Actions of the groups and on Stein varieties
Geometriae Dedicata(2009)
Abstract
In this paper we present recent results concerning global aspects of $${\mathbb C}$$ and $${\mathbb C^*}$$ -actions on Stein surfaces. Our approach is based on a byproduct of techniques from Geometric Theory of Foliations (holonomy,
stability), Potential theory (parabolic Riemann surfaces, Riemann-Koebe Uniformization theorem) and Several Complex Variables
(Hartogs’ extension theorems, Theory of Stein spaces). Our main motivation comes from the original works of M. Suzuki and
Orlik-Wagreich. Some of their results are extended to a more general framework. In particular, we prove some linearization
theorems for holomorphic actions of $${\mathbb C}$$ and $${\mathbb C^*}$$ on normal Stein analytic spaces of dimension two. We also add a list of questions and open problems in the subject. The underlying
idea is to present the state of the art of this research field.
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Key words
Stein manifold,Holomorphic flow,Quasi-homogeneous singularity,Foliation,32E10,32S65,37F75,32M25
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