ω)(z )<λ<∞. 0 On the other hand, if for some z0, Ent(M,ω)(z0) < λ, then HλDz0 6= {0} and by Lemma 2.2 we conclude that λω is balanced. (cid:3) Remark 2.5. Fromthe previousproof,weseethatifM issimply-connected,then Ent(M,ω)(z ) = λ for any z ∈ M, where λ is the positive constant defined in 0 0 0 0 Lemma 2.1. References [1] E.Calabi, Isometric Imbeddings of Complex Manifolds,Ann.ofMath.58(1953), 1-23. [2] S.C.Chen,M.C.Shaw,Partialdifferentialequationsinseveralcomplexvariables,AMS/IP Studies inAdvanced Mathematics, 19. AmericanMathematical Society, Providence, RI; In- ternationalPress,Boston,MA,(2001), xii+380pp. [3] A.J.DiScala, A.Loi,H.Hishi,K¨ahler immersions of homogeneous K¨ahler manifolds into complex space forms,toappear inAsianJournalofMathematics. [4] J.Dorfmeister,K.Nakajima,ThefundamentalconjectureforhomogeneousK¨ahlermanifolds, ActaMath.161(1988), no.1-2,23-70. [5] M. Engliˇs Berezin Quantization and Reproducing Kernels on Complex Domains, Trans. Amer.Math.Soc.vol.348(1996), 411-479. [6] S. G. Gindikin, I. I. Piatetskii-Shapiro, E. B. , Classification and canonical realization of complex bounded homogeneous domains, Trans.MoscowMath.Soc.12(1963), 404-437. [7] A. Loi, R. Mossa, F. Zuddas, The log-term of the disc bundle over a homogeneous Hodge manifold,arXiv:1402.2089. [8] A. Loi, R. Mossa, Berezin quantization of homogeneous bounded domains, Geom. Dedicata 161(2012), 119-128. [9] A.Loi,R.Mossa,The diastatic exponential of a symmetricspace, Math.Z.268(2011), 3-4, 1057-1068. [10] R. Mossa, The volume entropy of local Hermitian symmetric space of noncompact type, DifferentialGeom.Appl.31(2013), no.5,594-601. [11] R.MossaA bounded homogeneous domain and a projective manifold are not relatives, Riv. Mat.Univ.Parma4(2013), no.1,55-59. [12] R. Mossa, A note on diastatic entropy and balanced metrics, J. Geom. Phys. 86 (2014), 492-496. [13] R. Mossa, Upper and lower bounds for the first eigenvalue and the volume entropy of non- compact K¨ahler manifold, arXiv:1211.2705[math.DG]. [14] R. Mossa, Balanced metrics on homogeneous vector bundles, Int. J. Geom. Methods Mod. Phys.8(2011), no.7,1433-1438. [15] A.Loi,R.Mossa,Uniquenessofbalancedmetricsoncomplexvectorbundles,J.Geom.Phys. 61(2011), no.1,312-316. [16] J.Rawnsley, CoherentstatesandK¨ahlermanifolds,Quart.J.Math.Oxford(2),n.28(1977), 403-415. [17] J. Rosenberg, M. Vergne, Harmonically induced representations of solvable Lie groups J. Funct.Anal.,62(1985), pp.8-37 8 A.LOI,R.MOSSA [18] A.Loi,M.Zedda,K¨ahler-Einsteinsubmanifolds oftheinfinitedimensional projectivespace, Math.Ann.350(2011), 145-154.