Journal of Differential Geometry

Algebraic and Geometric Isomonodromic Deformations

Charles F. Doran

Full-text: Open access

Abstract

Using the Gauss-Manin connection (Picard-Fuchs differential equation) and a result of Malgrange, a special class of algebraic solutions to isomonodromic deformation equations, the geometric isomonodromic deformations, is defined from "families of families" of algebraic varieties. Geometric isomonodromic deformations arise naturally from combinatorial strata in the moduli spaces of elliptic surfaces over ℙ1. The complete list of geometric solutions to the Painlevé VI equation arising in this way is determined. Motivated by this construction, we define another class of algebraic isomonodromic deformations whose monodromy preserving families arise by "pullback" from (rigid) local systems. Using explicit methods from the theory of Hurwitz spaces, all such algebraic Painlevé VI solutions coming from arithmetic triangle groups are classified.

Article information

Source
J. Differential Geom., Volume 59, Number 1 (2001), 33-85.

Dates
First available in Project Euclid: 20 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1090349280

Digital Object Identifier
doi:10.4310/jdg/1090349280

Mathematical Reviews number (MathSciNet)
MR1909248

Zentralblatt MATH identifier
1043.34098

Citation

Doran, Charles F. Algebraic and Geometric Isomonodromic Deformations. J. Differential Geom. 59 (2001), no. 1, 33--85. doi:10.4310/jdg/1090349280. https://projecteuclid.org/euclid.jdg/1090349280


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