## 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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