  Let $D\subset\subset\mathbb{C}^n$ be a complex manifold of dimension $p\geq
2$ with $\C^2$ boundary in $\mathbb{C}^n$. Let $f$ be a $\C^1$ function on $bD$
and $V$ a generic and large enough family of complex $(n-p+1)$-planes. Let
suppose that for $\nu\in V$, no connected component of $bD\cap
\mathbb{C}^{n-p+1}_\nu$ is "almost" real analytic and that $f$ extends
holomorphically in $D\cap\mathbb{C}^{n-p+1}_\nu$. Then $f$ extend as a
holomorphic function in $D$. In a special case, this result gives a partial
answer to a conjecture of Globevnik-Stout. By generalizing the theorem of
Harvey-Lawson, we prove a Morera type theorem for the boundary problem in
$\mathbb{C}^n$ which answer to a problem asked by Dolbeault and Henkin.

Complex Variables