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Algebro-geometric solutions to Painleve VI and Schlesinger systems Shramchenko, Vasilisa


A method of constructing algebro-geometric solutions of rank two Schlesinger systems is presented. For an elliptic curve given as a ramified double covering of $CP^1$, a meromorphic differential is constructed with the following property: the common projection of its two zeros on the base of the covering, regarded as a function of the only moving branch point of the covering, is a solution of the Painleve VI $(1/8,-1/8,1/8,3/8)$ equation. This differential provides an invariant formulation of a classical Okamoto transformation for the Painleve VI equations. This approach is motivated by an observation of Hitchin connecting algebraic solutions of a Painleve VI equation to the Poncelet polygons in the plane. The research is partially supported by the NSF grant 1444147 and NSERC discovery grant.

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