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

在这项研究中，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.合流构造圈：我们已经构造了一维圈。推广了合流过程中极限收敛的定理。2.刚性局部系统的截面：我们用构造性的方法证明了刚性局部系统截面的存在性。这一结果与Katz等人的结果不同。我们还分析了Katz积分表示，并注意到它们是Selberg型积分的特例。3.Stokes乘子的求值：利用刚性局部系统截面的积分表示，计算拉普拉斯变换得到的系统的Stokes乘子。4.局部系统在高维空间上的可靠性：我们给出了高维空间上局部系统的例子，其刚性指数随坐标变化而变化。这意味着对于高维情形，我们需要一些内在的定义。5.应用于微分方程形变理论：构造刚性局部系统所用的运算可以应用于研究微分方程组的形变。得到了一些结果。

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.