Pure and Applied Mathematics Quarterly

Volume 16 (2020)

Number 2

Special Issue: In Honor of Eduard Looijenga, Part 3 of 3

Isomonodromic deformations of logarithmic connections and stable parabolic vector bundles

Pages: 191 – 227

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n2.a1

Authors

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

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

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

Abstract

We consider irreducible logarithmic connections $(E, \delta)$ over compact Riemann surfaces $X$ of genus at least two. The underlying vector bundle $E$ inherits a natural parabolic structure over the singular locus of the connection $\delta$; the parabolic structure is given by the residues of $\delta$. We prove that for the universal isomonodromic deformation of the triple $(X, E, \delta)$, the parabolic vector bundle corresponding to a generic parameter in the Teichmüller space is parabolically stable. In the case of parabolic vector bundles of rank two, the general parabolic vector bundle is even parabolically very stable.

Keywords

logarithmic connection, isomonodromic deformation, parabolic bundle, stability, very stability, Teichmüller space

2010 Mathematics Subject Classification

32G08, 14H60, 34M56, 53B05

Received 22 February 2019

Published 20 March 2020