A weakly complete space is a (connected) complex space endowed with a (smooth) plurisubharmonic exhaustion function. In this paper, we classify the weakly complete surfaces (i.e. weakly complete manifolds of dimension 2) for which such exhaustion function can be chosen to be real analytic: they can be modifications of Stein spaces or proper (i.e. endowed with a proper surjective holomorphic map onto) a non-compact (possibly singular) complex curve or surfaces of Grauert type i.e. foliated with real analytic Levi flat hypersurfaces whose Levi foliation has dense complex leaves. In the last case, we also show that such Levi flat hypersurfaces are in fact level sets of a global proper pluriharmonic function, up to passing to a holomorphic double covering.

On weakly complete surfaces [Sur les surfaces faiblement complètes]

MONGODI, SAMUELE;
2015

Abstract

A weakly complete space is a (connected) complex space endowed with a (smooth) plurisubharmonic exhaustion function. In this paper, we classify the weakly complete surfaces (i.e. weakly complete manifolds of dimension 2) for which such exhaustion function can be chosen to be real analytic: they can be modifications of Stein spaces or proper (i.e. endowed with a proper surjective holomorphic map onto) a non-compact (possibly singular) complex curve or surfaces of Grauert type i.e. foliated with real analytic Levi flat hypersurfaces whose Levi foliation has dense complex leaves. In the last case, we also show that such Levi flat hypersurfaces are in fact level sets of a global proper pluriharmonic function, up to passing to a holomorphic double covering.
Un espace complexe est dit faiblement complet s'il est muni d'une fonction d'exhaustion plurisousharmonique. Dans ce papier on classie les surfaces complexes faiblement complètes qui admettent une fonction d'exhaustion plurisousharmonique et analytique réelle. Elles sont des types suivants : modications des espaces de Stein, surfaces complexes propres sur des courbes complexes non compactes, ou bien surfaces complexes de type Grauert i.e. feuilletées par des hypersurfaces Levi plates dont les feuilles du feuilletage de Levi sont partout denses. Dans ce dernier cas on montre aussi que, sauf à passer à un double revêtement, les hypersurfaces Levi plates sont en fait les niveaux d'une fonction pluriharmonique globale.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11311/1010754
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