Algebro-Geometric Solutions of the Schlesinger Systems and the Poncelet-Type Polygons in Higher Dimensions

Abstract

A new form of algebro-geometric solutions of Rank 2 Schlesinger systems is presented. The solutions are written in terms of a particular meromorphic differential of the third type on hyperelliptic curves represented as a ramified double coverings of CP1. As was shown in the authors' earlier paper, in the case of genus one, this differential has its zeros at a solution of a Painlevé VI equation and provides an invariant formulation of a particular Okamoto transformation for the Painlevé VI equations. In the hyperelliptic case, positions of zeros of the differential provide part of a solution of the multidimensional Garnier system. The construction of the differential starts with a choice of a point in the Jacobian of the curve. This point is later assumed to move in the Jacobian in a way that its coordinates with respect to the lattice of the Jacobian are constant as the branch points of the curve vary. The case when the coordinates of the point are rational corresponds to periodic trajectories of the billiard ordered games associated with g confocal quadrics in g + 1 dimensional space. This generalizes a relationship between solutions of a Painlevé VI equation and the Poncelet polygons discovered by Hitchin.