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

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 A_n 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 A_n variety or the set of all bounded pseudoconvex complete Reinhardt domains with real analytic boundaries in A_n variety. In particular the moduli space of these domains in A_n variety is constructed explicitly as the image of this complete family of numerical invariants. It is well known that A_n 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.