General hypergeometric functions and geometry of the space of arrangements of points with infinitesimal neighborhoods

一般超几何函数(GHF)和TWSTED上同调群的结构。GL(N)的正则元的中心化子的共轭类由N的划分决定。GHF是Grassman流形Gr(n+1，N)上的多值函数，定义为中心化子的泛覆盖群的一个特征标的Radon变换。对于一个整数q&gt；0，考虑N的一个划分(q，1，…，1)。为了阐明一般超几何系统解空间的结构，我们计算了相应的de Rham上同调群的秩和基。当GHF由n维积分给出时，我们发现当k不同于n时，k个上同调群为零，且n个上同调群的秩为(N-2)！/n！(n-n-2)！我们显式地利用Schur函数、Schlesinger系统及其推广给出了这个群的基础。我们从扭曲理论的角度出发，开始了对这一推广的研究。当人们考虑到…属时在Grassman流形Gr(2，N)上进一步推广了反自对偶Yang-Mills方程(GASDYM)，它的解通过Ward对应对应于扭曲空间Pn-1上的一个全纯向量丛，当仅限于扭曲直线时，这种对应是平凡的.设H是GL(N)的极大交换子群，如文1)，考虑其在扭曲空间Pn-1上的自然作用。此外，我们假设H的作用可以提升到对应于GASYM方程的解的全纯向量丛。然后，这一作用确定了束上的扁平连接，并且当限制为加捻线时，该扁平连接描述了ODE的保持单行的变形。我们给出了平坦联系的显式，并通过这个显式明确了与GHF定义的类比。从这一观点出发，我们将一般的Schlesinger系统统一地导出为Gr(2，N)上的微分方程组，它对应于Painleve方程(包括退化的方程)。我们还明确了与H相联系的Weyl群描述了一般Schlesinger系统的对称性群。由此，我们可以对Painleve方程中的参数个数在简并后减少这一事实给予群论上的理解。我们还可以构造一般Schlesinger系统的退化(汇流)过程。较少

The general hypergeometric functions(GHF) and the structure of the twsted cohomology group. The conjugacy classes of the centralizers of regular elements of GL(N) are determined by partitions of N. GHF is a multi-valued function on the Grassmannian manifold Gr(n+1, N) defined as a Radon transform of a character of the universal covering group of the centralizer. For an integer q > 0, consider a partition (q, 1,...,1) of N. To clarify the structure of the solution space of general hypergeometric system, we computed the rank and a basis of the associated de Rham cohomology group. When GHF is given by n dimensional integral, we found that the k-th cohomology group vanishes for k different from n, and the rank of the n-th cohomology group is (N-2)!/n!(N-n-2)!. We gave a basis for this group explicitly using Schur functions.Schlesinger system and its generalizations. We started the research of giving this generalizations from the point of view of twistor theory. When one consider the genera … More lized anti-self dual Yang-Mills equation(GASDYM) on the Grassmannian manifold Gr(2, N), its solution corresponds to a holomorphic vector bundle on the twistor space PN-1 via the Ward correspondence which is trivial when restricted to twistor lines. Let H be a maximal abelian subgroup of GL(N) as in 1) and consider its natural action on the twistor space PN-1. Moreover we assume that the action of H can be lifted to the holomorphic vector bundle corresponding to a solution to the GASYM equation. Then this action determines a flat connection on the bundle and when restricted to twistor lines, this flat connection describes a monodromy preserving deformation of ODEs. We gave the explicit form of the flat connection and by this explicit expression we made clear the analogy to the definition of GHF. We derived in a unified way the general Schlesinger systems from this point of view as the differential equations on Gr(2,N) which corresponds to the Painleve equations(including the degenerated ones). We also made clear that the Weyl group associated with H describes a group of symmetry of the general Schlesinger system. By this, we can give the group theoretic understanding for the fact that the number of parameters in the Painleve equations deceases after the degeneration. We could also construct the process of degeneration (confluence) for the general Schlesinger systems. Less