Isomonodromic deformations along a stratum of the coalescence locus

TL;DR

This paper studies conditions for deformations of certain differential systems to preserve monodromy, providing explicit criteria and extending previous results to more general cases with applications in geometric structures.

Contribution

It offers necessary and sufficient conditions for strongly isomonodromic deformations of systems with specific singularities, generalizing prior work by relaxing generic assumptions.

Findings

Derived explicit Pfaffian system for isomonodromic deformations

Established nonlinear PDE conditions on residue matrices

Extended previous results to non-generic cases

Abstract

We consider deformations of a differential system with Poincare' rank 1 at infinity and Fuchsian singularity at zero along a stratum of a coalescence locus. We give necessary and sufficient conditions for the deformation to be strongly isomonodromic, both as an explicit Pfaffian system (integrable deformation) and as a non linear system of PDEs on the residue matrix A at the Fuchsian singularity. This construction is complementary to that of [13]. For the specific system here considered, the results generalize those of [26], by giving up the generic conditions, and those of [3], by giving up the Lidskii generic assumption. The importance of the case here considered originates form its applications in the study of strata of Dubrovin-Frobenius manifolds and F-manifolds.