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Addendum to our characterization of the unit polydisc. (English) Zbl 1206.32010

In a previous paper [Mich. Math. J. 56, No. 1, 173–181 (2008; Zbl 1171.32011)], the authors proved that, given an \(n\)-dimensional manifold \(M\), which is holomorphically separable and with smooth envelope of holomorphy (this occurs for instance when \(M\) is Stein and connected or is a domain in \(\mathbb C^n\)), if \(\operatorname{Aut}(M)\) is isomorphic (as topological group) to the automorphism group \(\operatorname{Aut}(\Delta^n)\) of the polydisc \(\Delta^n\), then \(M\) is biholomorphic to \(\Delta^n\).
In this paper, they show that the same claim is true under the weaker hypothesis that \(\operatorname{Aut}(M)\) contains a topological subgroup isomorphic to the identity component of \(\operatorname{Aut}(\Delta^n)\).
They also prove that, if \(M\) is a connected complex manifold of dimension \(n\) and \(\operatorname{Aut}(M)\) contains a subgroup \(G\), which is isomorphic (as topological group) to the identity component of \(\operatorname{Aut}(D)\) for some bounded symmetric domain \(D\) in \(\mathbb C^n\) and such that all isotropy subgroups are compact, then \(M\) is biholomorphic to the bounded symmetric domain \(D\).
This generalizes a previous result of A. V. Isaev [J. Geom. Anal. 18, No. 3, 795–799 (2008; Zbl 1146.32009)], originally proved for the case \(D = \Delta^n\).

MSC:

32M05 Complex Lie groups, group actions on complex spaces
32Q28 Stein manifolds
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
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