Isomonodromic deformations of irregular connections and stability of bundles

TL;DR

This paper studies the deformation of irregular connections on complex curves and proves that, under certain conditions, the associated bundles are stable or semistable for general deformations.

Contribution

It establishes the stability and semistability of bundles arising from universal isomonodromic deformations of irregular connections on complex curves.

Findings

For genus g ≥ 2, general deformations yield stable bundles.

For genus g ≥ 1, general deformations yield semistable bundles.

The results connect isomonodromic deformations with bundle stability properties.

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 Teichm\"uller space we describe. We show that if the genus $g$ of $X_0$ satisfies $g\geq 2$, then for a general parameter $t\in \mathcal{T}$, the $G$-bundle $E_t\to X_t$ is stable. For $g\geq 1$, we are able to show that for a general parameter $t\in \mathcal{T}$, the $G$-bundle $E_t\to X_t$ is semistable.