Deformation theory of Fuchsian equations and logarithmic connections

TL;DR

This paper characterizes the tangent space of Fuchsian equations within the moduli space of logarithmic connections, introducing a Hodge structure that links deformations of these equations to a specific component.

Contribution

It constructs a weight 1 Hodge structure on the tangent space, providing a new geometric perspective on deformations of Fuchsian equations.

Findings

Tangent space of Fuchsian equations characterized within the moduli space.

A Hodge structure on the tangent space is constructed.

Deformations correspond to the (1,0)-part of the Hodge structure.

Abstract

Motivated by a remark and a question of Nicholas Katz, we characterize the tangent space of the space of Fuchsian equations with given generic exponents inside the corresponding moduli space of logarithmic connections: we construct a weight 1 Hodge structure on the tangent space of the moduli of logarithmic connections such that deformations of Fuchsian equations correspond to the $(1,0)$-part.