Rational Surfaces Associated with Affine Root Systems¶and Geometry of the Painlevé Equations
Rational Surfaces Associated with Affine Root Systems¶and Geometry of the Painlevé Equations
复制标题
DOI：
10.1007/s002200100446
复制
发表时间：
2001-06
影响因子：
2.4
通讯作者：
H. Sakai
中科院分区：
文献类型：
--
作者：
H. Sakai
我们提出了一种基于有理曲面的Painlevé方程理论的几何方法。我们的出发点是一个紧致光滑有理曲面X，它具有唯一的正则型反正则因子D。我们把所有这样的表面X.对于每个X，在X的Picard格中对应一个E（1）8的根子。我们实现了相应的仿射Weyl群的克雷莫纳行动的家庭这些表面上的行动。我们证明了仿射Weyl群的平移部分产生离散的Painlevé方程，并且上述作用通过Bäcklund变换构成了它们的对称群。这六个Painlevé微分方程似乎是这种构造的退化情况。在后一种情况下，X是冈本的初始条件空间和D是定义哈密顿结构的辛形式的极因子。
We present a geometric approach to the theory of Painlevé equations based on rational surfaces. Our starting point is a compact smooth rational surfaceXwhich has a unique anti-canonical divisorDof canonical type. We classify all such surfacesX. To eachX, there corresponds a root subsystem ofE(1)8inside the Picard lattice ofX. We realize the action of the corresponding affine Weyl group as the Cremona action on a family of these surfaces. We show that the translation part of the affine Weyl group gives rise to discrete Painlevé equations, and that the above action constitutes their group of symmetries by Bäcklund transformations. The six Painlevé differential equations appear as degenerate cases of this construction. In the latter context,Xis Okamoto's space of initial conditions andDis the pole divisor of the symplectic form defining the Hamiltonian structure.