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

一般超几何函数（GHF）和扭曲上同调群的结构 GL(N) 正则元素的中心化子的共轭类由 N 的划分确定。GHF 是格拉斯曼流形 Gr( 上的多值函数。 n+1, N) 定义为中心化器通用覆盖群的字符的 Radon 变换 对于整数 q > 0，考虑分区 (q, 1,...,1) N。为了阐明一般超几何系统解空间的结构，我们计算了相关的 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 与 n 不同时，第 k 个上同调群消失，第 n 个上同调群的秩为 (N-2)!/n!(N-n-2)! 我们明确地使用该群给出了基础。舒尔当考虑格拉斯曼流形 Gr( 上的广义反自对偶杨-米尔斯方程 (GASDYM) 时，我们开始从扭量理论的角度进行这一推广的研究。 2，N），其解通过 Ward 对应对应于扭量空间 PN-1 上的全纯向量丛，当限制为扭量线时，该全纯向量束是微不足道的。 GL(N) 的阿贝尔子群如 1) 所示，并考虑其在扭量空间 PN-1 上的自然作用，此外，我们假设 H 的作用可以提升到对应于 GASYM 方程的解的全纯向量丛。这个作用决定了束上的平坦连接，当限制于扭量线时，这种平坦连接描述了常微分方程的单性保留变形，我们给出了平坦连接的显式形式，通过这个显式表达式，我们清楚了类比。从这个角度出发，我们统一导出了一般的施莱辛格系统，即对应于Painlevel方程（包括退化方程）的Gr(2,N)上的微分方程。与 H 相关的 Weyl 群描述了一般施莱辛格系统的一组对称性，由此，我们可以对 Painlevel 方程中的参数数量在退化后减少这一事实给出群论的理解。我们还可以构建一般施莱辛格系统的退化（汇合）过程。

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。

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。

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

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）。