Derivative of the Riemann–Hilbert map

IF 0.9 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2025-05-15 DOI:10.1112/blms.70092

Vladimir Marković, Ognjen Tošić

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Abstract

Given a pair $(X, \nabla)$ , consisting of a closed Riemann surface $X$ and a holomorphic connection $\nabla$ on the trivial principal bundle $X \times {SL}_{2} (C) \to X$ , 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.

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黎曼-希尔伯特映射的导数

给定一对（X,∇）$(X,\nabla)$，由一个封闭黎曼曲面X $X$和一个平凡主束X × SL 2 (C)上的全纯连接∇$\nabla$组成→X $X\times \mathrm{SL}_2(\mathbb {C})\rightarrow X$， Riemann-Hilbert映射将（X,∇）$(X,\nabla)$发送到它的单态表示。我们计算了这个映射的导数，并提供了它被注入的轨迹的简单描述，在这个过程中恢复了几个先前得到的结果。

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