We prove a generalization to the context of real geometry of an intersection formula for the vanishing cycle functor, which in the complex context is due to Dubson, Lê, Ginsburg and Sabbah (after a conjecture of Deligne). It is also a generalization of similar results of Kashiwara and Schapira, where these authors work with a suitable assumption about the micro-support of the corresponding constructible complex of sheaves. We only use a similar assumption about the support of the corresponding characteristic cycle so that our result can be formulated in the language of constructible functions and Lagrangian cycles.

We will clarify the microlocal structure of the vanishing cycle of the solution complexes to $\cal D$-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.

We calculate l-adic nearby cycles in the étale cohomology for families with log smooth reduction using log étale cohomology. In particular, nearby cycles for log smooth families coincide with tame nearby cycles, as L. Illusie expected, and nearby cycles for semistable families depend only on the first infinitesimal neighborhood of the special fiber.