Isomonodromic deformations of logarithmic connections and stable parabolic vector bundles

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.