Balanced metrics on some Hartogs type domains over bounded symmetric domains

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• 作者：Zhiming Feng ; Zhenhan Tu
• 关键词：Balanced metrics ; Bergman kernels ; Bounded symmetric domains ; Cartan–Hartogs domains ; K?hler metrics ; 32A25 ; 32M15 ; 32Q15
• 刊名：Annals of Global Analysis and Geometry
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：47
• 期：4
• 页码：305-333
• 全文大小：380 KB
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• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Statistics for Business, Economics, Mathematical Finance and Insurance
  Mathematical and Computational Physics
  Group Theory and Generalizations
  Geometry
  
• 出版者：Springer Netherlands
• ISSN：1572-9060

文摘

The definition of balanced metrics was originally given by Donaldson in the case of a compact polarized K?hler manifold in 2001, who also established the existence of such metrics on any compact projective K?hler manifold with constant scalar curvature. Currently, the only noncompact manifolds on which balanced metrics are known to exist are homogeneous domains. The generalized Cartan–Hartogs domain \((\prod _{j=1}^k\Omega _j)^{{\mathbb {B}}^{d_0}}(\mu )\) is defined as the Hartogs type domain constructed over the product \(\prod _{j=1}^k\Omega _j\) of irreducible bounded symmetric domains \(\Omega _j\) \((1\le j \le k)\) , with the fiber over each point \((z_1,\ldots ,z_k)\in \prod _{j=1}^k\Omega _j\) being a ball in \(\mathbb {C}^{d_0}\) of the radius \(\prod _{j=1}^kN_{\Omega _j}(z_j,\overline{z_j})^{\frac{\mu _j}{2}}\) of the product of positive powers of their generic norms. Any such domain \((\prod _{j=1}^k\Omega _j)^{{\mathbb {B}}^{d_0}}(\mu )\) \((k\ge 2)\) is a bounded nonhomogeneous domain. The purpose of this paper was to obtain necessary and sufficient conditions for the metric \(\alpha g(\mu )\) \((\alpha >0)\) on the domain \((\prod _{j=1}^k\Omega _j)^{{\mathbb {B}}^{d_0}}(\mu )\) to be a balanced metric, where \(g(\mu )\) is its canonical metric. As the main contribution of this paper, we obtain the existence of balanced metrics for a class of such bounded nonhomogeneous domains.