Complex analysis
On weakly complete surfaces
Comptes Rendus. Mathématique, Volume 353 (2015) no. 11, pp. 969-972.

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 (connexe) est dit faiblement complet s'il est muni d'une fonction d'exhaustion (lisse et) pluri-sous-harmonique. Dans cet article, on classifie les surfaces complexes faiblement complètes (c'est-à-dire les variétés complexes de dimension 2 faiblement complètes) qui admettent une fonction d'exhaustion pluri-sous-harmonique et analytique réelle. Elles sont des types suivants : modifications des espaces de Stein, surfaces complexes propres sur des courbes complexes (en général singulières) non compactes, ou bien surfaces complexes de type Grauert, c'est-à-dire feuilletées par des hypersurfaces Levi plates dont les feuilles du feuilletage de Levi sont partout denses. Dans ce dernier cas, on montre aussi que, quitte à passer à un double revêtement, les hypersurfaces Levi plates sont en fait les niveaux d'une fonction pluriharmonique globale.

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DOI: 10.1016/j.crma.2015.08.009
Mongodi, Samuele 1; Slodkowski, Zbigniew 2; Tomassini, Giuseppe 3

1 Dipartimento di Matematica, Università di “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy
2 Department of Mathematics, University of Illinois at Chicago, 851 South Morgan Street, Chicago, IL 60607, USA
3 Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy
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Mongodi, Samuele; Slodkowski, Zbigniew; Tomassini, Giuseppe. On weakly complete surfaces. Comptes Rendus. Mathématique, Volume 353 (2015) no. 11, pp. 969-972. doi : 10.1016/j.crma.2015.08.009. http://www.numdam.org/articles/10.1016/j.crma.2015.08.009/

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