Truncated microsupport and holomorphic solutions of D-modules

Abstract

We study the truncated microsupport SS k of sheaves on a real manifold. Applying our results to the case of F=R Hom 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 C -constructible sheaves in terms of their truncated microsupports.