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Algebraic and geometric isomonodromic deformations. (English) Zbl 1043.34098

It is well known that the sixth Painlevé equation (PVI) follows from isomonodromic deformations of Fuchsian differential equations with four regular singularities \(0, 1, \infty, t\) and one apparent regular singularity \(\lambda\), where \(t\) is a deformation parameter. This paper deals with, in particular, algebraic and geometric isomonodromic deformations, and gives some classes of algebraic solutions to (PVI). First, the author considers isomonodromic deformations of Picard-Fuchs differential equations corresponding to moduli spaces of elliptic surfaces over \(\mathbb P^1\) with four singular fibers; and using classification results for such elliptic surfaces, he determines a complete list of algebraic solutions of the corresponding type together with Weierstrass presentations \((g_2,g_3).\) In this elliptic case, there exists a pair \((\mathcal J(z), \Lambda)\) of Kodaira’s functional invariant and certain hypergeometric function such that the isomonodromic family is obtained by pulling back \(\Lambda\) by \(\mathcal J(z).\) Generalizing this method of constructing isomonodromic families, the author characterizes the topological types of all pullback algebraic solutions to (PVI), and classifies the topological types of algebraic solutions obtained by pullback of hypergeometric equations for arithmetic Fuchsian triangles and for spherical and planar triangles.

MSC:

34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
12H99 Differential and difference algebra
32G34 Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation)
34M35 Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms
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