  We study the truncated microsupport $Ss_k$ of sheaves on a real manifold.
Applying our results to the case of $F=RHom_D(M,O)$, the complex of holomorphic
solutions of a coherent $D$-module $M$, we show that $Ss_k(F)$ is completely
determined by the characteristic variety of $M$. As an application, we obtain
an extension theorem for the sections of $H^j(F)$, $j<d$, defined on an open
subset whose boundary is non characteristic outside of a complex analytic
subvariety of codimension $d$. We also give a characterization of the
perversity for ${\bf C}$-constructible sheaves in terms of their truncated
microsupports.
