Balanced metrics and Berezin quantization on Hartogs triangles

Abstract

In this paper, we study balanced metrics and Berezin quantization on a class of Hartogs domains defined by $$\varOmega _n=\{(z_1,\ldots ,z_n)\in {\mathbb {C}}^n:\vert z_1\vert<\vert z_2\vert<\cdots<\vert z_n\vert <1\}$$ Ω n = { ( z 1 , … , z n ) ∈ C n : | z 1 | < | z 2 | < ⋯ < | z n | < 1 } which generalize the so-called classical Hartogs triangle. We introduce a Kähler metric $$g(\nu )$$ g ( ν ) associated with the Kähler potential $$\varPhi _n(z):=-\sum _{k=1}^{n-1}\nu _k\ln (\vert z_{k+1}\vert ^2-\vert z_k\vert ^2)-\nu _n\ln (1-\vert z_n\vert ^2)$$ Φ n ( z ) : = - ∑ k = 1 n - 1 ν k ln ( | z k + 1 | 2 - | z k | 2 ) - ν n ln ( 1 - | z n | 2 ) on $$\varOmega _n$$ Ω n . As main contributions, on one hand we compute the explicit form for Bergman kernel of weighted Hilbert space, and then, we obtain the necessary and sufficient condition for the metric $$g(\nu )$$ g ( ν ) on the domain $$\varOmega _n$$ Ω n to be a balanced metric. On the other hand, by using the Calabi’s diastasis function, we prove that the Hartogs triangles admit a Berezin quantization.

Canonical metrics on generalized Hartogs triangles

“…In 2017, Zapałowski [17] characterized the existence of proper holomorphic mappings between generalized Hartogs triangles and gave their explicit forms. Later on, Bi-Su [2] explored the geometric properties of a special Hartogs triangle (that is Ω 1, m ). There are also some other beautiful works on Hartogs triangles, the reader is referred to [4,8,16] and references therein.…”

“…Recently, Bi-Su [2] proved that there exist a balanced metric on Ω 1, m . Inspired by this, we want to solve the existence of balanced metric on generalized Hartogs domain Ω k, m for general k. In fact, we also obtain the following result.…”