Czechoslovak Mathematical Journal, Vol. 73, No. 2, pp. 633-647, 2023
Remarks on the balanced metric on Hartogs triangles with integral exponent
Qiannan Zhang, Huan Yang
Received May 16, 2022. Published online February 14, 2023.
Abstract: In this paper we study the balanced metrics on some Hartogs triangles of exponent $\gamma\in\mathbb{Z}^+$, i.e., $\Omega_n(\gamma)= \{z=(z_1,\dots,z_n)\in\mathbb{C}^n |z_1|^{1/\gamma}< |z_2| <\dots < |z_n| < 1 \}$ equipped with a natural Kähler form $\omega_{g(\mu)} := \frac12({\rm i}/\pi)\partial \overline{\partial}\Phi_n$ with $\Phi_n(z)=-\mu_1{\ln(|z_2|^{2\gamma}- |z_1 |^2)}-\sum_{i=2}^{n-1} {\mu_i{\ln(|z_{i+1}|^2-|z_i|^2)}}-\mu_n{\ln(1-{|z_n|^2})},$
where $\mu=(\mu_1,\cdots,\mu_n)$, $\mu_i > 0$, depending on $n$ parameters. The purpose of this paper is threefold. First, we compute the explicit expression for the weighted Bergman kernel function for $(\Omega_n(\gamma),g(\mu))$ and we prove that $g(\mu)$ is balanced if and only if $\mu_1 > 1$ and $\gamma\mu_1$ is an integer, $\mu_i$ are integers such that $\mu_i\geq2$ for all $i=2,\ldots,n-1$, and $\mu_n > 1$. Second, we prove that $g(\mu)$ is Kähler-Einstein if and only if $\mu_1=\mu_2=\cdots=\mu_n=2\lambda$, where $\lambda$ is a nonzero constant. Finally, we show that if $g(\mu)$ is balanced then $(\Omega_n(\gamma),g(\mu))$ admits a Berezin-Engliš quantization.
Affiliations: Qiannan Zhang, School of Mathematics and Statistics, Qingdao University, Qingdao, Shandong 266071, P. R. China, e-mail: 2020020246@qdu.edu.cn; Huan Yang (corresponding author), College of Economic and Management, Qingdao University of Science and Technology, Qingdao, Shandong 266061, P. R. China, e-mail: huanyang@whu.edu.cn
