Symplectic geometry of Cartan-Hartogs domains

This paper studies the geometry of Cartan-Hartogs domains from the symplectic point of view. Inspired by duality between compact and noncompact Hermitian symmetric spaces, we construct a dual counterpart of Cartan-Hartogs domains and give explicit expression of global Darboux coordinates for both Cartan-Hartogs and their dual. Further, we compute their symplectic capacity and show that a Cartan-Hartogs admits a symplectic duality if and only if it reduces to be a complex hyperbolic space.

C n+1 , given by: M Ω,µ := {(z, w) ∈ Ω × C | |w| 2 < N µ Ω (z, z)}, (1.1) where Ω ⊂ C n is a bounded symmetric domain, N µ Ω (z, z) is its generic norm and µ > 0 is a positive real parameter. We endow M Ω,µ with its Kobayashi Kähler form ω Ω,µ whose global Kähler potential reads: (1.2) ϕ Ω,µ (z, w;z,w) = − log(N µ Ω (z, z) − |w| 2 ). Our first result answer positively to Q 1 when X is a Cartan-Hartogs domain: The global coordinates we exhibit generalize those of the hermitian symmetric space of noncompact type Ω the Cartan-Hartogs is based on. Indeed, if we restrict the map Ψ Ω,µ to the base Ω, we obtain the symplectic coordinates for Ω constructed by A. Di Scala and A. Loi in [6] (see also [28]). Further, we prove that as well as Di Scala and Loi's map, Ψ Ω,µ well-behaves with respect to the action of the automorphism group of Ω and enjoys a nice hereditary property (see Remarks 1 and 2).
We construct (see Lemma 4 below) a symplectic dual of Cartan-Hartogs domains, that is, M * Ω,µ = C n+1 equipped with the dual Kähler form ω * Ω,µ , that we show to be strictly plurisubharmonic on C n+1 . In a natural way, the dual counterpart of Ψ Ω,µ defines global Darboux coordinates for M * Ω,µ , and we have the following: We prove also that Φ Ω,µ is compatible with the action of the automorphism group of Ω and enjoys of an hereditary property analogously to Ψ Ω,µ (see Remark 5).
As third result we compute the symplectic capacity of (M Ω,µ , ω 0 ) and M * Ω,µ , ω * Ω,µ , answering for these domains to Q 3: Theorem 3. Let c be a symplectic capacity. Then for a Cartan-Hartogs domain M Ω,µ equipped with the flat form ω 0 , and for its dual M Ω,µ endowed with the dual form ω * Ω,µ , one has: The proof of the first part of Theorem 3 is based on the results in [27], on the symplectic capacity of Hermitian symmetric spaces of noncompact type. To prove the second part we apply Theorem 2.
Unfortunately, it can be proven that Ψ Ω,µ of Theorem 1 is not a symplectic duality unless the Cartan-Hartogs reduces to be a complex hyperbolic space, i.e. when Ω = CH n and µ = 1. With Theorem 4 below we show that this is not a peculiarity of our map, giving a negative answer to Q 4 that characterizes the complex hyperbolic space among Cartan-Hartogs domains: Theorem 4. There exists a symplectic duality between a Cartan-Hartogs domain (M Ω,µ , ω Ω,µ ) and its dual C n+1 , ω * Ω,µ if and only if (M Ω,µ , ω Ω,µ ) = CH n+1 , ω hyp . This is equivalent to Ω,µ , and in this case Ψ Ω,µ realize a symplectic duality.
The proof is obtained when µ < 1 as direct consequence of Theorem 3 while for µ ≥ 1 it is a consequence of a volume comparison.
The paper is organized as follows. In the next section we describe the geometry of Cartan-Hartogs domains, proving Theorem 1. Sections 3 is devoted to the construction of dual Cartan-Hartogs and the proof of Theorem 2. Finally in Section 4 we prove Theorem 3 and Theorem 4.
The authors are grateful to Prof. Andrea Loi for his interest in their work and for the useful comments.

Cartan-Hartogs domains and the proof of Theorem 1
Throughout this section we use the Jordan triple system theory, referring the reader to [6,23,27,28,29,33,34] for details and further applications.
2.1. Definition and geometric properties. Consider an Hermitian symmetric space of noncompact type (from now on HSSNT) Ω and the associated Hermitian positive Jordan triple systems (from now on HPJTS) (V, {, , }) (see e.g. [6, Section 2.2]). Recall that there is a natural identification between V equipped with the flat form ω 0 := i 2 ∂∂m 1 (x, x), where m 1 is the generic trace of V , and C n equipped with the standard flat form ω 0 = n j=1 dz j ∧ dz j . By mean of this identification, from now on we will always consider Ω as a bounded symmetric domain of C n in its (unique up to linear isomorphism) circled realization, which is usually called a Cartan domain when Ω is irreducible. Analogously, we will consider the Bergman operator B Ω as operator on C n , and its generic norm N Ω (z, z) as a polynomial of C n (see e.g. [6, Section 2.1]).

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Define: , for a, b the numerical invariants of Ω, r its rank and s ∈ R + . In [38, Prop. 2.1] W. Yin, K. Lu and G. Roos prove that: where Θ is the induced volume form on Fürstenberg-Satake boundary F of Ω (see e.g. [5, (1.28)] for its definiton). Thus: i.e. the volume of M Ω,µ with respect to the flat form induced by C n+1 is given by: The following is a key example for our analysis: Example 1. (Hartogs-polydisc) Let ∆ n be the n-dimensional polydisc: the generic norm is given by (Hua [19]): It follows that its hyperbolic metric reads: and that the associated Cartan-Hartogs domain, which we call Hartogs-polydisc, is given by: whose Kobayashi metric is ω ∆ n ,µ = i 2 ∂∂ϕ ∆ n ,µ , with:

Holomorphic isometries between Cartan-Hartogs domains.
A totally geodesic complex immersion f : Ω ′ , ω ′ hyp → (Ω, ω hyp ) between two HSSNCT equipped with their hyperbolic metrics, preserve the triple products {, , } ′ and {, , } of the associated HPJTS V ′ and V (see e.g. [6, Proposition 2.1]), i.e.: Hence, also the generic norm is preserved, that is, N µ Ω (f (z), f (z)) = N µ Ω ′ (z, z). Thus, the natural lift is a holomorphic isometric embedding with respect to the Kobayashi metrics defined by (2.1), i.e.: Thus we get: Then, by the argument above, the holomorphic isometric action of K on Ω induces in a natural way an holomorphic isometric action of K on M Ω , by: Moreover as a consequence of Propositon 1 and of the Polydisc Theorem for HSSNCT (see [15]), a Cartan-Hartogs domain can be realized as a union of Kähler embedded Hartogs-Polydiscs M ∆ r ,µ : where r is the rank of Ω and ∆ r ⊂ Ω is a r-dimensional complex polydisc totally geodesically embedded in Ω. 6 2.3. Proof of Theorem 1. Let M Ω,µ be an (n + 1)-dimensional Cartan-Hartogs domain and (C n , {, , } Ω ) the HJPTS associated to Ω. Define the map Ψ Ω,µ : M Ω,µ → C n+1 by: where B Ω and N Ω are respectively the Bergman operator and the generic norm associated to {, , } Ω . In order to prove Theorem 1 we will show that Ψ Ω,µ satisfies the following properties: Let us start with the following two lemmata.
Lemma 2. Let F : (Ω, ω hyp ) → (C n , w 0 ) be a holomorphic map satisfying F * ω 0 = ω hyp , and: Then: Proof. We start by proving (2.17). Observe first that from Then, expanding the right hand side of (2.17) we get: At this point, (2.17) is satisfied if and only if: where we used that ∂N µ = 2N µ/2 ∂N µ/2 . This last equivalence can be rewritten as: that is: which holds true once (2.16) does. Finally, the following computation proves (2.18): Now we can proceed with the proof of (A). In [6, Theorem 1.1], A. Loi and A. Di Scala show that the holomorphic map F : (Ω, ω hyp ) → (C n , w 0 ) defined by: is a global symplectomorphism. Thus, by Lemma 1 and Lemma 2, in order to prove (A) we need only to check that such F satisfies (2.16). Denote by D(x, y) the operator on (V, {, , }) defined by D(x, y)z = {x, y, z} and denote by z = j λ j c j the spectral decomposition of z. We have (see [6, (28)]) that: where we used the identity: and the fact that z z is self-adjoint with respect to the Hermitian metric m 1 . Using [6, (34)] we get: and thus: as wished.

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Thus: Setting: and: where V is the column vector with k-th entry z k 1+|z k | 2 , the Hessian H reads: and since A+B is positive definite (being the sum of a positive definite matrix A and a semipositive one B), H is positive definite iff its determinant is. A long but straightforward computation gives: det(H) = µ n n h=1 (1 + |z h | 2 ) µ(n+1)−2 ( n h=1 (1 + |z h | 2 ) µ + |w| 2 ) n+2 , and we are done.

Remark 4.
Observe that it turns out (see [6,Subsec. 2.4]) that the Hermitian symmetric space of compact type (Ω * , ω F S ) dual to (Ω, ω hyp ) is a compactification of C n , ω * hyp . Further, (z, w) ∈ M * Ω,µ | z = 0 is totally geodesic in M * Ω,µ and has CP 1 equipped with the Fubini-Study metric as compactification, therefore M * Ω,µ is not complete for any µ. The authors believe that C n+1 , ω * Ω,µ admits a completion only when M Ω,µ is itself a Hermitian symmetric space of noncompact type, which actually happens only when it reduces to be the (n + 1)-dimensional complex hyperbolic space, i.e. when µ = 1 and rank(Ω) = 1.
Proof. Observe first that since (see e.g. [21]): det(µω * hyp ) = µ n (N * ) −γ , a long but straightforward computation gives: Thus, using the polar coordinates of HSSNT (see e.g. ), we get: where γ = b + 2 + (r − 1) a 2 is the genus of Ω, F (s) is given in (2.3), and last equality follows by: where we performed in turn the change of variables λ 2 j = t j and t j = s j /(1 − s j ), and the last equality follows by (3.2).

3.2.
Holomorphic isometries between dual Cartan-Hartogs domains. Consider a totally geodesic complex immersion f : Ω ′ → Ω between HSSNCT. Identify Ω ′ with its image f (Ω ′ ) ⊂ Ω and observe that f trivially extends to an injective morphism f : V ′ → V of the associated HPJTS V ′ and V (see [6,Prop. 2.2]). Hence, the map: satisfies: (3.5) Let us identify V ∼ = C n and V ′ ∼ = C m , as in the beginning of this section, we just proved the following result: Let Ω be an HSSNCT, then any totally geodesic complex immersion f : Ω ′ → Ω extends to the Kähler embedding f : to the corresponding duals Cartan-Hartogs domains, given by (3.4).
As in the proof of Theorem 1, we start with the following two lemmata, where to shorten the notation we set N µ Ω * (z, z) := N µ Ω (z, −z). where h := (h 1 , . . . , h n ) satisfies: and Then Proof. The proof is totally similar to that of Lemma 1.
In [6, Theorem 1.1], A. Loi and A. Di Scala show that G : C n , i 2 ∂∂ log N µ Ω * → (C n , w 0 ) defined by: is a global symplectomorphism. Thus, by Lemma 6 and Lemma 7, in order to prove (A ′ ) we need only to check that such G satisfies (3.11). Also here the proof is very similar to that of (A), once substituting F with G and D(z, z) with − D(z, −z).
If the claim holds, since the left hand side of (4.7) is strictly decreasing in µ while the right hand side is strictly increasing, we can see that the only positive solution to (4.7) must lie in (0, 1), concluding the proof.