In this paper, we study balanced metrics and Berezin quantization on a class of Hartogs domains defined by \(\varOmega _n=\{(z_1,\ldots ,z_n)\in {\mathbb {C}}^n:\vert z_1\vert<\vert z_2\vert<\cdots<\vert z_n\vert <1\}\) which generalize the so-called classical Hartogs triangle. We introduce a Kähler metric \(g(\nu )\) associated with the Kähler potential \(\varPhi _n(z):=-\sum _{k=1}^{n-1}\nu _k\ln (\vert z_{k+1}\vert ^2-\vert z_k\vert ^2)-\nu _n\ln (1-\vert z_n\vert ^2)\) on \(\varOmega _n\). 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 \(g(\nu )\) on the domain \(\varOmega _n\) 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.

在本文中，我们研究了{\ mathbb {C}} ^ n：\ vert z_1中\\\\ varOmega _n = \ {（z_1，\ ldots，z_n）\定义的一类Hartogs域上的平衡度量和Berezin量化。\ vert <\ vert z_2 \ vert <\ cdots <\ vert z_n \ vert <1 \} \）来概括所谓的经典Hartogs三角形。我们引入与Kähler势\（\ varPhi _n（z）：=-\ sum _ {k = 1} ^ {n-1} \ nu _k \ ln（）相关的Kähler度量\ {g（\ nu）\）\ （\ varOmega _n \）上的\ vert z_ {k + 1} \ vert ^ 2- \ vert z_k \ vert ^ 2）-\ nu _n \ ln（1- \ vert z_n \ vert ^ 2）\）。作为主要的贡献，一方面我们计算用于加权Hilbert空间的Bergman核的明确形式，并且然后，我们获得必要和充分条件度量\（克（\ NU）\）域上的\（\ varOmega _n \）为平衡指标。另一方面，通过使用Calabi的diastasis函数，我们证明了Hartogs三角形接受Berezin量化。