Deformation of Okamoto–Painlevé pairs and Painlevé equations
Authors:
MasaHiko Saito, Taro Takebe and Hitomi Terajima
Journal:
J. Algebraic Geom. 11 (2002), 311362
DOI:
https://doi.org/10.1090/S1056391101003162
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 algebrogeometric 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
MasaHiko Saito
Affiliation:
Department of Mathematics, Faculty of Science, Kobe University, Kobe, Rokko, 6578501, Japan
Email:
mhsaito@math.kobeu.ac.jp
Taro Takebe
Affiliation:
Department of Mathematics, Faculty of Science, Kobe University, Kobe, Rokko, 6578501, Japan
Email:
takebe@math.kobeu.ac.jp
Hitomi Terajima
Affiliation:
Department of Mathematics, Faculty of Science, Kobe University, Kobe, Rokko, 6578501, Japan
Email:
terajima@math.kobeu.ac.jp
Received by editor(s):
July 7, 2000
Published electronically:
December 19, 2001
Additional Notes:
Partly supported by GrantinAid for Scientific Research (B09440015), (B12440008) and (C11874008), the Ministry of Education, Science and Culture, Japan