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

We introduce a way to represent pairs (E,a double dagger), where E is a bundle on a Riemann surface and a double dagger is a logarithmic connection in E, based on a representation of the surface as the quotient of the exterior of the unit disc. In this representation, we write the local isomonodromic deformation conditions for the pairs (E,a double dagger). These conditions are written as a modified Schlesinger system on a Riemann sphere (reduced to the ordinary Schlesinger system in the typical case) supplemented by a certain system of linear equations.