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