Deformation of Okamoto-Painlevé Pairs and Painlevé equations

Details Deformation of Okamoto-Painlevé Pairs and Painlevé equations Deformation of Okamoto-Painlevé Pairs and Painlevé equations

2000-06-05

arxiv.org/abs/math/0006026v2

J. Algebraic Geom. 11 (2002), 311-362.

Mathematics

Algebraic Geometry

38 pages, 4 figures, minor corrections, typos

Scientific paper

In this paper, we introduce the notion of generalized rational Okamoto-Painlev\'e pair (S, Y) by generalizing the notion of the spaces of initial conditions of Painlev\'e equations. After classifying those pairs, we will establish an algebro-geometric approach to derive the Painlev\'e differential equations from the deformation of Okamoto-Painlev\'e pairs by using the local cohomology groups. Moreover the reason why the Painlev\'e equations can be written in Hamiltonian systems is clarified by means of the holomorphic symplectic structure on S - Y. Hamiltonian structures for Okamoto-Painlev\'e 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.

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