Isomonodromic deformations of logarithmic connections and stable parabolic vector bundles

TL;DR

This paper proves that generic parabolic vector bundles arising from universal isomonodromic deformations of logarithmic connections on compact Riemann surfaces are parabolically stable, with rank two cases being very stable.

Contribution

It establishes the stability of parabolic vector bundles in the context of isomonodromic deformations, extending understanding of their geometric properties.

Findings

Generic parabolic bundles are parabolically stable in isomonodromic deformations.

Rank two parabolic bundles are parabolically very stable.

Stability holds over the Teichmüller space for generic parameters.

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 Teichm\"uller space is parabolically stable. In the case of parabolic vector bundles of rank two, the general parabolic vector bundle is even parabolically very stable.