Microlocal Vanishing Cycles and Ramified Cauchy Problems in the Nilsson Class

We will clarify the microlocal structure of the vanishing cycle of the solution complexes to D-modules. In particular, we find that the object introduced by D'Agnolo and Schapira is a kind of the direct product (with a monodromy structure) of the sheaf of holomorphic microfunctions. By this result, a totally new proof (that does not involve the use of the theory of microlocal inverse image) of the theorem of D'Agnolo and Schapira will be given. We also give an application to the ramified Cauchy problems with growth conditions, i.e., the problems in the Nilsson class functions of Deligne.