Higher order bergman functions and explicit construction of moduli space for complete reinhardt domains

Rong Du University of Illinois at Chicago Stephen Yau University of Illinois at Chicago

Differential Geometry mathscidoc:1609.10126

Journal of Differential Geometry, 82, (3), 567-610, 2009
In this article we introduce higher order Bergman functions for bounded complete Reinhardt domains in a variety with possi- bly isolated singularities. These Bergman functions are invariant under biholomorhic maps. We use Bergman functions to deter- mine all the biholomorhic maps between two such domains. As a result, we can construct an infinite family of numerical invari- ants from the Bergman functions for such  domains in An variety (x, y, z) ∈ C3 : xy = zn+1 . These infinite family of numerical invariants are actually a complete set of invariants for either the set of all bounded strictly pseudoconvex complete Reinhardt domain in An variety or the set of all bounded pseudoconvex complete Reinhardt domains with real analytic boundaries in An variety. In particular the moduli space of these domains in An variety is constructed explicitly as the image of this complete family of numerical invariants. It is well known that An variety is the quo- tient of cyclic group of order n + 1 on C2. We prove that the moduli space of bounded complete Reinhardt domains in An vari- ety coincides with the moduli space of the corresponding bounded complete Reinhardt domains in C2. Since our complete family of numerical invariants are computable, we have solved the biholo- morphically equivalent problem for large family of domains in C2.
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@inproceedings{rong2009higher,
  title={HIGHER ORDER BERGMAN FUNCTIONS AND EXPLICIT CONSTRUCTION OF MODULI SPACE FOR COMPLETE REINHARDT DOMAINS},
  author={Rong Du, and Stephen Yau},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911185411258071783},
  booktitle={Journal of Differential Geometry},
  volume={82},
  number={3},
  pages={567-610},
  year={2009},
}
Rong Du, and Stephen Yau. HIGHER ORDER BERGMAN FUNCTIONS AND EXPLICIT CONSTRUCTION OF MODULI SPACE FOR COMPLETE REINHARDT DOMAINS. 2009. Vol. 82. In Journal of Differential Geometry. pp.567-610. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911185411258071783.
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