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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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