Communications in Analysis and Geometry

Volume 29 (2021)

Number 1

Isomonodromic deformations of irregular connections and stability of bundles

Pages: 1 – 18



Indranil Biswas (School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India)

Viktoria Heu (Institut de Recherche Mathématique Avancée, Strasbourg, France)

Jacques Hurtubise (Department of Mathematics, McGill University, Montreal, Quebec, Canada)


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.

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