Isomonodromic deformations of irregular connections and stability of bundles

Abstract

Let $G$ be a reductive affine algebraic group defined over $\mathbb{C}$, and let ${\nabla }_0$ be a meromorphic $G$-connection on a holomorphic $G$-bundle $E_0$, over a smooth complex curve $X_0$, with polar locus $P_0\subset X_0$. We assume that ${\nabla }_0$ is irreducible in the sense that it does not factor through some proper parabolic subgroup of $G$. We consider the universal isomonodromic deformation $(E_t\to X_t,{\nabla }_t,P_t{)}_{t\in \mathcal{T}}$ of $(E_0\to X_0,{\nabla }_0,P_0)$, where $\mathcal{T}$ is a certain quotient of a certain framed Teichmüller space we describe. We show that if the genus $g$ of $X_0$ satisfies $g\ge 2$, then for a general parameter $t\in \mathcal{T}$, the $G$-bundle $E_t\to X_t$ is stable. For $g\ge 1$, we are able to show that for a general parameter $t\in \mathcal{T}$, the $G$-bundle $E_t\to X_t$ is semistable.