Open Access
September, 2001 Algebraic and Geometric Isomonodromic Deformations
Charles F. Doran
J. Differential Geom. 59(1): 33-85 (September, 2001). DOI: 10.4310/jdg/1090349280


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.


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Charles F. Doran. "Algebraic and Geometric Isomonodromic Deformations." J. Differential Geom. 59 (1) 33 - 85, September, 2001.


Published: September, 2001
First available in Project Euclid: 20 July 2004

zbMATH: 1043.34098
MathSciNet: MR1909248
Digital Object Identifier: 10.4310/jdg/1090349280

Rights: Copyright © 2001 Lehigh University

Vol.59 • No. 1 • September, 2001
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