Sur une caract\'erisation des D-modules holonomes r\'eguliers

TL;DR

This paper characterizes regular holonomic D-modules on complex manifolds by establishing that if the Riemann-Hilbert correspondence is an isomorphism for a module, then the module is regular.

Contribution

It provides a converse to the known fully-faithfulness of the Riemann-Hilbert correspondence, showing that isomorphism implies regularity of the D-module.

Findings

If RH_{M,M} is an isomorphism, then M is regular.

The paper establishes a new characterization of regular holonomic D-modules.

It deepens the understanding of the Riemann-Hilbert correspondence in D-module theory.

Abstract

Let X be a smooth complex manifold. Let Sol denote the solution functor for D-modules on X. Traditionally, the fully-faithfulness of Riemann-Hilbert correspondance is proved by showing that if M_1 and M_2 are regular holonomic D_X modules, then the canonical morphism of complexes of sheaves RH_{M_1,M_2} : RHom(M_1,M_2) ---> RHom(Sol(M_2),Sol(M_1)) is an isomorphism, in a derived sense. This paper has to do with the converse statement. We prove that if M is an holonomic D_X module for which RH_{M,M} is an isomorphism, then M is regular.