Regularity and b-functions for D-modules

TL;DR

This paper investigates the existence of regular b-functions for holonomic D-modules, establishing that regularity along a submanifold guarantees the existence of a regular b-function in that context.

Contribution

It proves that holonomic D-modules regular along a submanifold admit regular b-functions, extending the understanding of b-functions in the context of regularity.

Findings

Holonomic D-modules always admit a b-function along any submanifold.

Regular D-modules admit regular b-functions with specific order conditions.

Modules regular along a submanifold admit regular b-functions along that submanifold.

Abstract

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