Napier, Terrence; Ramachandran, Mohan
Generically strongly q-convex complex manifolds  [ Variétés complexes génériquement fortement q-convexes ]
Annales de l'institut Fourier, Tome 51 (2001) no. 6 , p. 1553-1598
Zbl 0996.32004 | MR 1870640
doi : 10.5802/aif.1866
URL stable : http://www.numdam.org/item?id=AIF_2001__51_6_1553_0

Classification:  32E40,  32F10
Mots clés: cycles analytiques, convexe holomorphiquement, q complet
On suppose que ϕ 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 ϕ n’est pas fortement q-convexe est de dimension 2q+1 ou moins, alors presque tous les sous-niveaux assez grands de ϕ 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 ϕ est analytique réelle avec coins.
Suppose ϕ 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 ϕ is not strongly q-convex is of dimension at most 2q+1, then almost every sufficiently large sublevel of ϕ 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 ϕ real analytic with corners is also obtained.

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