Regularity and $b$-functions for $D$-modules (Microlocal Analysis and Singular Perturbation Theory)

Abstract

A holonomic D-module on a complex analytic manifold admits always a b-function along any submanifold. If the module is regular, it admits also a regular b-function, that is a bfunction with a condition on the order of the lower terms of the equation. There is a weaker condition of regularity: regularity along a submanifold. We prove that a module which is regular along a submanifold admits a regular b-function along this submanifold.