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.
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.
Reviewer: M.Colţoiu (Bucureşti)
MSC:
| 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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