Isomonodromic deformations of $\mathfrak{sl}(2)$ Fuchsian systems on the Riemann sphere

This paper is devoted to the two geometric constructions provided by the isomonodromic method for Fuchsian systems. We develop the subject in the sense of geometric representation theory following Drinfeld's ideas. Thus we identify the initial data space of the $\mathfrak{sl}(2)$ Schlesinger system with the moduli space of the Frobenius–Hecke (FH-)sheaves originally introduced by Drinfeld. First, we perform the procedure of separation of variables in terms of the Hecke correspondences between moduli spaces. In this way we present a geometric interpretation of the Flashka–McLaughlin, Gaudin and Sklyanin formulas. In the second part of the paper, we construct the Drinfeld compactification of the initial data space and describe the compactifying divisor in terms of certain FH-sheaves. Finally, we give a geometric presentation of the dynamics of the isomonodromic system in terms of deformations of the compactifying divisor and explain the role of apparent singularities for Fuchsian equations. To illustrate the results and methods, we give an example of the simplest isomonodromic system with four marked points known as the Painlevé-VI system.