zbMATH — the first resource for mathematics

Finite dimensionality of the groups of CR-automorphisms of a standard CR- manifold and proper holomorphic mappings of Siegel domains. (Russian) Zbl 0655.32026
Let M and M’ be standard CR-manifolds, defined by non-degenerate vector valued Hermitian forms with linearly independent components. The author shows that each local CR-isomorphism of class \(C^ 1\) between M and M’ is rational. He derives several interesting results from the theorem. In particular, he proves that the group of all CR-automorphisms of M is a finite dimensional Lie group. He also generalizes Alexander’s theorem about proper holomorphic mappings between open balls in \({\mathbb{C}}^ n\) [H. Alexander, Math. Ann. 209, 249-256 (1974; Zbl 0272.32006)]; namely, he shows that if D and D’ are Siegel domains of type II, whose defining Hermitian forms have linearly independent components, then every proper holomorphic mapping f: \(D\to D'\) is biholomorphic and rational.
Reviewer: M.Klimek

32M05 Complex Lie groups, group actions on complex spaces
32H35 Proper holomorphic mappings, finiteness theorems
22E10 General properties and structure of complex Lie groups