Stratification, exterior power structure, asymptotic behavior and connection coefficients for confluenthypergeometric functions

合流超几何函数的分层、外部幂结构、渐近行为和连接系数

• 批准号: 13440057
• 负责人: HARAOKA Yoshishige
• 金额: $ 3.9万
• 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: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-13440057/
• 关键词: hypergeometric function; rigid local system; arrangement of hyperplanes; deformation of differential equations; accessory parameter; confluence; 超幾何変数; rigid local system; 積分表示

In this research we planed to study confluent hypergeometric functions as deformations of weighted arrangements of hypersurfaces. In particular, we planed to analyse the behavior of confluent hypeigeometric functions via geometrical structures of the moduli space. We obtained the following results.1.Construction of cycles by confluence : We had already constructed 1-dimensional cycles. We generalized the theorem on convergence of the limit in the process of confluence. Also we tried to construct higher-dimensional cycles by confluence, and noticed that they should be constructed in the flag variety.2.Section of rigid local systems : We proved the existence of integral representations of sections of rigid local systems in a constructive way. The result is different from one by Katz et al. We also analysed Katz integral representations, and noticed that they are special cases of Selberg type integrals. We studied on the basis of the (co) homology.3.Evaluation of Stokes multipliers : By using integral representations of sections of rigid local systems, we evaluated the Stokes multipliers for the system obtaind by Laplace transform.4.Rigidity of local systems on higher dimensional spaces : We came to examples of local systems on higher dimensional spaces whose indexes of rigidity vary by coordinate changes. This implies that we need some intrinsic definition of the ridigity for the higher dimensional case.5.Application to the deformation theory of differential equations : The operations used in constructing rigid local systems can be applied to the study of deformations of differential equations. Some results are obtained.

在本研究中，我们计划研究合流超几何函数作为超曲面的加权排列的变形。特别地，我们计划通过模空间的几何结构来分析合流超几何函数的行为。我们得到了以下结果：1.用汇合构造圈：我们已经构造了一维圈。推广了合流过程中的极限收敛定理。我们还尝试了用汇流法构造高维循环，并注意到它们应该在旗变式中构造。2.刚性局部系统的截面：我们构造性地证明了刚性局部系统截面的积分表示的存在性。我们还分析了Katz积分表示，并注意到它们是Selberg型积分的特殊情况。3. Stokes乘子的计算：利用刚性局部系统截面的积分表示，计算了由拉普拉斯变换得到的系统的Stokes乘子。4.高维空间上局部系统的刚性：给出了高维空间上局部系统的刚性指数随坐标变化的例子。这意味着我们需要对高维情况下的刚性进行某种内在的定义。5.微分方程变形理论的应用：构造刚性局部系统所用的运算可以应用于微分方程变形的研究。得到了一些结果。

项目成果

Construction of rigid local systems and integral representation of their sections

刚性局部系统的构建及其部分的整体表示

• 期刊: Math.Nachr. (発表予定)
• 作者: Y.Haraoka

原岡喜重: "超幾何関数"朝倉書店. 197 (2002)

原冈义茂：《超几何函数》朝仓书店 197 (2002)。

Singular Fano compactification of C^3 (1)

C^3 的奇异 Fano 紧化 (1)

• 发表时间: 2004
• 期刊: Math.Z. 248
• 作者: M.Furushima

Singular Fano compactification of C^3(1)

C^3(1) 的奇异 Fano 紧化

• 发表时间: 2004
• 期刊: Math.Z. 248
• 作者: M.Furushima

Singular Fano compactifications of C^3(I)

C^3(I) 的奇异 Fano 紧化

• 发表时间: 2004
• 期刊: Math.Z. 248
• 作者: Masashi Misawa;Masashi Misawa;Yoshihiro Tonegawa;Masashi Misawa;Masashi Misawa;三沢 正史;利根川 吉広;Masashi Misawa;Masashi Misawa;Mikio Furushima
• 通讯作者: Mikio Furushima