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 \(\dim G\leq \frac{1}{2}j(j+1)+(n-j)^ 2\).


32M05 Complex Lie groups, group actions on complex spaces
32E10 Stein spaces
Full Text: DOI EuDML


[1] Bedford, E.: Holomorphic mapping of products of annuli in ? n . Pac. J. Math.87, 271-281 (1980) · Zbl 0449.32024
[2] Bedford, E.: Invariant forms on complex manifolds with application to holomorphic mappings. Math. Ann. · Zbl 0532.32015
[3] Bedford, E., Burns, D.: Holomorphic mapping of annuli in ? n and the associated extremal function. Ann. Scuola Norm. Sup. Pisa3, 381-414 (1979) · Zbl 0422.32021
[4] Cartan, H.: Sur les groupes de transformations analytiques. Act. Sci. Ind. 198. Paris: Hermann 1935 · JFM 61.0370.02
[5] Cartan, H.: Sur les fonctions de plusieurs variables complexes. Math. Z.35, 760-773 (1932) · JFM 58.0349.02
[6] Duchamp, T.: Classification of Legendre imbeddings. Preprint
[7] Heath, L., Suffridge, T.: Holomorphic retracts in complexn-space. III. J. Math.35, 125-135 (1981) · Zbl 0463.32008
[8] Kaup, W.: Reelle Transformationsgruppen und invariante Metriken auf komplexen Räumen. Invent. Math.3, 43-70 (1947) · Zbl 0157.13401
[9] Kobayashi, S.: Geometry of bounded domains. Trans. A.M.S.93, 267-290 (1959) · Zbl 0136.07102
[10] Kobayashi, S.: Transformation groups in differential geometry. Berlin, Heidelberg, New York: Springer 1972 · Zbl 0246.53031
[11] Milnor, J., Stasheff, J.: Characteristic classes. Princeton: Princeton University Press 1974 · Zbl 0298.57008
[12] Mok, N.: Rigidity of holomorphic self-mappings and the automorphism groups of hyperbolic Stein spaces (to appear) · Zbl 0574.32021
[13] Narashimhan, R.: Several complex variables. Chicago: University of Chicago Press 1971
[14] Stout, E.L., Zame, W.: A Stein manifold topologically but not holomorphically equivalent to a domain in ? n . Preprint · Zbl 0589.32026
[15] Whitney, H.: Complex analytic varieties. Reading: Addison-Wesley 1972 · Zbl 0265.32008
[16] Wu, H.: Normal families of holomorphic mappings. Acta Math.119, 193-233 (1967) · Zbl 0158.33301
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.