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Remarks on the balanced metric on Hartogs triangles with integral exponent
Czechoslovak Mathematical Journal, Tome 73 (2023) no. 2, pp. 633-647
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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.
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.
DOI : 10.21136/CMJ.2023.0208-22
Classification : 32A25, 32Q15, 53C55
Keywords: balanced metric; Kähler-Einstein metric; Berezin-Engliš quantization 
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     title = {Remarks on the balanced metric on {Hartogs} triangles with integral exponent},
     journal = {Czechoslovak Mathematical Journal},
     pages = {633--647},
     year = {2023},
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Zhang, Qiannan; Yang, Huan. Remarks on the balanced metric on Hartogs triangles with integral exponent. Czechoslovak Mathematical Journal, Tome 73 (2023) no. 2, pp. 633-647. doi: 10.21136/CMJ.2023.0208-22

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