Roberto Mossa

Berezin quantization of homogeneous bounded domains

We prove that a homogeneous bounded domain admits a Berezin quantization.

The diastatic exponential of a symmetric space

AbstractLet (M, g) be a real analytic Kähler manifold. We say that a smooth map Expp : W → M from a neighbourhood W of the origin of TpM into M is a diastatic exponential at p if it satisfies

Some remarks on homogeneous Kähler manifolds

\begin{array}{lll} &\,\,\left(d{\rm Exp}_p\right)_0 & = {\rm id}_{T_pM},\\ D_p&\left({\rm Exp}_p \left(v\right) \right) & = g_p\left(v, v\right),\,\,\forall v\in W, \end{array}

The log-term of the Bergman kernel of the disc bundle over a homogeneous Hodge manifold

where Dp is Calabi’s diastasis function at p (the usual exponential expp obviously satisfied these equations when Dp is replaced by the square of the geodesics distance from p). In this paper we prove that for every point p of an Hermitian symmetric space of noncompact type M there exists a globally defined diastatic exponential centered in p which is a diffeomorphism and it is uniquely determined by its restriction to polydisks. An analogous result holds true in an open dense neighbourhood of every point of M*, the compact dual of M. We also provide a geometric interpretation of the symplectic duality map (recently introduced in Di Scala and Loi (Adv Math 217:2336–2352, 2008)) in terms of diastatic exponentials. As a byproduct of our analysis we show that the symplectic duality map pulls back the reproducing kernel of M* to the reproducing kernel of M.

Diastatic entropy and rigidity of complex hyperbolic manifolds

Inspired by the work of G. Lu on pseudo symplectic capacities we obtain several results on the Gromov width and the Hofer--Zehnder capacity of Hermitian symmetric spaces of compact type. Our results and proofs extend those obtained by Lu for complex Grassmannians to Hermitian symmetric spaces of compact type. We also compute the Gromov width and the Hofer--Zehnder capacity for Cartan domains and their products.

BALANCED METRICS ON HOMOGENEOUS VECTOR BUNDLES

In this paper we provide a positive answer to a conjecture due to Di Scala et al. (Asian J Math, 2012, Conjecture 1) claiming that a simply-connected homogeneous Kähler manifold M endowed with an integral Kähler form

Minimal symplectic atlases of Hermitian symmetric spaces

\mu _0\omega