Let D1 and D2 be two irreducible bounded symmetric domains in the complex spaces V1 and V2 respectively. Let E be the Euclidean metric on V2 and h the Bergman metric on V1. The Bloch constant b(D1,D2) is defined to be the supremum of E(f ′(z)x, f ′(z)x) 1 2 /hz(x, x)1/2, taken over all the holomorphic functions f : D1 → D2 and z ∈ D1, and nonzero vectors x ∈ V1. We find the constants for all th...

The Ricci curvature of the Bergman metric on a bounded domain $D\subset \mathbb{C}^n$ is strictly above by $n+1$ and consequently $\log (K_D^{n+1}g_{B,D})$, where $K_D$ kernel for $D$ diagonal $g_{B, D}$ Riemannian volume element $D$, potential K\"ahler known as Kobayashi--Fuks metric. In this note we study localization near holomorphic peak points also show that shares several properties with ...

We study the complete Kähler-Einstein metric of a Hartogs domain Ω̃ built on an irreducible bounded symmetric domain Ω, using a power N of the generic norm of Ω. The generating function of the Kähler-Einstein metric satisfies a complex Monge-Ampère equation with boundary condition. The domain Ω̃ is in general not homogeneous, but it has a subgroup of automorphisms, the orbits of which are paramet...

The study of holomorphic and isometric immersions of a Kähler manifold (M,g) into a Kähler manifold (N,G) started with Calabi. In his famous paper [3] he considered the case when the ambient space (N,G) is a complex space form, i.e. its holomorphic sectional curvature KN is constant. There are three types of complex space forms : flat, hyperbolic or elliptic according as the holomorphic section...

We show that the classical Bernstein polynomials BN(f)(x) on the interval [0, 1] (and their higher dimensional generalizations on the simplex Σm ⊂ R) may be expressed in terms of Bergman kernels for the Fubini-Study metric on CP: BN(f)(x) is obtained by applying the Toeplitz operator f(N−1Dθ) to the Fubini-Study Bergman kernels. The expression generalizes immediately to any toric Kähler variety...

In a previous paper, [1], we have studied a property of subharmonic dependence on a parameter of Bergman kernels for a family of weighted L-spaces of holomorphic functions. Here we prove a result on the curvature of a vector bundle defined by this family of L-spaces itself, which has the earlier results on Bergman kernels as a corollary. Applying the same arguments to spaces of holomorphic sect...

We show that the classical Bernstein polynomials BN (f)(x) on the interval [0, 1] (and their higher dimensional generalizations on the simplex Σm ⊂ R) may be expressed in terms of Bergman kernels for the Fubini-Study metric on CP: BN (f)(x) is obtained by applying the Toeplitz operator f(N−1Dθ) to the Fubini-Study Bergman kernels. The expression generalizes immediately to any toric Kähler varie...

We show that the classical Szasz analytic function SN (f)(x) is related to the Bergman kernel for the Bargmann-Fock space. Then we generalize this relation to any noncompact toric Kähler manifold, defining the generalized Szasz analytic function ShN (f)(x). Then we will prove the complete asymptotic expansion of ShN (f)(x) and its scaling limit property. As examples, we will compute the general...