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 $\Gamma$ of X which are boundaries of analytic sub-varieties of $X \backslash \Gamma$. Our principal result treat the case $X=U\times\omega$ where $U$ is a connected complex manifold and $\omega$ is a disk-convex K\"ahler 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 K\"ahler manifold is embeddable in $\CC^n$ if and only if it has a non constant CR function.