Deformations of singularities of meromorphic $\mathfrak{sl}_2(\mathbb{C})$-connections and meromorphic quadratic differentials

Abstract: This paper contributes to the theory of singularities of meromorphic linear ODEs in traceless $2\times2$ cases, focusing on their deformations and confluences. It is divided into two parts: The first part addresses individual singularities without imposing restrictions on their type or degeneracy. The main result establishes a correspondence between local formal invariants and jets of meromorphic quadratic differentials. This result is then utilized to describe the parameter space of universal isomonodromic deformation of meromorphic $\mathfrak{sl}_2(\mathbb{C})$-connections over Riemann surfaces. The second part examines the confluence of singularities in a fully general setting, accommodating all forms of degeneracies. It explores the relationship between the geometry of the unfolded Stokes phenomenon and the horizontal and vertical foliations of parametric families of quadratic differentials. The local moduli space is naturally identified with a specific space of local monodromy and Stokes data, presented as a space of representations of certain fundamental groupoids associated with the foliations. This is then used for studying degenerations of isomonodromic deformations in parametric families.