  We prove a relation between the $\bar\partial_M$ cohomology of a minimal
orbit $M$ of a real form $G_0$ of a complex semisimple Lie group $G$ in a flag
manifold $G/Q$ and the Dolbeault cohomology of the Matsuki dual open orbit $X$
of the complexification $K$ of a maximal compact subgroup $K_0$ of $G_0$, under
the assumption that $M$ is Levi-flat.
