Actions of the groups $${\mathbb C}$$ and $${\mathbb C^*}$$ on Stein varieties

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.