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Rigidity of holomorphic self-mappings and the automorphism groups of hyperbolic Stein spaces. (English) Zbl 0574.32021

Suppose X is an n-dimensional Stein space having the homotopy type of a CW-complex with real dimension exactly n. The author seeks conditions such that an injective holomorphic self-map \(f: X\to X\) which induces an automorphism on \(H_ n(X,{\mathbb{Z}})\) will be an automorphism of X. For example, if X is a manifold such that the bounded holomorphic functions give a local embedding near each point, then summing over a countable family yields a bounded strictly plurisubharmonic function on X. Also assuming the existence of a bounded weakly plurisubharmonic exhaustion function and that \(H^ n(X,{\mathbb{C}})\) is finite dimensional, the author uses the \(L^ 2\)-estimates for \({\bar \partial}\) with weights, due to A. Andreotti and E. Vesentini [Publ. Math., Inst. Haut. Étud. Sci. 25, 81-130 (1965; Zbl 0138.066)] and L. Hörmander [Acta Math. 113, 89-152 (1965; Zbl 0158.110)], to show that every de Rham class in \(H^ n(X,{\mathbb{C}})\) can be represented by square-integrable holomorphic n-forms. This along with a modification of the iteration technique of H. Cartan yields \(f\in Aut(X)\). He also shows \(f\in Aut(X)\) assuming X is complete with respect to its Caratheodory pseudometric.
The proof for spaces is more delicate (without assuming a bound on the embedding dimension). By embedding one obtains holomorphic n-forms on a relatively compact subset and by solving a suitable \({\bar \partial}\)- problem one can extend to \(L^ 2\) holomorphic n-forms on Reg(X). That Stokes’ theorem holds for \(L^ 2\) holomorphic n-forms on Reg(X) when the cycles intersect Sing(X) is proved using the normal decomposition of semi-analytic sets due to S. Lojasiewicz [Ann. Sc. Norm Super. Pisa, Sci. Fis. Mat., III. Ser. 18(1964), 449-474 (1965; Zbl 0128.171)]. In the complete case another Runge approximation argument is needed.
As well the author proves a rigidity theorem for weakly pseudoconvex domains with smooth boundary without the injectivity assumption on the map.
Reviewer: B.Gilligan

MSC:

32E10 Stein spaces
32M05 Complex Lie groups, group actions on complex spaces
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
55P15 Classification of homotopy type
32U05 Plurisubharmonic functions and generalizations
32H99 Holomorphic mappings and correspondences
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References:

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