Remarks on the balanced metric on Hartogs triangles with integral exponent

Zhang, Qiannan, and Yang, Huan. "Remarks on the balanced metric on Hartogs triangles with integral exponent." Czechoslovak Mathematical Journal 73.2 (2023): 633-647. <http://eudml.org/doc/299418>.

@article{Zhang2023,
abstract = {In this paper we study the balanced metrics on some Hartogs triangles of exponent $\gamma \in \mathbb \{Z\}^\{+\}$, i.e., \[ \Omega \_n(\gamma )= \lbrace z=(z\_1,\dots ,z\_n)\in \mathbb \{C\}^n\colon |z\_1|^\{1/\gamma \}<|z\_2|<\dots <|z\_n|<1 \rbrace \] equipped with a natural Kähler form $\omega _\{g(\mu )\} := \frac\{1\}\{2\}(i /\pi )\partial \overline\{\partial \}\Phi _n$ 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 $\mu _i\ge 2$ 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.},
author = {Zhang, Qiannan, Yang, Huan},
journal = {Czechoslovak Mathematical Journal},
keywords = {balanced metric; Kähler-Einstein metric; Berezin-Engliš quantization},
language = {eng},
number = {2},
pages = {633-647},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Remarks on the balanced metric on Hartogs triangles with integral exponent},
url = {http://eudml.org/doc/299418},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Zhang, Qiannan
AU - Yang, Huan
TI - Remarks on the balanced metric on Hartogs triangles with integral exponent
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 2
SP - 633
EP - 647
AB - In this paper we study the balanced metrics on some Hartogs triangles of exponent $\gamma \in \mathbb {Z}^{+}$, i.e., \[ \Omega _n(\gamma )= \lbrace z=(z_1,\dots ,z_n)\in \mathbb {C}^n\colon |z_1|^{1/\gamma }<|z_2|<\dots <|z_n|<1 \rbrace \] equipped with a natural Kähler form $\omega _{g(\mu )} := \frac{1}{2}(i /\pi )\partial \overline{\partial }\Phi _n$ 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 $\mu _i\ge 2$ 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.
LA - eng
KW - balanced metric; Kähler-Einstein metric; Berezin-Engliš quantization
UR - http://eudml.org/doc/299418
ER -