Algebro-geometric approach to an Okamoto transformation, the Painlevé VI and Schlesinger equations
In this paper the authors provide an essentialy new method of building algebro-geometric solutions of the Schlesinger system with eigenvalues $\pm \frac{1}{4}$ of the residue matrix $A^{(i)}$ in the case of four singular points. Such a system yields the sixth Painlevé equation $P_{VI}$ with parameters \[ \alpha=-\beta=\gamma=\frac{1}{8},\quad \delta=\frac{3}{8}. \] The authors gives a new description of solutions of the pointed Schlesinger system in terms of a certain abelian differential of third kind $\Omega$, the normalized holomorphic differential $\omega$ and the Picard solution $y_0$ of $P_{VI}$. These differentials lie on an elliptic curve considered as a twofold ramified covering of $\mathbb{C}\mathbb{P}^1$. The differential $\Omega$ is constructed in such a way that its two zeros project to the same point on the base of the covering and this point regarded as a function of the moving branch point satisfies the pointed $P_{VI}$. Moreover, this differential gives an invariant characterization of the classical Okamoto transformation. ``The initial data for construction of the meromorphic differential include a point in the Jacobian of the curve, under the assumption that this point has non-variable coordinates with respect to the lattice of the Jacobian while the branch points vary. It appears that the cases where the coordinates of the point are rational correspond to the Poncelet polygons inscribed and circumscribed in a pair of conics. Thus, this is a generalization of situation studied by Hitchin who related algebraic solutions of a Painlevé VI equation with the Poncelet polygons.''
scientific article; zbMATH DE number 7044355
scientific article