Geometriae Dedicata

, Volume 139, Issue 1, pp 5–14 | Cite as

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

  • César Camacho
  • Bruno Scárdua
Original Paper


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.


Stein manifold Holomorphic flow Quasi-homogeneous singularity Foliation 

Mathematics Subject Classification (2000)

32E10 32S65 37F75 32M25 


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Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.IMPA-Estrada D. CastorinaRio de JaneiroBrazil
  2. 2.Instituto de MatemáticaUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil

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