Explicit construction of moduli space of bounded complete Reinhardt domains in ℂⁿ

One of the most fundamental problems in complex geometry is to determine when two bounded domains in Cn are biholomorphically equivalent. Even for complete Reinhardt domains, this fundamental problem remains unsolved completely for many years. Using the Bergmann function theory, we construct an infinite family of numerical invariants from the Bergman functions for complete Reinhardt domains in Cn. These infinite family of numerical invariants are actually a complete set of invariants if the domains are pseudoconvex with C1 boundaries. For bounded complete Reinhardt domains with real analytic boundaries, the complete set of numerical invariants can be reduced dramatically although the set is still infinite. As a consequence, we have constructed the natural moduli spaces for these domains for the first time.