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On the automorphism groups of algebraic bounded domains. (English) Zbl 0823.14005
Let \(D\) be a bounded domain in \(\mathbb{C}^ n\). By the theorem of H. Cartan, the group \(\operatorname{Aut} (D)\) of all biholomorphic automorphisms of \(D\) has a unique structure of a real Lie group such that the action \(\operatorname{Aut} (D)\times D\to D\) is real analytic. This structure is defined by the embedding \(C_ v: \operatorname{Aut} (D) \hookrightarrow D\times \text{Gl}_ n (\mathbb{C})\), \(f\mapsto (f(v), f_{*v})\), where \(v\in D\) is arbitrary. Here we restrict our attention to the class of domains \(D\) defined by finitely many polynomial inequalities. The appropriate category for studying automorphism of such domains is the Nash category. Therefore we consider the subgroup \(\operatorname{Aut}_ a (D) \subset \operatorname{Aut} (D)\) of all algebraic biholomorphic automorphisms which in many cases coincides with \(\operatorname{Aut} (D)\). Assume that \(n>1\) and \(D\) has a boundary point where the Levi form is non-degenerate. Our main result is that the group \(\operatorname{Aut}_ a (D)\) carries a unique structure of an affine Nash group such that the action \(\operatorname{Aut}_ a (D)\times D\to D\) is Nash. This structure is defined by the embedding \(C_ v: \operatorname{Aut}_ a (D) \hookrightarrow D\times \text{Gl}_ n (\mathbb{C})\) and is independent of the choice of \(v\in D\).
Reviewer: D.Zaitsev (Bochum)

14H37 Automorphisms of curves
14P20 Nash functions and manifolds
32M05 Complex Lie groups, group actions on complex spaces
14P10 Semialgebraic sets and related spaces
Full Text: DOI EuDML arXiv
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