The Monodromy Map from Differential Systems to the Character Variety Is Generically Immersive

Abstract

Let $G$ be a connected reductive affine algebraic group defined over $\mathbb{C}$ and $\mathfrak{g}$ its Lie algebra. We consider all pairs of the form $(Y,D)$, where $Y$ is a complex structure on a compact oriented $C^{\infty }$ surface $\Sigma$, and $D$ is a holomorphic connection on the trivial holomorphic principal $G$-bundle $Y\times G$ on $Y$; these are known as $\mathfrak{g}$-differential systems. We study the monodromy map from the space of $\mathfrak{g}$-differential systems to the character variety of $G$-representations of the fundamental group of $\Sigma$. If the complex dimension of $G$ is at least three, and $\mathrm{genus}(\Sigma )\ge 2$, we show that the monodromy map is an immersion at the generic point.