What question did this study set out to answer?

The aim is to characterize the space obtained by omitting coordinate hyperplanes in complex Euclidean space, particularly using holomorphic automorphism groups.

April 1, 2026Open Access

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

Key Points

The aim is to characterize the space obtained by omitting coordinate hyperplanes in complex Euclidean space, particularly using holomorphic automorphism groups.
Proved non-isomorphism of holomorphic automorphism groups for specified spaces.
Established conditions for biholomorphic equivalence between manifolds.
Utilized properties of connected complex manifolds.
Holomorphic automorphism groups of Ck × (C*)n-k and (Ck - {0}) × (C*)n-k are not isomorphic.
If holomorphic automorphism groups are isomorphic, manifolds are biholomorphically equivalent.
Clear characterization of spaces where these conditions hold.

Abstract

In this paper, we prove that the holomorphic automorphism groups of the spaces Ck × (C*) n-k and (Ck - 0) × (C*) n-k are not isomorphic as topological groups. By making use of this fact, we establish the following characterization of the space Ck × (C*) n-k: Let M be a connected complex manifold of dimension n that is holomorphically separable and admits a smooth envelope of holomorphy. Assume that the holomorphic automorphism group of M is isomorphic to the holomorphic automorphism group of Ck × (C*) n-k as topological groups. Then M itself is biholomorphically equivalent to Ck × (C*) n-k. This was first proved by us in 5 under the stronger assumption that M is a Stein manifold. 全文公開200907

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