Balanced metrics on the Fock–Bargmann–Hartogs domains

Abstract

The Fock-Bargmann-Hartogs domain D n,m (µ) (µ > 0) in C n+m is defined by the inequality w 2 < e −µ z 2 , where (z, w) ∈ C n × C m , which is an unbounded non-hyperbolic domain in C n+m . This paper introduces a Kähler metric αgThe 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 (µ), g(µ; ν)) with the weight exp{−αΦ} for α > 0. Secondly, using the explicit expression of the Bergman kernel, we obtain the necessary and sufficient condition for the metric αg(µ; ν) (α > 0) on the domain D n,m (µ) to be a balanced metric. So we obtain the existence of balanced metrics for a class of Fock-Bargmann-Hartogs domains.

The regular quantizations of certain holomorphic bundles

“…For the case of non-compact manifolds, Loi-Mossa in [28] have shown that a bounded homogeneous domain admits a regular quantization. For the nonhomogeneous setting, we give the existence of regular quantizations on some Hartogs domains in [6] and [18].…”

Strongly not relatives Kähler manifolds

“…Finally, we describe what we need about the 1-parameter family of Fock-Bargmann-Hartogs domains, referring the reader to [1] and reference therein for details and further results. For any value of µ > 0, a Fock-Bargmann-Hartogs domain D n,m (µ) is a strongly pseudoconvex, nonhomogeneous unbounded domains in C n+m with smooth real-analytic boundary, given by:…”

“…In [1], E. Bi, Z. Feng and Z. Tu prove that when n = 1 and ν = − 1 m+1 , the metric ω(µ; ν) is infinite projectively induced whenever it is rescaled by a big enough constant. More precisely they prove the following: Recall that a balanced Kähler metric is a particular projectively induced metric such that the immersion map is defined by a orthonormal basis of a weighted Hilbert space (see e.g.…”

Regularity of the Weighted Bergman Projection on the Fock–bargmann–hartogs Domain

“…11671306). c 2020 Australian Mathematical Publishing Association Inc. [2] L p regularity of the weighted Bergman projection 283…”

“…Thus, A 2 (Ω, η) is a subspace of holomorphic functions in L 2 (Ω, η). From [18], if η is continuous and never vanishes inside Ω, then A p (Ω, η) is a closed subspace of L 2 (Ω, η) and there is an orthogonal projection, called the weighted Bergman projection,…”