November 14, 2005

Dan Volok

Title: On the Painlevé property of the Schlesinger system

Abstract:

In 1905 L. Schlesinger has formulated a theorem that a holomorphic deformation of Fuchsian linear differential systems

$\displaystyle \frac{dY}{dx}=\sum_{1\leq j\leq n}\frac{Q_j(t)} {x-t_j}Y$

parameterized by the pole loci $ t=(t_1,\dots,t_n)$ is isomonodromic if and only if the residues $ Q_j$ satisfy with respect to $ t$ the non-linear system

\begin{displaymath} \left\{ \begin{aligned} \dfrac{\partial Q_j}{\partial t_k... ...k\not=j} \dfrac{[Q_j,Q_k]} {t_j-t_k}, \end{aligned}\right. \end{displaymath}

which is known today as the Schlesinger system.

In 1981 in a paper by T. Miwa it was stated that isomonodromic deformations of Fuchsian systems enjoy the Painlevé property: they are globally meromorphic with respect to the parameter.

In the formulated generality these two well-known results, which played an important role in the study of differential equations in the complex domain, are false: they hold under certain generic assumptions on the spectra of the residues $ Q_j$, but not in general. Nevertheless, the corollary that the Schlesinger system enjoys the Painlevé property holds true without any restrictions on the initial data. We shall discuss a proof of this fact and its generalization in the case of linear systems with singularities of higher Poincaré rank.