Generically strongly $q$-convex complex manifolds

[Variétés complexes génériquement fortement $q$-convexes]

Napier, Terrence ; Ramachandran, Mohan

Annales de l'Institut Fourier, Tome 51 (2001), p. 1553-1598 / Harvested from Numdam

Résumé

On suppose que $\varphi $ est une fonction analytique-réelle plurisousharmonique sur une variété complexe connexe et non-compacte $X$. Le résultat principal démontre que si l’ensemble analytique-réel des points où $\varphi $ n’est pas fortement $q$-convexe est de dimension $2q+1$ ou moins, alors presque tous les sous-niveaux assez grands de $\varphi $ sont des variétés complexes fortement $q$-convexes. Pour $X$ de dimension 2, c’est un cas spécial d’un théorème de Diederich et Ohsawa. Nous obtenons aussi une version de ce résultat dans le cas où $\varphi $ est analytique réelle avec coins.

Abstract

Suppose $\varphi $ is a real analytic plurisubharmonic exhaustion function on a connected noncompact complex manifold $X$. The main result is that if the real analytic set of points at which $\varphi $ is not strongly $q$-convex is of dimension at most $2q+1$, then almost every sufficiently large sublevel of $\varphi $ is strongly $q$-convex as a complex manifold. For $X$ of dimension $2$, this is a special case of a theorem of Diederich and Ohsawa. A version for $\varphi $ real analytic with corners is also obtained.

Détail

Publié le : 2001-01-01

MR 1870640 | Zbl 0996.32004

DOI : https://doi.org/10.5802/aif.1866
Classification:  32E40,  32F10
Mots clés: cycles analytiques, convexe holomorphiquement, $q$ complet

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