Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Deformation of Okamoto-Painlevé pairs and Painlevé equations

Authors: Masa-Hiko Saito, Taro Takebe and Hitomi Terajima
Journal: J. Algebraic Geom. 11 (2002), 311-362
Published electronically: December 19, 2001
MathSciNet review: 1874117
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Abstract | References | Additional Information

Abstract: In this paper, we introduce the notion of a 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.

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Additional Information

Masa-Hiko Saito
Affiliation: Department of Mathematics, Faculty of Science, Kobe University, Kobe, Rokko, 657-8501, Japan

Taro Takebe
Affiliation: Department of Mathematics, Faculty of Science, Kobe University, Kobe, Rokko, 657-8501, Japan

Hitomi Terajima
Affiliation: Department of Mathematics, Faculty of Science, Kobe University, Kobe, Rokko, 657-8501, Japan

Received by editor(s): July 7, 2000
Published electronically: December 19, 2001
Additional Notes: Partly supported by Grant-in-Aid for Scientific Research (B-09440015), (B-12440008) and (C-11874008), the Ministry of Education, Science and Culture, Japan