Article
| Title: | Remarks on the balanced metric on Hartogs triangles with integral exponent (English) |
| Author: | Zhang, Qiannan |
| Author: | Yang, Huan |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 73 |
| Issue: | 2 |
| Year: | 2023 |
| Pages: | 633-647 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | In this paper we study the balanced metrics on some Hartogs triangles of exponent $\gamma \in \mathbb {Z}^{+}$, i.e., $$ \Omega _n(\gamma )= \{z=(z_1,\dots ,z_n)\in \mathbb {C}^n\colon |z_1|^{1/\gamma }<|z_2|<\dots <|z_n|<1 \} $$ equipped with a natural Kähler form $\omega _{g(\mu )} := \frac 12(i /\pi )\partial \overline {\partial }\Phi _n$ 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 $\mu _i\geq 2$ 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. (English) |
| Keyword: | balanced metric |
| Keyword: | Kähler-Einstein metric |
| Keyword: | Berezin-Engliš quantization |
| MSC: | 32A25 |
| MSC: | 32Q15 |
| MSC: | 53C55 |
| idZBL: | Zbl 07729529 |
| idMR: | MR4586916 |
| DOI: | 10.21136/CMJ.2023.0208-22 |
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| Date available: | 2023-05-04T17:52:50Z |
| Last updated: | 2023-09-13 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151679 |
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