Communications in Analysis and Geometry

Volume 18 (2010)

Number 3

Explicit construction of moduli space of bounded complete Reinhardt domains in $\mathbb{C}^n$

Pages: 601 – 626

DOI: http://dx.doi.org/10.4310/CAG.2010.v18.n3.a7

Authors

Rong Du (Department of Mathematics, East China Normal University, Shanghai, China)

Yun Gao (Department of Mathematics, Shanghai Jiao Tong University, Shanghai, China)

Stephen Yau (S. T.)

Abstract

One of the most fundamental problems in complex geometry isto determine when two bounded domains in $\mathbb{C}^n$ arebiholomorphically equivalent. Even for complete Reinhardtdomains, this fundamental problem remains unsolvedcompletely for many years. Using the Bergmann functiontheory, we construct an infinite family of numericalinvariants from the Bergman functions for completeReinhardt domains in $\mathbb{C}^n$. These infinite familyof numerical invariants are actually a complete set ofinvariants if the domains are pseudoconvex with $C^1$boundaries. For bounded complete Reinhardt domains withreal analytic boundaries, the complete set of numericalinvariants can be reduced dramatically although the set isstill infinite. As a consequence, we have constructed thenatural moduli spaces for these domains for the first time.

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