Isomonodromic deformations of irregular connections and stability of bundles

Let $G$ be a reductive affine algebraic group defined over $\mathbb{C}$, and let $\nabla_0$ be a meromorphic $G$-connection on a holomorphic principal $G$-bundle $E_0$, over a smooth complex projective 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 $g \geq 2$, then for a general parameter $t \in \mathcal{T}$, the principal $G$-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 principal $G$-bundle $Et \to X_t$ is semistable.

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The first-named author thanks McGill University for hospitality while a part of the work was carried out. He is supported by a J. C. Bose Fellowship. The second-named author is supported by the ANR project Foliage ANR-16-CE40-0008. The third author is supported by an NSERC Discovery grant and an FRQNT Subvention d’Équipe.

Received 16 December 2016

Accepted 15 January 2018

Published 11 March 2021