Standardization of certain compact group actions and the automorphism group of the complex Euclidean space

Abstract

For a connected compact Lie group of rank n, we study its effective action on an n-dimensional complex manifold as a holomorphic transformation group. We give a result on the standardization of such an action, called the generalized standardization theorem, by using the theory of Reinhardt domains and a result of Barrett, Bedford and Dadok. The generalized standardization theorem has various applications. As one of them, we apply it to the study of the structure of the holomorphic automorphism group Aut(C n ) of C n , and prove two theorems on subgroups of Aut(C n ). One is a fact on the conjugacy of compact subgroups of Aut(C n ) whose rank is n, and the other is about the nonexistence of a certain noncompact subgroup of Aut(C n ).