A new formulation of regular singularity

TL;DR

This paper introduces a new, more explicit way to define and determine regular singularities in meromorphic connections, simplifying previous approaches and extending classical criteria to higher dimensions.

Contribution

It offers an alternative, filtration-free definition of regular singularity and an explicit algorithm for its detection, extending Fuchs' criterion to multidimensional Weyl algebra modules.

Findings

New formulation avoids derived categories and special filtrations.

Provides an explicit algorithm for regular singularity detection.

Extends Fuchs criterion to higher-dimensional Weyl algebra modules.

Abstract

We provide an alternative definition for the familiar concept of regular singularity for meromorphic connections. Our new formulation does not use derived categories, and it also avoids the necessity of finding a special good filtration as in the formulation due to Kashiwara--Kawai. Moreover, our formulation provides an explicit algorithm to decide the regular singularity of a meromorphic connection. An important intermediary result, interesting in its own right, is that taking associated graded modules with respect to (not necessarily canonical) $V$-filtrations commutes with non-characteristic restriction. This allows us to reduce the proof of the equivalence of our formulation with the classical concept to the one-dimensional case. In that situation, we extend the well-known one-dimensional Fuchs criterion for ideals in the Weyl algebra to arbitrary holonomic modules over the Weyl algebra equipped with an arbitrary $(-1,1)$-filtration.