Actions of the groups $${\mathbb C}$$ and $${\mathbb C^*}$$ on Stein varieties |
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Authors: | César Camacho Bruno Scárdua |
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Institution: | (1) IMPA-Estrada D. Castorina, 110 Jardim Botanico, Rio de Janeiro, CEP. 22460-320, RJ, Brazil;(2) Instituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, Rio de Janeiro, 21.945-970, RJ, Brazil |
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Abstract: | In this paper we present recent results concerning global aspects of and -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 and 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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Keywords: | Stein manifold Holomorphic flow Quasi-homogeneous singularity Foliation |
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