Moduli of Stokes torsors and singularities of differential equations

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.

New application of the main theorem to the boundedness of turning loci (Theorem 2). New description of the geometry of the restriction morphisms for the moduli of Stokes torsors (Theorem 3). Application to Cotti, Dubrovin and Guzzetti's injectivity theorem generalized in any dimension (Theorem 4). Rareness of Stokes structures in higher dimension and explicit computations of the moduli of Stokes torsors will appear in a separate note