Probleme Plateau complexe dans les varietes Kahleriennes

Abstract

The ``complex Plateau problem'' (or boundary problem) in a complexe manifold X is the problem of characterizing the real submanifolds $Γ$ of X which are boundaries of analytic sub-varieties of $X \backslash Γ$. Our principal result treat the case $X=U\timesω$ where $U$ is a connected complex manifold and $ω$ is a disk-convex Kähler manifold. As a consequence, we obtain results of Harvey-Lawson, Dolbeault-Henkin and Dinh. We give generalizations of Hartogs-Levi and Hartogs-Bochner theorems. Finally, we prove that a strictly pseudo-convex CR structure embeddable in a compact Kähler manifold is embeddable in $\CC^n$ if and only if it has a non constant CR function.