Derivative of the Riemann-Hilbert map

TL;DR

This paper computes the derivative of the Riemann-Hilbert map for holomorphic connections on Riemann surfaces, identifying where it is injective and recovering known results in the process.

Contribution

It provides an explicit computation of the derivative of the Riemann-Hilbert map and characterizes the injectivity locus, enhancing understanding of the map's local behavior.

Findings

Computed the derivative of the Riemann-Hilbert map.

Identified the locus where the map is injective.

Recovered several known results about the map's properties.

Abstract

Given a pair $(X,\nabla)$, consisting of a closed Riemann surface $X$ and a holomorphic connection $\nabla$ on the trivial principal bundle $X\times\mathrm{SL}_2(\mathbb{C})\to X$, the Riemann-Hilbert map sends $(X,\nabla)$ to its monodromy representation. We compute the derivative of this map, and provide a simple description of the locus where it is injective, recovering in the process several previously obtained results.