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A group-theoretic characterization of the Fock-Bargmann-Hartogs domains. (English) Zbl 1439.32060
Summary: Let $$M$$ be a connected Stein manifold of dimension $$N$$ and let $$D$$ be a Fock-Bargmann-Hartogs domain in $$\mathbb{C}^N$$. Let $$\text{Aut}(M)$$ and $$\text{Aut}(D)$$ denote the groups of all biholomorphic automorphisms of $$M$$ and $$D$$, respectively, equipped with the compact-open topology. Note that $$\text{Aut}(M)$$ cannot have the structure of a Lie group, in general; while it is known that $$\text{Aut}(D)$$ has the structure of a connected Lie group. In this paper, we show that if the identity component of $$\text{Aut}(M)$$ is isomorphic to $$\text{Aut}(D)$$ as topological groups, then $$M$$ is biholomorphically equivalent to $$D$$. As a consequence of this, we obtain a fundamental result on the topological group structure of $$\text{Aut}(D)$$.

##### MSC:
 32Q02 Special domains (Reinhardt, Hartogs, circular, tube, etc.) in $$\mathbb{C}^n$$ and complex manifolds 32M05 Complex Lie groups, group actions on complex spaces
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