Deformation of Okamoto-Painleve pairs and Painleve equations (Proceedings of the Workshop "Algebraic Geometry and Integrable Systems related to String Theory")
Deformation of Okamoto-Painleve pairs and Painleve equations (Proceedings of the Workshop "Algebraic Geometry and Integrable Systems related to String Theory")
复制标题
Okamoto-Painleve 对和 Painleve 方程的变形（“与弦理论相关的代数几何和可积系统”研讨会论文集）
DOI：
10.1090/s1056-3911-01-00316-2
复制
发表时间：
2000
影响因子：
1.9
通讯作者：
Masahiko Saito
中科院分区：
文献类型：
--
作者：
Masahiko Saito
本文推广了Painlev方程初值空间的概念，引入了广义有理Okamoto-Painlev对(S，Y)的概念。在对这些对进行分类后，我们将建立一种代数几何方法，利用局部上同调群从Okamoto-Painlev对的形变导出Painlev微分方程。此外，借助于S-Y上的全纯辛结构，阐明了Painlev方程可以写成哈密顿系统的原因。作为我们理论的例子，我们显式地计算了类型为{E}_7(=P_(II))和{D}_8(=P_(III)^{D}_8)的Okamoto-Painlev对的哈密顿结构。
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.