Geometric invariant theory on Stein spaces. (English) Zbl 0728.32010

The aim of this paper is to present results on actions of compact Lie groups on Stein spaces. The main result is the following:
Complexification Theorem. Let K be a compact Lie group and \(K^{{\mathbb{C}}}\) a complexification of K. If K acts on a reduced Stein space X, then there exists a complex space \(X^{{\mathbb{C}}}\) with a holomorphic action \(K^{{\mathbb{C}}}\times X^{{\mathbb{C}}}\to X^{{\mathbb{C}}}\) and a K-equivariant holomorphic map i: \(X\to X^{{\mathbb{C}}}\) with the following properties:
(i) i: \(X\to X^{{\mathbb{C}}}\) is an open embedding and i(X) is a Runge subset of \(X^{{\mathbb{C}}}\) such that \(K^{{\mathbb{C}}}\cdot i(X)=X^{{\mathbb{C}}}.\)
(ii) \(X^{{\mathbb{C}}}\) is a Stein space.
(iii) If \(\Phi\) is a K-equivariant holomorphic map from X into a complex space Y on which \(K^{{\mathbb{C}}}\) acts holomorphically, then there exists a unique \(K^{{\mathbb{C}}}\)-equivariant holomorphic map \(\Phi^{{\mathbb{C}}}: X^{{\mathbb{C}}}\to Y\) such that the diagram commutes.


32E10 Stein spaces
32M05 Complex Lie groups, group actions on complex spaces
22E10 General properties and structure of complex Lie groups
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