Derivative of the Riemann-Hilbert map

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.

11 pages, 0 figures