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Discrete symmetries of systems of isomonodromic deformations of second-order Fuchsian differential equations. (English. Russian original) Zbl 1084.32010
Funct. Anal. Appl. 38, No. 2, 111-124 (2004); translation from Funkts. Anal. Prilozh. 38, No. 2, 38-54 (2004).
The author studies discrete transformations of moduli spaces of logarithmic \(sl(2)\)-connections with singularities at distinct points \(\{ x_1,\dots,x_{n} \}\) on the Riemann sphere \(P^1\) and with given eigenvalues of the residues of the connection. Using the modification technique for vector bundles with connections this allows to compute the discrete affine group of Schlesinger transformations for isomonodromic deformations of a Fuchsian system of second order differential equations. The obtained result is applied to three examples of Fuchsian differential equations, the hypergeometric equation, the Heun equation and the sixth Painlevé equation, and therefore is the generalization of classical situation.

32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation)
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
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