Moduli of Stokes torsors and singularities of differential equations

TL;DR

This paper establishes a geometric link between the formal decomposition of meromorphic connections and the local system nature of their solution complexes using moduli of Stokes torsors.

Contribution

It proves the coincidence of the good formal decomposition locus with the locus where solution complexes restrict to local systems, via a geometric approach involving Stokes torsors.

Findings

The good formal decomposition locus matches the local system locus for solution complexes.

The proof uses the moduli space of Stokes torsors to connect algebraic and analytic properties.

The result provides a geometric perspective on the structure of meromorphic connections and their solutions.

Abstract

Let M be a meromorphic connection with poles along a smooth divisor D in a smooth algebraic variety. Let Sol M be the solution complex of M. We prove that the good formal decomposition locus of M coincides with the locus where the restrictions to D of Sol M and Sol End M are local systems. By contrast to the very different natures of these loci (the first one is defined via algebra, the second one is defined via analysis), the proof of their coincidence is geometric. It relies on the moduli of Stokes torsors.