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A note on diastatic entropy and balanced metrics
Journal of Geometry and Physics 86 (2014) 492–496
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Journal of Geometry and Physics journal homepage: www.elsevier.com/locate/jgp
A note on diastatic entropy and balanced metrics Roberto Mossa Dipartimento di Matematica e Informatica, Palazzo delle Scienze, via Ospedale 72 - 09124, Cagliari, Italy
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abstract
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We give an upper bound Entd (Ω , g ) < λ of the diastatic entropy Entd (Ω , g ) (defined by the author in Mossa (2012) of a complex bounded domain (Ω , g ) in terms of the balanced condition (in Donaldson terminology) of the Kähler metric λ g. When (Ω , g ) is a homogeneous bounded domain we show that the converse holds true, namely if Entd (Ω , g ) < 1 then g is balanced. Moreover, we explicitly compute Entd (Ω , g ) in terms of Piatetski-Shapiro constants. © 2014 Elsevier B.V. All rights reserved.
Article history: Received 15 March 2013 Received in revised form 30 September 2014 Accepted 6 October 2014 Available online 16 October 2014 Keywords: Kählerian manifolds Homogeneous domains Diastatic entropy Balanced metrics
1. Introduction and statements of the main results Let (M , g ) be a real analytic Kähler manifold with associated Kähler form ω. Fix a local coordinate system z = (z1 , . . . , zn ) on a neighborhood U of a point p ∈ M and let φ : U → R be a real analytic Kähler potential (i.e ω|U = 2i ∂ ∂¯ φ ). By shrinking U, we can assume that the potential φ (z ) can be analytically continued to U × U. Denote this extension by φ (z , w). Calabi’s diastasis function D : U × U → R (E. Calabi [1]) is defined by
D (z , w) = Dw (z ) = φ (z , z ) + φ (w, w) − φ (z , w) − φ (w, z ) . Assume that ω has global diastasis function Dp : M → R centered at p. The diastatic entropy at p is defined as
Entd (M , g ) (p) = min c > 0 |
M
e−c Dp
ωn <∞ . n!
(1)
This definition does not depend on the point p fixed, provided that for every p ∈ Ω the diastasis Dp is globally defined (see [2, Proposition 2.2]). The concept of diastatic entropy was defined by the author in [2] (following the ideas developed in [3]), where he obtained upper and lower bound for the first eigenvalue of a real analytic Kähler manifold. In this paper we study the link between the diastatic entropy Entd (Ω , g ) and the balanced condition of the metric g. S. Donaldson in [4], in order to obtain a link between the constant scalar curvature condition on a Kähler metric g and the Chow stability of a polarization L, gave the definition of a balanced metric on a compact manifold. Later, this definition has been generalized, by C. Arezzo and A. Loi in [5] (see also [6]), to the noncompact case. In this paper we are interested in complex domains Ω ⊂ Cn (connected open subset of Cn ) endowed with a real analytic Kähler metric g. Let ω be the Kähler form associated to g. Assume that ω admits a global Kähler potential φ : Ω → R. We can define the weighted Hilbert space Hφ of square integrable holomorphic functions on Ω with weight e−φ
Hφ = s ∈ Hol(Ω ) :
e−φ |s|2 Ω
ωn <∞ . n!
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.geomphys.2014.10.004 0393-0440/© 2014 Elsevier B.V. All rights reserved.
(2)
R. Mossa / Journal of Geometry and Physics 86 (2014) 492–496
493
If Hφ ̸= {0} we can pick an orthonormal basis {sj }j=1,...,N ≤∞ ,
Ω
e−φ sj sk
ωn = δjk n!
and define its reproducing kernel by Kφ (z , w) =
N
sj (z ) sj (w).
(3)
j =0
The so called ε -function is defined by
εg (z ) = e−φ(z ) Kφ (z , z¯ ) ,
(4)
as suggested by the notation εg depends only on the metric g and not on the choice of the Kähler potential φ (see, for example, [7, Lemma 1] for a proof). Definition 1. The metric g is balanced if and only if the function εg is a positive constant. In the literature the constancy of εg was studied first by J.H. Rawnsley in [8] under the name of η-function, later renamed as ε -function in [9]. It also appears under the name of distortion function for the study of Abelian varieties by J.R. Kempf [10] and S. Ji [11], and for complex projective varieties by S. Zhang [12]. (See also [13] and references therein.) The following theorem represents the first result of this paper. Theorem 1. Let Ω ⊂ Cn be a complex domain and g a Kähler metric such that λ g is balanced for some λ > 0. Then the diastatic entropy Entd (Ω , g ) is constant and Entd (Ω , g ) < λ. In particular (see Lemma 4) the diastatic entropy of a homogeneous bounded domain is constant. It is natural to ask when the converse of the previous theorem holds true. In the next theorem we show that this is the case when (Ω , g ) is assumed to be homogeneous. Theorem 2. Let (Ω , g ) be a homogeneous bounded domain, then g is balanced if and only if Entd (Ω , g ) < 1. We also compute the diastatic entropy of a homogeneous bounded domain in terms of the constants pk , bk , qk and γk defined by the Piatetski-Shapiro root structure (see (10) and (11) in the Appendix): Corollary 2. Let (Ω , g ) be a homogeneous bounded domain. Then the diastatic entropy of (Ω , g ) is the positive constant given by Entd (Ω , g ) (z ) = max
1 + pk + bk + qk /2
γk
1≤k≤r
,
∀ z ∈ Ω.
Example 3. If g is the Bergman metric on Ω , then γk = 2 + pk + qk + bk , k = 1, . . . , r (see [14, Theorem 5.1] or [15, (2.19)]), so by Corollary 2, Entd (Ω , g ) (z ) = max
1≤k≤r
1 + pk + bk + qk /2 2 + pk + qk + bk
,
∀ z ∈ Ω.
(5)
In particular, when (Ω , g ) is a bounded symmetric domain (i.e. is a homogeneous bounded domain and a symmetric space as Riemannian manifold), there exist integers a and b such that pk = (k − 1) a,
qk = (r − k) a,
bk = b,
γk = (r − 1) a + b + 2,
where r is the rank of (Ω , g ). Therefore, denoted by γ the genus of Ω , by (5) we obtain Entd (Ω , g ) (z ) = max
1 + (k − 1) a + b + (r − k) a/2
γ
1≤k≤r
=
1 + (r − 1) a + b
γ
=
γ −1 , γ
∀ z ∈ Ω.
See also [16] for a similar formula relating the volume entropy with the invariants a, b, and r. The table below summarizes the numerical invariants and the dimension of irreducible bounded symmetric domains according to their type (for a more detailed description of these invariants, see e.g. [17,12]). See Fig. 1.
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R. Mossa / Journal of Geometry and Physics 86 (2014) 492–496
Fig. 1. Numerical invariants of irreducible bounded symmetric domains.
2. Proofs of Theorems 1, 2 and Corollary 2 2.1. Proof of Theorem 1 Assume that λ g is balanced, namely that the ε -function is constant. Consider the Kähler potential λ φ (z ) of λ g, let λ φ (z , w) be its analytic continuation and let Kλ φ (z , w) be the reproducing kernel for the weighted Hilbert space Hλ φ . We have
ελ g (z , w) = e−λ φ(z , w) Kλ φ (z , w) = C > 0,
(6)
so Kλ φ (z , w) never vanish. Hence, fixed a point z0 , the function
ψ (z , w) =
Kλ φ (z , w) Kλ φ (z0 , z 0 ) Kλ φ (z , z 0 ) Kλ φ (z0 , w)
(7)
is well defined. Note that ψ (z0 , w) = ψ (z , z 0 ) = 1 and that
Dz0 (z ) =
1
λ
log ψ (z , z )
(8)
is the diastasis centered in z0 associated to g. So the diastasis Dz0 is globally defined and λ Dz0 is a global Kähler potential for λ g. Consider the Hilbert space Hλ Dz0 given by:
ωn |f | Hλ Dz0 = f ∈ Hol (Ω ) : e <∞ . n! Ω Let Kλ Dz0 (z , w) be its reproducing kernel, since the ε -function does not depend on the Kähler potential, by (6) we have
−λ Dz0
2
ελ g (z , w) = e−λ Dz0 (z , w) Kλ Dz0 (z , w) = C
(9)
where Dz0 (z , w) denotes the analytic continuation of Dz0 (z ) = Dz0 (z , z ). In particular C = e−λ Dz0 (z , z 0 ) Kλ Dz0 (z , z 0 ) = Kλ Dz0 (z , z 0 ) ∈ Hλ Dz0 , indeed by (7) and (8) we have Dz0 (z , z 0 ) = λ1 log ψ (z , z 0 ) = 0. It would be worth to say that Kλ Dz0 ∈ Hλ Dz0 by its very N definition (indeed Kλ Dz0 (z , z 0 ) = j=0 sj (z ) sj (z0 ) where {sj } is a basis for Hλ Dz0 ). Thus Hλ Dz0 contains the constant functions and
e−λ Dz0 Ω
ωn < ∞. n!
We conclude, by the very definition of the diastatic entropy, that Entd (Ω , g ) (z0 ) < λ. Moreover, by (8), the diastasis Dz0 is globally defined for any fixed z0 ∈ Ω and [2, Proposition 2.2] says us that Entd (Ω , g ) is constant. As wished. 2.2. Proof of Theorem 2 Recall that a homogeneous bounded domain (Ω , g ) is a bounded domain of Cn with a Kähler metric g such that the group G = Aut(Ω ) ∩ Isom (Ω , g ) acts transitively on it, where Aut (Ω ) denotes the group of biholomorphisms of Ω and ¯ Isom (Ω , g ) the group of isometries of (Ω , g ). It is well-known that such a domain Ω is contractible and that ω = 2i ∂ ∂φ for a globally defined Kähler potential φ . Indeed Ω is pseudoconvex being biholomorphically equivalent to a Siegel domain (see, e.g. [18] for a proof) and so the existence of a global potential follows by a classical result of Hormander (see [19]) asserting that the equation ∂¯ u = f with f ∂¯ -closed form has a global solution on pseudoconvex domains (see also [16,22] and the proof of Theorem 4 in [20], for an explicit construction of the potential φ following the ideas developed in [21]).
R. Mossa / Journal of Geometry and Physics 86 (2014) 492–496
495
We have to prove that if 1 > Entd (Ω , g ) (z ) then g is balanced (the converse follows by Theorem 1). Assume that 1 > Entd (Ω , g ) (z ) i.e.
e−Dz Ω
ωn < ∞, n!
then the Hilbert space HDz contains the constant functions and the associated ε -function is strictly positive. By the homogeneity of (Ω , g ) we conclude that εg has to be a positive constant (see the proof of [7, Lemma 1]), that is g is balanced. This concludes the proof of Theorem 2. 2.3. Proof of Corollary 2 In order to prove Corollary 2 we need the following result: Lemma 4 ([7, Theorem 2] and [7, (12)]). Let (Ω , λ g ) be a homogeneous bounded domain and let pk , bk , qk and γk be the constants defined by (10) and (11) in the Appendix. Then the metric λ g is balanced if and only if
λ>
1 + pk + bk + qk /2
γk
,
1 ≤ k ≤ r.
By the very definition of diastatic entropy, we have that Entd (Ω , λ g ) =
Entd (Ω , g )
λ
,
combining this with Theorem 2 we obtain that Entd (Ω , g ) < λ if and only if the metric λ g is balanced. Therefore Entd (Ω , g ) = min {λ > 0 | λ g is balanced} ,
∀ z ∈ Ω.
Together with Lemma 4, this concludes the proof of Corollary 2. Acknowledgment The author would like to thank Professor Andrea Loi for his interest in author’s work, his comments and various stimulating discussions. Appendix In this section we give the definition of the constants pk , bk , qk and γk associated to the Piatetski-Shapiro root structure (see [23] for details). Let (Ω ⊂ Cn , g ) be a homogeneous bounded domain, and set G = Aut(Ω ) ∩ Isom (Ω , g ), where Aut (Ω ) (resp. Isom (Ω , g )) denotes the group of invertible holomorphic maps (resp. g-isometries) of Ω . By [21, Theorem 2 (c)], there exists a connected split solvable Lie subgroup S ⊂ G acting simply transitively on the domain Ω . Fix a point p0 ∈ Ω , we have a diffeomorphism ι : S → Ω with ι (h) = h · p0 . By differentiation, we get a linear isomorphism between the Lie ∼
algebra s of S and the tangent space Tp0 M, given by X → X · p0 . Then the evaluation of the Kähler form ω on s is given by
ω (X · p0 , Y · p0 ) = β ([X , Y ])
X, Y ∈ s
with a certain linear form β ∈ s . Let j : s → s be the linear map defined in such a way that ∗
( j X ) · p0 =
√ −1 ( X · p 0 )
for X ∈ s.
We have g (X · p0 , Y · p0 ) = β ([jX , Y ]) for X , Y ∈ s, so the right-hand side defines a positive inner product on s. Let a be the orthogonal complement of [s, s] in s with respect to this inner product. Then a is a commutative Cartan subalgebra of s. Given a linear form α on the Cartan algebra a, we denote by sα the subspace
sα = {X ∈ s : [C , X ] = α (C ) X , ∀ C ∈ a} of s. We say that α is a root if sα ̸= {0} and α ̸= 0. Thanks to [23, Chapter 2, Section 3] or [24, Theorem 4.3], there exists a basis {α1 , . . . , αr }, (r := dim a) of a∗ such that every root is one of the following:
αk , αk /2,
(k = 1, . . . , r ),
(αl ± αk )/2,
1 ≤ k < l ≤ r.
For k = 1, . . . , r we define (see [24, Definition 4.7] and [15, (2.7)]) pk :=
dim s(αk −αi )/2 ,
i
qk :=
dim s(αl −αk )/2 ,
l >k
γk := 4 β ([jAk , Ak ]) , where {A1 , . . . , Ar } is the basis of a dual to {α1 , . . . , αr }.
bk :=
1 2
dim sαk /2 ,
(10) (11)
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R. Mossa / Journal of Geometry and Physics 86 (2014) 492–496
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
E. Calabi, Isometric imbeddings of complex manifolds, Ann. of Math. 58 (1953) 1–23. R. Mossa, Upper and lower bounds for the first eigenvalue and the volume entropy of noncompact Kähler manifold, 2012. arXiv:1211.2705 [math.DG]. A. Loi, R. Mossa, The diastatic exponential of a symmetric space, Math. Z. 268 (3-4) (2011) 1057–1068. S.K. Donaldson, Scalar curvature and projective embeddings. I, J. Differential Geom. 59 (3) (2001) 479–522. C. Arezzo, A. Loi, Moment maps, scalar curvature and quantization of Kähler manifolds, Comm. Math. Phys. 246 (2004) 543–549. C. Arezzo, A. Loi, Quantization of Kähler manifolds and the asymptotic expansion of Tian–Yau–Zelditch, J. Geom. Phys. 47 (2003) 87–99. A. Loi, R. Mossa, Berezin quantization of homogeneous bounded domains, Geom. Dedicata 161 (2012) 119–128. J. Rawnsley, Coherent states and Kählermanifolds, Quart. J. Math. Oxford (2) (28) (1977) 403–415. M. Cahen, S. Gutt, J.H. Rawnsley, Quantization of Kähler manifolds I: geometric interpretation of Berezin’s quantization, J. Geom. Phys. 7 (1990) 45–62. G.R. Kempf, Metrics on invertible shaves on abelian varieties, in: Topics in Algebraic Geometry (Guanajuato, 1989), in: Aportationes Mat. Notas Investigacion, vol. 5, Soc. Mat. Mexicana, Mexico, 1992, pp. 107–108. S. Ji, Inequality for distortion function of invertible shaves on Abelian varieties, Duke Math. J. 58 (1989) 657–667. S. Zhang, Heights and reductions of semi-stable varieties, Comp. Math. 104 (1996) 77–105. A. Loi, R. Mossa, Uniqueness of balanced metrics on holomorphic vector bundles, J. Geom. Phys. 61 (1) (2011) 312–316. S.G. Gindikin, Analysis in homogeneous domains, Russian Math. Surveys 19 (1964) 1–89. T. Nomura, Berezin transforms and Laplace-Beltrami operators on homogeneous Siegel domains, J. Lie Theory 11 (2001) 185–206. R. Mossa, The volume entropy of local Hermitian symmetric space of noncompact type, Differential Geom. Appl. 31 (5) (2013) 594–601. J. Arazy, A survey of invariant hilbert spaces of analytic functions on bounded symmetric domains, Contemp. Math. 185 (1995). É.B. Vinberg, S.G. Gindikin, I.I. Piatetski-Shapiro, Classification and canonical realization of complex homogeneous bounded domains, Trans. Moscow Math. Soc. 12 (1963) 359–388. S.-C. Chen, M.-C. Shaw, Partial Differential Equations in Several Complex Variables, in: AMS/IP Studies in Advanced Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2001, International Press, Boston, MA, xii+380 pp. A.J. Di Scala, H. Ishi, A. Loi, Kähler immersions of homogeneous Kähler manifolds into complex space forms, Asian J. Math. 16 (3) (2012) 479–487. J. Dorfmeister, Simply transitive groups and Kähler structures on homogeneous Siegel domains, Trans. Amer. Math. Soc. 288 (1985) 293–305. R. Mossa, A bounded homogeneous domain and a projective manifold are not relatives, Riv. Mat. Univ. Parma 4 (1) (2013) 55–59. I.I. Piatetski-Shapiro, Automorphic Functions and The Geometry of Classical Domains, Gordon and Breach, New York, 1969. H. Rossi, M. Vergne, Representations of certain solvable Lie groups on Hilbert spaces of holomorphic functions and the application to the holomorphic discrete series of a semisimple Lie group, J. Funct. Anal. 13 (1973) 324–389.
Contents lists available at ScienceDirect
Journal of Geometry and Physics journal homepage: www.elsevier.com/locate/jgp
A note on diastatic entropy and balanced metrics Roberto Mossa Dipartimento di Matematica e Informatica, Palazzo delle Scienze, via Ospedale 72 - 09124, Cagliari, Italy
article
abstract
info
We give an upper bound Entd (Ω , g ) < λ of the diastatic entropy Entd (Ω , g ) (defined by the author in Mossa (2012) of a complex bounded domain (Ω , g ) in terms of the balanced condition (in Donaldson terminology) of the Kähler metric λ g. When (Ω , g ) is a homogeneous bounded domain we show that the converse holds true, namely if Entd (Ω , g ) < 1 then g is balanced. Moreover, we explicitly compute Entd (Ω , g ) in terms of Piatetski-Shapiro constants. © 2014 Elsevier B.V. All rights reserved.
Article history: Received 15 March 2013 Received in revised form 30 September 2014 Accepted 6 October 2014 Available online 16 October 2014 Keywords: Kählerian manifolds Homogeneous domains Diastatic entropy Balanced metrics
1. Introduction and statements of the main results Let (M , g ) be a real analytic Kähler manifold with associated Kähler form ω. Fix a local coordinate system z = (z1 , . . . , zn ) on a neighborhood U of a point p ∈ M and let φ : U → R be a real analytic Kähler potential (i.e ω|U = 2i ∂ ∂¯ φ ). By shrinking U, we can assume that the potential φ (z ) can be analytically continued to U × U. Denote this extension by φ (z , w). Calabi’s diastasis function D : U × U → R (E. Calabi [1]) is defined by
D (z , w) = Dw (z ) = φ (z , z ) + φ (w, w) − φ (z , w) − φ (w, z ) . Assume that ω has global diastasis function Dp : M → R centered at p. The diastatic entropy at p is defined as
Entd (M , g ) (p) = min c > 0 |
M
e−c Dp
ωn <∞ . n!
(1)
This definition does not depend on the point p fixed, provided that for every p ∈ Ω the diastasis Dp is globally defined (see [2, Proposition 2.2]). The concept of diastatic entropy was defined by the author in [2] (following the ideas developed in [3]), where he obtained upper and lower bound for the first eigenvalue of a real analytic Kähler manifold. In this paper we study the link between the diastatic entropy Entd (Ω , g ) and the balanced condition of the metric g. S. Donaldson in [4], in order to obtain a link between the constant scalar curvature condition on a Kähler metric g and the Chow stability of a polarization L, gave the definition of a balanced metric on a compact manifold. Later, this definition has been generalized, by C. Arezzo and A. Loi in [5] (see also [6]), to the noncompact case. In this paper we are interested in complex domains Ω ⊂ Cn (connected open subset of Cn ) endowed with a real analytic Kähler metric g. Let ω be the Kähler form associated to g. Assume that ω admits a global Kähler potential φ : Ω → R. We can define the weighted Hilbert space Hφ of square integrable holomorphic functions on Ω with weight e−φ
Hφ = s ∈ Hol(Ω ) :
e−φ |s|2 Ω
ωn <∞ . n!
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.geomphys.2014.10.004 0393-0440/© 2014 Elsevier B.V. All rights reserved.
(2)
R. Mossa / Journal of Geometry and Physics 86 (2014) 492–496
493
If Hφ ̸= {0} we can pick an orthonormal basis {sj }j=1,...,N ≤∞ ,
Ω
e−φ sj sk
ωn = δjk n!
and define its reproducing kernel by Kφ (z , w) =
N
sj (z ) sj (w).
(3)
j =0
The so called ε -function is defined by
εg (z ) = e−φ(z ) Kφ (z , z¯ ) ,
(4)
as suggested by the notation εg depends only on the metric g and not on the choice of the Kähler potential φ (see, for example, [7, Lemma 1] for a proof). Definition 1. The metric g is balanced if and only if the function εg is a positive constant. In the literature the constancy of εg was studied first by J.H. Rawnsley in [8] under the name of η-function, later renamed as ε -function in [9]. It also appears under the name of distortion function for the study of Abelian varieties by J.R. Kempf [10] and S. Ji [11], and for complex projective varieties by S. Zhang [12]. (See also [13] and references therein.) The following theorem represents the first result of this paper. Theorem 1. Let Ω ⊂ Cn be a complex domain and g a Kähler metric such that λ g is balanced for some λ > 0. Then the diastatic entropy Entd (Ω , g ) is constant and Entd (Ω , g ) < λ. In particular (see Lemma 4) the diastatic entropy of a homogeneous bounded domain is constant. It is natural to ask when the converse of the previous theorem holds true. In the next theorem we show that this is the case when (Ω , g ) is assumed to be homogeneous. Theorem 2. Let (Ω , g ) be a homogeneous bounded domain, then g is balanced if and only if Entd (Ω , g ) < 1. We also compute the diastatic entropy of a homogeneous bounded domain in terms of the constants pk , bk , qk and γk defined by the Piatetski-Shapiro root structure (see (10) and (11) in the Appendix): Corollary 2. Let (Ω , g ) be a homogeneous bounded domain. Then the diastatic entropy of (Ω , g ) is the positive constant given by Entd (Ω , g ) (z ) = max
1 + pk + bk + qk /2
γk
1≤k≤r
,
∀ z ∈ Ω.
Example 3. If g is the Bergman metric on Ω , then γk = 2 + pk + qk + bk , k = 1, . . . , r (see [14, Theorem 5.1] or [15, (2.19)]), so by Corollary 2, Entd (Ω , g ) (z ) = max
1≤k≤r
1 + pk + bk + qk /2 2 + pk + qk + bk
,
∀ z ∈ Ω.
(5)
In particular, when (Ω , g ) is a bounded symmetric domain (i.e. is a homogeneous bounded domain and a symmetric space as Riemannian manifold), there exist integers a and b such that pk = (k − 1) a,
qk = (r − k) a,
bk = b,
γk = (r − 1) a + b + 2,
where r is the rank of (Ω , g ). Therefore, denoted by γ the genus of Ω , by (5) we obtain Entd (Ω , g ) (z ) = max
1 + (k − 1) a + b + (r − k) a/2
γ
1≤k≤r
=
1 + (r − 1) a + b
γ
=
γ −1 , γ
∀ z ∈ Ω.
See also [16] for a similar formula relating the volume entropy with the invariants a, b, and r. The table below summarizes the numerical invariants and the dimension of irreducible bounded symmetric domains according to their type (for a more detailed description of these invariants, see e.g. [17,12]). See Fig. 1.
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R. Mossa / Journal of Geometry and Physics 86 (2014) 492–496
Fig. 1. Numerical invariants of irreducible bounded symmetric domains.
2. Proofs of Theorems 1, 2 and Corollary 2 2.1. Proof of Theorem 1 Assume that λ g is balanced, namely that the ε -function is constant. Consider the Kähler potential λ φ (z ) of λ g, let λ φ (z , w) be its analytic continuation and let Kλ φ (z , w) be the reproducing kernel for the weighted Hilbert space Hλ φ . We have
ελ g (z , w) = e−λ φ(z , w) Kλ φ (z , w) = C > 0,
(6)
so Kλ φ (z , w) never vanish. Hence, fixed a point z0 , the function
ψ (z , w) =
Kλ φ (z , w) Kλ φ (z0 , z 0 ) Kλ φ (z , z 0 ) Kλ φ (z0 , w)
(7)
is well defined. Note that ψ (z0 , w) = ψ (z , z 0 ) = 1 and that
Dz0 (z ) =
1
λ
log ψ (z , z )
(8)
is the diastasis centered in z0 associated to g. So the diastasis Dz0 is globally defined and λ Dz0 is a global Kähler potential for λ g. Consider the Hilbert space Hλ Dz0 given by:
ωn |f | Hλ Dz0 = f ∈ Hol (Ω ) : e <∞ . n! Ω Let Kλ Dz0 (z , w) be its reproducing kernel, since the ε -function does not depend on the Kähler potential, by (6) we have
−λ Dz0
2
ελ g (z , w) = e−λ Dz0 (z , w) Kλ Dz0 (z , w) = C
(9)
where Dz0 (z , w) denotes the analytic continuation of Dz0 (z ) = Dz0 (z , z ). In particular C = e−λ Dz0 (z , z 0 ) Kλ Dz0 (z , z 0 ) = Kλ Dz0 (z , z 0 ) ∈ Hλ Dz0 , indeed by (7) and (8) we have Dz0 (z , z 0 ) = λ1 log ψ (z , z 0 ) = 0. It would be worth to say that Kλ Dz0 ∈ Hλ Dz0 by its very N definition (indeed Kλ Dz0 (z , z 0 ) = j=0 sj (z ) sj (z0 ) where {sj } is a basis for Hλ Dz0 ). Thus Hλ Dz0 contains the constant functions and
e−λ Dz0 Ω
ωn < ∞. n!
We conclude, by the very definition of the diastatic entropy, that Entd (Ω , g ) (z0 ) < λ. Moreover, by (8), the diastasis Dz0 is globally defined for any fixed z0 ∈ Ω and [2, Proposition 2.2] says us that Entd (Ω , g ) is constant. As wished. 2.2. Proof of Theorem 2 Recall that a homogeneous bounded domain (Ω , g ) is a bounded domain of Cn with a Kähler metric g such that the group G = Aut(Ω ) ∩ Isom (Ω , g ) acts transitively on it, where Aut (Ω ) denotes the group of biholomorphisms of Ω and ¯ Isom (Ω , g ) the group of isometries of (Ω , g ). It is well-known that such a domain Ω is contractible and that ω = 2i ∂ ∂φ for a globally defined Kähler potential φ . Indeed Ω is pseudoconvex being biholomorphically equivalent to a Siegel domain (see, e.g. [18] for a proof) and so the existence of a global potential follows by a classical result of Hormander (see [19]) asserting that the equation ∂¯ u = f with f ∂¯ -closed form has a global solution on pseudoconvex domains (see also [16,22] and the proof of Theorem 4 in [20], for an explicit construction of the potential φ following the ideas developed in [21]).
R. Mossa / Journal of Geometry and Physics 86 (2014) 492–496
495
We have to prove that if 1 > Entd (Ω , g ) (z ) then g is balanced (the converse follows by Theorem 1). Assume that 1 > Entd (Ω , g ) (z ) i.e.
e−Dz Ω
ωn < ∞, n!
then the Hilbert space HDz contains the constant functions and the associated ε -function is strictly positive. By the homogeneity of (Ω , g ) we conclude that εg has to be a positive constant (see the proof of [7, Lemma 1]), that is g is balanced. This concludes the proof of Theorem 2. 2.3. Proof of Corollary 2 In order to prove Corollary 2 we need the following result: Lemma 4 ([7, Theorem 2] and [7, (12)]). Let (Ω , λ g ) be a homogeneous bounded domain and let pk , bk , qk and γk be the constants defined by (10) and (11) in the Appendix. Then the metric λ g is balanced if and only if
λ>
1 + pk + bk + qk /2
γk
,
1 ≤ k ≤ r.
By the very definition of diastatic entropy, we have that Entd (Ω , λ g ) =
Entd (Ω , g )
λ
,
combining this with Theorem 2 we obtain that Entd (Ω , g ) < λ if and only if the metric λ g is balanced. Therefore Entd (Ω , g ) = min {λ > 0 | λ g is balanced} ,
∀ z ∈ Ω.
Together with Lemma 4, this concludes the proof of Corollary 2. Acknowledgment The author would like to thank Professor Andrea Loi for his interest in author’s work, his comments and various stimulating discussions. Appendix In this section we give the definition of the constants pk , bk , qk and γk associated to the Piatetski-Shapiro root structure (see [23] for details). Let (Ω ⊂ Cn , g ) be a homogeneous bounded domain, and set G = Aut(Ω ) ∩ Isom (Ω , g ), where Aut (Ω ) (resp. Isom (Ω , g )) denotes the group of invertible holomorphic maps (resp. g-isometries) of Ω . By [21, Theorem 2 (c)], there exists a connected split solvable Lie subgroup S ⊂ G acting simply transitively on the domain Ω . Fix a point p0 ∈ Ω , we have a diffeomorphism ι : S → Ω with ι (h) = h · p0 . By differentiation, we get a linear isomorphism between the Lie ∼
algebra s of S and the tangent space Tp0 M, given by X → X · p0 . Then the evaluation of the Kähler form ω on s is given by
ω (X · p0 , Y · p0 ) = β ([X , Y ])
X, Y ∈ s
with a certain linear form β ∈ s . Let j : s → s be the linear map defined in such a way that ∗
( j X ) · p0 =
√ −1 ( X · p 0 )
for X ∈ s.
We have g (X · p0 , Y · p0 ) = β ([jX , Y ]) for X , Y ∈ s, so the right-hand side defines a positive inner product on s. Let a be the orthogonal complement of [s, s] in s with respect to this inner product. Then a is a commutative Cartan subalgebra of s. Given a linear form α on the Cartan algebra a, we denote by sα the subspace
sα = {X ∈ s : [C , X ] = α (C ) X , ∀ C ∈ a} of s. We say that α is a root if sα ̸= {0} and α ̸= 0. Thanks to [23, Chapter 2, Section 3] or [24, Theorem 4.3], there exists a basis {α1 , . . . , αr }, (r := dim a) of a∗ such that every root is one of the following:
αk , αk /2,
(k = 1, . . . , r ),
(αl ± αk )/2,
1 ≤ k < l ≤ r.
For k = 1, . . . , r we define (see [24, Definition 4.7] and [15, (2.7)]) pk :=
dim s(αk −αi )/2 ,
i
qk :=
dim s(αl −αk )/2 ,
l >k
γk := 4 β ([jAk , Ak ]) , where {A1 , . . . , Ar } is the basis of a dual to {α1 , . . . , αr }.
bk :=
1 2
dim sαk /2 ,
(10) (11)
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
E. Calabi, Isometric imbeddings of complex manifolds, Ann. of Math. 58 (1953) 1–23. R. Mossa, Upper and lower bounds for the first eigenvalue and the volume entropy of noncompact Kähler manifold, 2012. arXiv:1211.2705 [math.DG]. A. Loi, R. Mossa, The diastatic exponential of a symmetric space, Math. Z. 268 (3-4) (2011) 1057–1068. S.K. Donaldson, Scalar curvature and projective embeddings. I, J. Differential Geom. 59 (3) (2001) 479–522. C. Arezzo, A. Loi, Moment maps, scalar curvature and quantization of Kähler manifolds, Comm. Math. Phys. 246 (2004) 543–549. C. Arezzo, A. Loi, Quantization of Kähler manifolds and the asymptotic expansion of Tian–Yau–Zelditch, J. Geom. Phys. 47 (2003) 87–99. A. Loi, R. Mossa, Berezin quantization of homogeneous bounded domains, Geom. Dedicata 161 (2012) 119–128. J. Rawnsley, Coherent states and Kählermanifolds, Quart. J. Math. Oxford (2) (28) (1977) 403–415. M. Cahen, S. Gutt, J.H. Rawnsley, Quantization of Kähler manifolds I: geometric interpretation of Berezin’s quantization, J. Geom. Phys. 7 (1990) 45–62. G.R. Kempf, Metrics on invertible shaves on abelian varieties, in: Topics in Algebraic Geometry (Guanajuato, 1989), in: Aportationes Mat. Notas Investigacion, vol. 5, Soc. Mat. Mexicana, Mexico, 1992, pp. 107–108. S. Ji, Inequality for distortion function of invertible shaves on Abelian varieties, Duke Math. J. 58 (1989) 657–667. S. Zhang, Heights and reductions of semi-stable varieties, Comp. Math. 104 (1996) 77–105. A. Loi, R. Mossa, Uniqueness of balanced metrics on holomorphic vector bundles, J. Geom. Phys. 61 (1) (2011) 312–316. S.G. Gindikin, Analysis in homogeneous domains, Russian Math. Surveys 19 (1964) 1–89. T. Nomura, Berezin transforms and Laplace-Beltrami operators on homogeneous Siegel domains, J. Lie Theory 11 (2001) 185–206. R. Mossa, The volume entropy of local Hermitian symmetric space of noncompact type, Differential Geom. Appl. 31 (5) (2013) 594–601. J. Arazy, A survey of invariant hilbert spaces of analytic functions on bounded symmetric domains, Contemp. Math. 185 (1995). É.B. Vinberg, S.G. Gindikin, I.I. Piatetski-Shapiro, Classification and canonical realization of complex homogeneous bounded domains, Trans. Moscow Math. Soc. 12 (1963) 359–388. S.-C. Chen, M.-C. Shaw, Partial Differential Equations in Several Complex Variables, in: AMS/IP Studies in Advanced Mathematics, vol. 19, American Mathematical Society, Providence, RI, 2001, International Press, Boston, MA, xii+380 pp. A.J. Di Scala, H. Ishi, A. Loi, Kähler immersions of homogeneous Kähler manifolds into complex space forms, Asian J. Math. 16 (3) (2012) 479–487. J. Dorfmeister, Simply transitive groups and Kähler structures on homogeneous Siegel domains, Trans. Amer. Math. Soc. 288 (1985) 293–305. R. Mossa, A bounded homogeneous domain and a projective manifold are not relatives, Riv. Mat. Univ. Parma 4 (1) (2013) 55–59. I.I. Piatetski-Shapiro, Automorphic Functions and The Geometry of Classical Domains, Gordon and Breach, New York, 1969. H. Rossi, M. Vergne, Representations of certain solvable Lie groups on Hilbert spaces of holomorphic functions and the application to the holomorphic discrete series of a semisimple Lie group, J. Funct. Anal. 13 (1973) 324–389.