On automorphism groups of homogeneous complex manifolds

Abstract

The reproducing property of the Bergman kernel function [1 ] plays an essential role in our proof. As we consider not only bounded domains in Cn, but also general complex manifolds, we replace the kernel function by the kernel form (see [2 ] for the definition and properties of the kernel form). If AT is a domain in Cn, then F can be identified with the space F* of square integrable holomorphic functions (with respect to the Euclidean measure of Cn). For our purpose, it is, however, desirable to use F even for a domain in Cn, because every holomorphic transformation of M induces, in a natural way, a unitary transformation of F. We recall that, by a square integrable holomorphic n-form f, we mean