Painlevé equations and deformations of rational surfaces with rational double points

Abstract

In this paper, we show that the Bäcklund transformations of Painlevé equations come from birational maps of rational surfaces constructed by Okamoto as the spaces of initial conditions. The simultaneous resolutions of rational double points of the families of rational surfaces give rise to flops which are origins of symmetry groups of Painlevé equation. Moreover we point out that Kodaira--Spencer theory of deformations of rational surfaces explains a geomrtric meaning of Painlevé equations.

45 pages

Cited by in corpus (9)

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• Dynamics of the Sixth Painlevé Equation
• Deformation of Okamoto-Painlevé Pairs and Painlevé equations
• Nodal curves and Riccati solutions of Painlevé equations
• Moduli spaces for linear differential equations and the Painlevé equations
• A Riemann--Hilbert approach to Painlevé IV
• Integrability of $n$-dimensional dynamical systems of type $E_7^{(1)}$ and $E_8^{(1)}$