Moduli of Stokes torsors and singularities of differential equations

Abstract

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