Truncated microsupport and holomorphic solutions of D-modules

Abstract

We study the truncated microsupport $S{s}_k$ of sheaves on a real manifold. Applying our results to the case of $F=RHo{m}_D(M,O)$, the complex of holomorphic solutions of a coherent $D$-module $M$, we show that $S{s}_k(F)$ 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 $\mathbf{C}$-constructible sheaves in terms of their truncated microsupports.

22 pages