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On the automorphism group of a Stein manifold. (English) Zbl 0532.32014

Let \(\Omega\) be a Stein manifold such that \(\Omega\subset \subset {\tilde \Omega}\) for some other Stein manifold \({\tilde \Omega}\). This paper considers properties of Au\(t(\Omega)\) in terms of \(H_ n(\Omega,{\mathbb{R}})\) from two points of view. The first point of view is that a ”nontrivial” self-mapping of \(\Omega\) is an automorphism. A typical theorem is as follows: Under some hypotheses (e.g. \(\partial \Omega\) is strongly pseudoconvex) a self-mapping f:\(\Omega\to \Omega\) such that the induced mapping \(f_*:H_*(\Omega)\to H_*(\Omega)\) is nonzero, must be an automorphism.
The second point of view is that if \(H_ j(\Omega,{\mathbb{R}})\neq 0\) for some 1\(\leq j\leq n\), then the dimension of Au\(t(\Omega)\) is restricted. Classically, it is known that \(\dim Aut(\Omega)\leq n(n+2)\). Here this is extended to the following: if G is a compact Lie group acting on a Stein manifold \(\Omega\) with \(H_ j(\Omega,{\mathbb{R}})\neq 0\), then \(\dim G\leq \frac{1}{2}j(j+1)+(n-j)^ 2\).

MSC:

32M05 Complex Lie groups, group actions on complex spaces
32E10 Stein spaces
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References:

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