Seminars## Explicit Construction of Moduli Space for Complete Reinhardt Domains via Bergman Functions

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Stephen S-T. Yau

2008-05-26

14:00:00 - 15:00:00

308 , Mathematics Research Center Building (ori. New Math. Bldg.)

Explicit Construction of Moduli Space for Complete Reinhardt Domains via Bergman Functions" We introduce higher order Bergman functions for complete Reinhardt domains in a variety with isolated singularities. These Bergman functions are invariant under biholomorphic maps. We use Bergman functions to determine all the biholomorphic maps between two such domains. We construct an infinite family of numerical invariants from the Bergman functions for such domains in A_n-variety {(x, y, z) in C3 : xy = z^(n+1)}. These infinite family of numerical invariants are actually a complete set of invariants for either the set of all strictly pseudoconvex domains or the set of all pseudoconvex 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. Recall that A_n-variety is the quotient of cyclic group of order n + 1 on C^2. We prove that the moduli space of complete Reinhardt domains in A_n variety coincides with the moduli space of the corresponding complete Reinhardt domains in C^2. Since our complete family of numerical invariants are explicitly computable, we have solved the biholomorphically equivalent problem for large family of domains in C^2.