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

DOI:
https://doi.org/10.1090/S1056-3911-01-00316-2

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 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 . Hamiltonian structures for Okamoto-Painlevé pairs of type and 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

Email:
mhsaito@math.kobe-u.ac.jp

**Taro Takebe**

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

Email:
takebe@math.kobe-u.ac.jp

**Hitomi Terajima**

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

Email:
terajima@math.kobe-u.ac.jp

DOI:
https://doi.org/10.1090/S1056-3911-01-00316-2

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