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

一般的超几何函数（GHF）和旋转共同体学组的结构。 GHF的分区确定了常规元素的中央元素的共轭类别，这是Grassmannian歧管GR（N+1，N）的多价值函数，该功能定义为中央覆盖群的特征的ra。对于整数Q> 0，请考虑N。N。为了阐明一般超测量系统的解决方案空间的结构，我们计算了相关的DE RHAM共同学组的级别和基础。当GHF由n维积分给出时，我们发现K-TH的同胞组消失了与n不同的k，而n-th colomology群的等级为（n-2）！/n！（n-n-n-n-2）！。我们为该组提供了使用Schur函数的基础。Schlesinger系统及其概括。我们从扭曲理论的角度开始研究了这种概括的研究。当人们考虑将军……更liz的抗自我双阳米尔斯方程（gasdym）上的格拉斯曼尼亚歧管GR（2，n）时，其溶液对应于扭曲器空间PN-1上的holomorthic vector Bundle通过病房对应关系，当限制到Twistor线时，这是微不足道的。令H为1中的GL（N）的最大ABELIAN亚组，并考虑其对Twistor空间PN-1的自然作用。此外，我们假设H的作用可以将其提升到对应于气体方程溶液的圆锥形载体束。然后，此动作决定了捆绑包上的平坦连接，并且在二线线时，这种平坦的连接描述了一种单轨道保留ODE的变形。我们给出了平坦连接的明确形式，通过这种明确的表达方式，我们明确了与GHF定义的类比。我们以统一的方式得出了一般的施莱辛格系统，从这个角度来看，GR（2，n）上的微分方程对应于Painleve方程（包括退化的方程）。我们还明确指出，与H相关的Weyl组描述了一组一般施莱辛格系统的对称性。通过这种情况，我们可以为群体理论理解，即变性后，painleve方程中的参数数量逐渐消失。我们还可以为一般的Schlesinger系统构建变性过程（汇合）。较少的

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）。