Operations with regular holonomic D-modules with support a normal crossing

The aim of this work is to describe some operations in the category of regular holonomic \cD-modules with support a normal crossing and variation zero introduced in [J.Alvarez Montaner, R.Garcia Lopez and S.Zarzuela, "Local cohomology, arrangements of subspaces and monomial ideals ", Adv. in Math. 174 (2003), 35--56]. These operations will allow us to compute the characteristic cycle of the local cohomology supported on homogeneous prime ideals of these modules. In particular, we will be able to describe their Bass and dual Bass numbers

Operations with regular holonomic D-modules with support a normal crossing

AbstractThe aim of this work is to describe some operations in the category of regular holonomicD-modules with support a normal crossing and variation zero introduced in [Àlvarez Montaner, J., García López, R., Zarzuela, S., 2003. Local cohomology, arrangements of subspaces and monomial ideals. Adv. Math. 174 (1), 35–56]. These operations will allow us to compute the characteristic cycle of the local cohomology supported on homogeneous prime ideals of these modules. In particular, we will be able to describe their Bass and dual Bass numbers

Regular and irregular holonomic D-modules

This is a survey paper based on a series of lectures given at the IHES in February/March 2015. In a first part, we recall the main results on the tempered holomorphic solutions of D-modules in the language of indsheaves and, as an application, the Riemann-Hilbert correspondence for regular holonomic modules. In a second part, we present the enhanced version of the first part, treating along the same lines the irregular holonomic case.Comment: 114 page

Riemann-Hilbert correspondence for holonomic D-modules

The classical Riemann-Hilbert correspondence establishes an equivalence between the triangulated category of regular holonomic D-modules and that of constructible sheaves. In this paper, we prove a Riemann-Hilbert correspondence for holonomic D-modules which are not necessarily regular. The construction of our target category is based on the theory of ind-sheaves by Kashiwara-Schapira and influenced by Tamarkin's work. Among the main ingredients of our proof is the description of the structure of flat meromorphic connections due to Mochizuki and Kedlaya.Comment: 114pages; v.2 minor changes, 114 p

Irregular holonomic kernels and Laplace transform

Given a (not necessarily regular) holonomic D-module defined on the product of two complex manifolds, we prove that the associated correspondence commutes (in some sense) with the De Rham functor. We apply this result to the study of the classical Laplace transform. The main tools used here are the theory of ind-sheaves and its enhanced version.Comment: 62 pages. 2nd version typoes correcte