The Schlesinger system and isomonodromic deformations of bundles with connections on Riemann surfaces

Abstract

We introduce a way of presentation of pairs $(E,\nabla)$, where $E$ is a bundle on a Riemann surface and $\nabla$ is a logarithmic connection in $E$, which is based on a presentation of the surface as a factor of the exterior of the unit disc. In this presentation we write the local equation of isomonodormic deformation of pairs $(E,\nabla)$. These conditions are written as a modified Schlesinger system on a Riemann sphere (and in the typical case just as an ordinary Schlesinger system) plus some linear system.