Isomonodromic deformations of the sl(2) Fuchsian systems on the Riemann sphere
Isomonodromic deformations of the sl(2) Fuchsian systems on the Riemann sphere
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黎曼球上 sl(2) Fuchsian 系统的等单向变形
DOI：
10.17323/1609-4514-2005-5-2-415-441
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
S. Oblezin
中科院分区：
文献类型：
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作者：
S. Oblezin
本文讨论了与等同构法有关的两个几何结构。我们遵循德林菲尔德的思想，并在曲线$X=\mathbb{P}^1\setminus\{a_1，…，a_n\}$的情况下发展它们。因此，我们将Arinkin和Lysenko的结果推广到任意数目$n$点的情况。首先，我们根据模空间之间的Hecke对应构造分离的达布坐标。这样，我们就给出了斯克里亚宁公式的几何解释。在论文的第二部分，我们构造了初始数据空间的Drinfeld紧化，并在一定的fh - shaves上描述了紧化因子。最后，我们用紧化因子的变形给出了等同构系统动力学的几何表示，并解释了在Fuchsian方程中，视奇点的作用。为了说明结果和方法，我们给出了一个最简单的具有四个标记点的等单调系统的例子，称为Painlev ' e-VI系统。
This paper is devoted to two geometric constructions related to the isomonodromic method. We follow the Drinfeld ideas and develop them in the case of the curve $X=\mathbb{P}^1\setminus\{a_1,...,a_n\}$. Thus we generalize the results of Arinkin and Lysenko to the case of arbitrary number $n$ of points. First, we construct separated Darboux coordinated in terms of the Hecke correspondences between moduli spaces. In this way we present a geometric interpretation of the Sklyanin formulas. In the second part of the paper, we construct Drinfeld's compactification of the initial data space and describe the compactifying divisor in terms of certain FH-sheaves. Finally, we give a geometric presentation of the dynamics of the isomonodromic system in terms of deformations of the compactifying divisor and explain the role of apparent singularities for Fuchsian equations. To illustrate the results and methods, we give an example of the simplest isomonodromic system with four marked points known as the Painlev´e-VI system.