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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)

##### MSC:
 14H37 Automorphisms of curves 14P20 Nash functions and manifolds 32M05 Complex Lie groups, group actions on complex spaces 14P10 Semialgebraic sets and related spaces
##### Keywords:
biholomorphic automorphisms; affine Nash group
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##### References:
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