Garnier系统的经典解的几何
The geometry of the classical solutions of the Garnier systems
摘要：
我们的目的是找到一个通用的方法理论的经典解的Garnier系统的$n$ -变量，${\cal G}_n$，黎曼-希尔伯特问题的基础上的几何空间的isomonodromy变形。我们的方法包括在确定相应的Fuchsian系统，保证有一个经典的解决方案的Garnier系统${\cal G}_n$的单值数据。这就引出了Garnier系统的简化思想。证明了若Garnier系统${\cal G}_{n}$的解使得Fuchsian系统的$l$单值矩阵等于$\pm\ID$，则该解可经典地化为具有$n-l$变量${\cal G}_{n-l}$的Garnier系统的解.当$n$单值矩阵等于$\pm\ID$时，我们得到了${\cal G}_n$的经典解.我们还给出了产生经典解的另一种机制：证明了具有可约单值群的Garnier系统的解可以用Lauricella超几何函数约化为Okamoto和Kimura发现的经典解.对于1-变量的Garnier系统，即Painlevé VI方程，我们证明了所有的经典非代数解要么有可约的单值群，要么至少有一个单值矩阵等于$\pm\ID$ .
英文摘要：
Our aim is to find a general approach to the theory of classical solutions of the Garnier system in $n$-variables, ${\cal G}_n$, based on the Riemann-Hilbert problem and on the geometry of the space of isomonodromy deformations. Our approach consists in determining the monodromy data of the corresponding Fuchsian system that guarantee to have a classical solution of the Garnier system ${\cal G}_n$. This leads to the idea of the reductions of the Garnier systems. We prove that if a solution of the Garnier system ${\cal G}_{n}$ is such that the associated Fuchsian system has $l$ monodromy matrices equal to $\pm\ID$, then it can be reduced classically to a solution of a the Garnier system with $n-l$ variables ${\cal G}_{n-l}$. When $n$ monodromy matrices are equal to $\pm\ID$, we have classical solutions of ${\cal G}_n$. We give also another mechanism to produce classical solutions: we show that the solutions of the Garnier systems having reducible monodromy groups can be reduced to the classical solutions found by Okamoto and Kimura in terms of Lauricella hypergeometric functions. In the case of the Garnier system in 1-variables, i.e. for the Painlevé VI equation, we prove that all classical non-algebraic solutions have either reducible monodromy groups or at least one monodromy matrix equal to $\pm\ID$.
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