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Remarks on the balanced metric on Hartogs triangles with integral exponent

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Abstract

In this paper we study the balanced metrics on some Hartogs triangles of exponent γ ∈ ℤ+, i.e. equipped with a natural Kähler form with where μ = (μ1, …, μn), μi > 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 (Ωn(γ),g(μ)) and we prove that g(μ) is balanced if and only if μ1 > 1 and γμ1 is an integer, μi are integers such that μi ≽ 2 for all i = 2, …, n − 1, and μn > 1. Second, we prove that g(μ) is Kähler-Einstein if and only if μ1 = μ2 = … = μn = 2λ, where λ is a nonzero constant. Finally, we show that if g(μ) is balanced then (Ωn(γ),g(μ)) admits a Berezin-Engliš quantization.

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Acknowledgments

We sincerely thank the referees, who have read the paper very carefully and made many useful suggestions.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Qingdao University, Qingdao, Shandong, 266071, China

    Qiannan Zhang

  2. College of Economic and Management, Qingdao University of Science and Technology, Qingdao, Shandong, 266061, China

    Huan Yang

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  1. Qiannan Zhang
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  2. Huan Yang
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Correspondence to Huan Yang.

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Zhang, Q., Yang, H. Remarks on the balanced metric on Hartogs triangles with integral exponent. Czech Math J 73, 633–647 (2023). https://doi.org/10.21136/CMJ.2023.0208-22

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  • DOI: https://doi.org/10.21136/CMJ.2023.0208-22

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