On the algebro-geometric integration of the Schlesinger equations
Let \(a_1,\dots, a_{2g+1}\) be points on the \(\lambda\)-plane \(\mathbb{C}\). For a given (very special) \(2\times 2\) Riemann-Hilbert data, depending on \(2g\) parameters, on the polygon \(\infty- a_1-\cdots- a_{2g+1}-\infty\), the solution \(Y(\lambda)\) to the R-H problem is explicitly given in terms of the period integrals and the Riemann theta function associated with the hyperelliptic curve branching at \(\{a_1,\dots, a_{2g+1},\infty\}\). The solution \(Y(\lambda)\) satisfies the linear differential equation \[ {dY(\lambda)\over d\lambda}= \sum^{2g+1}_{j= 1} {A_j\over \lambda- a_j} Y(\lambda). \] The expression of \(Y(\lambda)\) yields an explicit representation of \(A_j(a)\) giving a \(2g\)-parameter family of solutions to the so-called Schlesinger equation governing the isomonodromic deformation of the linear equation above. When \(g= 1\), this coincides with the solution to the special 6th Painlevé equation obtained by Hitchin, which can be transformed into Picard's solution under the Weyl group symmetry.
scientific article; zbMATH DE number 1379951