Convexity of complements of limit sets for holomorphic foliations on surfaces

Let F be a holomorphic foliation on a compact Kähler surface with hyperbolic singularities and no foliation cycle. We prove that if the limit set of F has zero Lebesgue measure, then its complement is a modification of a Stein domain. This applies for the case of suspensions of Kleinian representati...

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Published inMathematische annalen Vol. 388; no. 3; pp. 2727 - 2753
Main Authors Deroin, Bertrand, Dupont, Christophe, Kleptsyn, Victor
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2024
Springer Nature B.V
Springer Verlag
Subjects
Complex Variables
Convexity
Dynamical Systems
Mathematical analysis
Mathematics
Mathematics and Statistics
43A46
37F75
32E10
32U40
32M25
holomorphic foliations
Stein domains
Brownian motion
thin sets
harmonic currents
2020 Mathematics Subject Classification. 32M25
43A46 holomorphic foliations
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Summary:Let F be a holomorphic foliation on a compact Kähler surface with hyperbolic singularities and no foliation cycle. We prove that if the limit set of F has zero Lebesgue measure, then its complement is a modification of a Stein domain. This applies for the case of suspensions of Kleinian representations, answering a question asked by Brunella. The proof consists in building, in several steps, a metric of positive curvature for the normal bundle of F near the limit set. Then we construct a proper strictly plurisubharmonic exhaustion function for the complement of the limit set by extending Brunella’s method to our singular context. The arguments hold more generally when the limit set is thin, a property relying on Brownian motion.
Bibliography:ObjectType-Article-1
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ISSN:0025-5831
1432-1807
DOI:10.1007/s00208-023-02590-1