Deformation of Okamoto-Painlevé Pairs and Painlevé equations

Abstract

In this paper, we introduce the notion of generalized rational Okamoto-Painlevé pair (S, Y) by generalizing the notion of the spaces of initial conditions of Painlevé equations. After classifying those pairs, we will establish an algebro-geometric approach to derive the Painlevé differential equations from the deformation of Okamoto-Painlevé pairs by using the local cohomology groups. Moreover the reason why the Painlevé equations can be written in Hamiltonian systems is clarified by means of the holomorphic symplectic structure on S - Y. Hamiltonian structures for Okamoto-Painlevé pairs of type ${\tilde{E}}_7(=P_{II})$ and ${\tilde{D}}_8(=P_{III}^{{\tilde{D}}_8})$ are calculated explicitly as examples of our theory.

38 pages, 4 figures, minor corrections, typos

References in corpus (2)

• Affine Weyl groups, discrete dynamical systems and Painleve equations
• Painlevé equations and deformations of rational surfaces with rational double points

Cited by in corpus (3)

• Moduli of unramified irregular singular parabolic connections on a smooth projective curve
• Full-parameter discrete Painlevé systems from non-translational Cremona isometries
• Isomonodromy for the Degenerate Fifth Painlevé Equation