Algebraic and geometric isomonodromic deformations.
type: Item
instance of: scholarly article
title: Algebraic and geometric isomonodromic deformations.
zbMATH Open document ID: 1043.34098
author: C. F. Doran
published in: Journal of Differential Geometry
publication date: 2002-08-15T00:00:00Z
review text: 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.
reviewed by: Shun Shimomura
Mathematics Subject Classification ID: 34M55
Mathematics Subject Classification ID: 12H99
Mathematics Subject Classification ID: 32G34
Mathematics Subject Classification ID: 34M35
zbMATH DE Number: 1782701
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1090349280
OpenAlex ID: W1496679112
Wikidata QID: Q115202167
DOI: 10.4310/JDG/1090349280
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description: scientific article; zbMATH DE number 1782701