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

一般超几何函数和无穷小邻域点排列空间的几何

• 批准号: 15340058
• 负责人: KIMURA Hironobu
• 金额: $ 8.51万
• 依托单位: Kumamoto University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-15340058/
• 关键词: Twistor Theory; Generalized anti-self-dual Yang Mills; general Schlesinger system; Painleve equation; monodromy preserving deformation; general hypergeometric function; de Rham Theory; generalized Airy function; 一般反自己双対Yang-Mills方程式; Radon変換; Sc

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

广义超几何函数与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的分拆决定。GHF是Grassmannian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特征标的Radon变换.对于整数q > 0，考虑分区（q，1，.，1)瑟菌的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不等于n时，k阶上同调群为零，且n阶上同调群的秩为（N-2）！/你好！（N-n-2）！我们明确地用Schur函数给出了这个群的基.Schlesinger系统及其推广.我们从扭量理论的观点出发，开始了给出这种推广的研究。当一个人考虑 ...更多信息 Grassmannian流形Gr（2，N）上的反自对偶Yang-Mills方程（GASDYM），其解通过Ward对应对应于扭量空间PN-1上的一个全纯向量丛，当仅限于扭量线时，Ward对应是平凡的.设H是GL（N）的极大交换子群，并考虑它在扭量空间PN-1上的自然作用.此外，我们假设H的作用可以提升到GASYM方程的解所对应的全纯向量丛。然后，这个行动确定了一个平面连接的丛，当限制到扭线，这个平面连接描述了一个单值保持变形的常微分方程。我们给出了平坦联络的显式表达式，并通过这个显式表达式，我们清楚地表明了与GHF定义的相似性。从这一观点出发，我们统一地导出了一般Schlesinger系统为Gr（2，N）上的微分方程，它对应于Painleve方程（包括退化的）。我们还清楚地表明，与H相关的外尔群描述了一般施莱辛格系统的一组对称性。由此，我们可以给出Painleve方程退化后参数个数减少的群论理解。我们还可以构造一般Schlesinger系统的退化（汇合）过程。少

项目成果

Masashi Misawa: "Existence for a Cauchy-Dirichlet problem for evolutional p-Laplacian systems."Applicationes Math.. (To appear). (2004)

Masashi Misawa：“进化 p-拉普拉斯系统的柯西-狄利克雷问题的存在性。”应用数学..（出现）。

Katsunori Iwasaki: "Cohomology groups for recurrence relations and contiguity relations of hypergeometric systems"Journal of the Mathematical Society of Japan. 55. 185-219 (2003)

Katsunori Iwasaki：“超几何系统的递归关系和邻接关系的上同调群”日本数学会杂志。

Backlund transformation of the sixth Painleve equation in terms of Riemann-Hilbert correspondence

第六 Painleve 方程的黎曼-希尔伯特对应关系的贝克兰德变换

• 发表时间: 2004
• 期刊: Int. Math. Res. Notice 2004・1
• 作者: M.Inaba;K.Iwasaki;M.Saito
• 通讯作者: M.Saito

Hironobu Kimura: "Generalized Airy functions and the cohomological intersection numbers"Contemporary Mathematics. Fundamental direction. Proceedings of the sattelite conference of ICM 2002. 2. 83-94 (2003)

Hironobu Kimura：“广义艾里函数和上同调交集数”当代数学。

Masaki Suzuki, Nobuhiko Tahara, Kyoichi Takano: "Hierarchy of B"acklund transformation groups of the Painlev'e systems"Journal of the Mathematical Society of Japan. (To appear).

Masaki Suzuki、Nobuhiko Tahara、Kyoichi Takano：“Hierarchy of B”acklund conversion groups of the Painleve systems”Journal of the Mathematical Society of Japan.（待发表）。