On orders of movable poles of the Schlesinger equation
The author treats movable singular points of the Schlesinger equation of isomonodromic deformations as the points of jumping of the splitting type of the vector bundle constructed from the initial Fuchsian equation. The Fuchsian system considered has the form \[ {dy\over dx}= \Biggl( \sum^n_{i= 1} {B^0_i\over x- a^0_i}\Biggr) y,\;\sum^n_{i=1} B^0_i= 0, \] on the Riemann sphere determined by the Schlesinger equation \[ dB_i(a)=- \sum^n_{j=1} {[B_i(a), B_j(a)]\over a_i- a_j} d(a_i- a_j). \] The main result cleverly constructed including many minor details is that the solutions to the Schlesinger equation of isomonodromic deformations for the case of irreducible monodromy have poles of order two at \({\mathcal M}\). This means that the solutions to the Schlesinger equation being multiplied by \(C^2(t)\) become holomorphic at \({\mathcal M}\), where the local ideal of the analytic subvariety \({\mathcal M}\) is determined by \(C(t)\). A wonderful example of a Schlesinger isomonodromic deformation is included which is closely connected to a Painlevé VI equation.
scientific article; zbMATH DE number 1475292