Por Teresa Monteiro Fernandes (Departamento de Matemática | Ciências ULisboa).

Let X be a complex manifold. In this talk we first present an overview of the construction of Kashiwara’s Riemann-Hilbert functor as a quasi-inverse to the solution functor for regular holonomic modules. Then we will explain how this construction can be adapted to a relative (smooth) framework, where instead of differential operators on X we deal with relative differential operators associated to a projection p: X x S --->S, being S a complex manifold. This summarizes recent joint work with Claude Sabbah and Luisa Fiorot, where we treat the general case for the dimensions of X and S, completing a program started by myself with Claude Sabbah around 2010.

By that time, we aimed to study the case of modules underlying a mixed twistor D-module so that we only needed to assume that S was a complex curve.

Recent colaborations with Luisa Fiorot allowed to treat general regular holonomic D-modules and general dimensions. We will illustrate these rather technical constructions with simple examples.