New classes of domains with explicit Bergman kernel

Abstract

We introduce two classes of egg type domains, built on general bounded symmetric domains, for which we obtain the Bergman kernel in explicit formulas.1991 Mathematics Subject Classification. 32H10.

A Cartan-Hartogs version of the Polydisk Theorem

“…where Ω is a bounded symmetric domain not necessarily irreducible and N Ω (z, z) is its generic norm. Observe that originally [30] the domain Ω the Cartan-Hartogs is based on is a Cartan domain, i.e. an irreducible bounded symmetric domain.…”

“…In all the other cases it is a nonhomogeneous domain that inherits symmetric peculiarities from the symmetric bounded domain it based on. For this reason Cartan-Hartogs domains represent an important class of domains in C n , and since their first apparence in [30] they have been studied from different points of view, see e.g. [2,6,7,12,13,25,27,31,32].…”

On the Szegö kernel of Cartan–Hartogs domains

“…It is worth pointing out that in [13] it is shown that the logterm of the Szegö kernel of M d 0 Ω (µ) ⊂ C d+d 0 vanishes (in the sense of our Definition 3) when the Szegö kernel is obtained using the standard volume form of C d+d 0 restricted to ∂M d 0 Ω (µ) instead of the volume form dν = α ∧ (dα) d used in this paper. The reader is referred also to [24] for the proof of the vanishing of the log-term of the Bergman kernel.…”

Zeros of Bergman kernels on some Hartogs domains

“…From [12], we know the two weighted Bergman kernel on the right hand side are polynomial with respect to k. So by standard summation in the Bergman kernel of U , one gets the result that K U (ξ, Z, W ) is the product of some power of φ by a polynomial in…”