HIGHER ORDER BERGMAN FUNCTIONS ANDEXPLICIT CONSTRUCTION OF MODULI SPACE FORCOMPLETE REINHARDT DOMAINS

摘要

In this article we introduce higher order Bergman functionsfor bounded complete Reinhardt domains in a variety with possi-bly isolated singularities. These Bergman functions are invariantunder biholomorhic maps. We use Bergman functions to deter-mine all the biholomorhic maps between two such domains. Asa result, we can construct an infinite family of numerical invari-ants from the Bergman functions for such domains in An variety{(x, y, z) ∈C~3:xyz~(n+1)}.These infinite family of numericalinvariants are actually a complete set of invariants for either the setof all bounded strictly pseudoconvex complete Reinhardt domainin An variety or the set of all bounded pseudoconvex completeReinhardt domains with real analytic boundaries in A_n variety.In particular the moduli space of these domains in A_nvarietyis constructed explicitly as the image of this complete family ofnumerical invariants. It is well known that An variety is the quo-tient of cyclic group of order n + 1 on C~2.We prove that themoduli space of bounded complete Reinhardt domains in An vari-ety coincides with the moduli space of the corresponding boundedcomplete Reinhardt domains in C2. Since our complete family ofnumerical invariants are computable, we have solved the biholo-morphically equivalent problem for large family of domains in C~2.