Balanced metrics on the Fock-Bargmann-Hartogs domains

Abstract

The Fock-Bargmann-Hartogs domain $D_{n,m}(\mu )$ ($\mu >0$) in ${\mathbb{C}}^{n+m}$ is defined by the inequality $\Vert w{\Vert }^2<e^{-\mu \Vert z{\Vert }^2},$ where $(z,w)\in {\mathbb{C}}^n\times {\mathbb{C}}^m$, which is an unbounded non-hyperbolic domain in ${\mathbb{C}}^{n+m}$. This paper introduces a Kähler metric $\alpha g(\mu ;\nu )$ $(\alpha >0)$ on $D_{n,m}(\mu )$, where $g(\mu ;\nu )$ is the Kähler metric associated with the Kähler potential $\Phi (z,w):=\mu \nu {\Vert z\Vert }^2-\ln (e^{-\mu {\Vert z\Vert }^2}-\Vert w{\Vert }^2)$ ($\nu >-1$) on $D_{n,m}(\mu )$. The purpose of this paper is twofold. Firstly, we obtain an explicit formula for the Bergman kernel of the weighted Hilbert space of square integrable holomorphic functions on $(D_{n,m}(\mu ),g(\mu ;\nu ))$ with the weight $\exp \{-\alpha \Phi \}$ for $\alpha >0$. Secondly, using the explicit expression of the Bergman kernel, we obtain the necessary and sufficient condition for the metric $\alpha g(\mu ;\nu )$ $(\alpha >0)$ on the domain $D_{n,m}(\mu )$ to be a balanced metric. So we obtain the existence of balanced metrics for a class of Fock-Bargmann-Hartogs domains.

10 pages, to appear in Annals of Global Analysis and Geometry

References in corpus (2)

• Berezin quantization of homogeneous bounded domains
• Rigidity of proper holomorphic mappings between certain unbounded non-hyperbolic domains