A group-theoretic characterization of the space obtained by omitting the coordinate hyperplanes from the complex Euclidean space

TLDR: In this paper, it was shown that the holomorphic automorphism group Aut(C × (C) ) of a complex manifold has the structure of a Lie group and can be viewed as a topological group.

Abstract: In the study of the holomorphic automorphism group Aut( ) of a complex manifold , it seems to be natural to direct our attention not only t o the abstract group structure of Aut( ) but also to its topological group structu re equipped with the compact-open topology. In fact, a well-known theorem of H. C artan says that the topological group of the holomorphic automorphisms of a bounded omain in C has the structure of a Lie group, and this result enables us to make va rious kinds of detailed studies of bounded domains in C . On the other hand, in contrast to the case of bounded domains, the holomorphic automorphism group Aut (C × (C) ) of the unbounded domainC ×(C∗) is terribly big when +≥ 2, and cannot have the structure of a Lie group. But, by looking at topological subgroups of Au t(C × (C) ) with Lie group structures, we can find a lead to apply the Lie group theo ry t the investigation of the problems related to the structure of Aut( C × (C) ). In the present paper, we try to approach from this standpoint to the fundamental pr oblem of what complex manifold has the holomorphic automorphism group isomorphi c to Aut(C × (C) ) as topological groups. Namely, we prove the following result w i h the aid of the theory of Reinhardt domains developed in Shimizu [8], [9] (cf. Kruz hilin [6]).