1. Convexity of complements of limit sets for holomorphic foliations on surfaces
- Author
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Bertrand Deroin, Christophe Dupont, Victor Kleptsyn, CY Cergy Paris Université (CY), Université de Rennes (UR), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-Institut Agro Rennes Angers, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)
- Subjects
37F75 ,32M25 ,32U40 ,32E10 ,43A46 ,thin sets ,Mathematics - Complex Variables ,General Mathematics ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,holomorphic foliations ,harmonic currents ,Stein domains ,2020 Mathematics Subject Classification. 32M25 ,[MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV] ,Dynamical Systems (math.DS) ,FOS: Mathematics ,43A46 holomorphic foliations ,Mathematics - Dynamical Systems ,Complex Variables (math.CV) ,Brownian motion ,Mathematics::Symplectic Geometry - Abstract
Let $\mathcal F$ be a holomorphic foliation on a compact K\'ahler surface with hyperbolic singularities and no foliation cycle. We prove that if the limit set of $\mathcal F$ has zero Lebesgue measure, then its complement is a modification of a Stein domain. The proof consists in building, in several steps, a metric of positive curvature for the normal bundle of $\mathcal F$ near the limit set. Then we construct a proper strictly plurisubharmonic exhaustion function for the complement of the limit set, by adapting Brunella's ideas to our singular context. The arguments hold more generally when the limit set is thin, a property relying on Brownian motion.
- Published
- 2023
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