Topological theory of local systems having irregular singularities

研究合流超几何函数积分表示的拓扑结构，阐明其拓扑结构与函数的解析性质之间的关系。当一维空间上定义了与积分对应的局部系时，我们构造了扭(余)同调群的良好基。正则奇异超几何函数的扭黎曼周期关系可推广到合流超几何函数的扭黎曼周期关系.在环上积分所给函数的扭环与渐近行为之间的关系上，我们注意到，如果在合流过程中存在一个具有纯渐近行为的环，则合流后的环也将具有纯行为.对于定义在多维空间上的局部系统，我们利用外部幂结构构造了扭(余)同调群的基。我们注意到，由消失圈给出的函数的渐近行为可以用对应于超平面退化排列的函数来刻画，对应于超曲面退化排列的汇合超几何函数的研究将是非常重要的，它们将与刚性局部系统和GKZ超几何函数密切相关。

The aim of this research was to study the topological structure of integral representations of confluent hypergeometric functions, and to clarify the relation between the topological structure and the analytic behavior of the functions.When the local system corresponding to the integral is defined over a 1-dimensional space, we have constructed good bases of twisted (co) homology group. Since we have constructed them by confluence, the twisted Riemann's period relation for regular singular hypergeometric functions can be extended to ones for confluent hypergeometric functions.On the relation between twisted cycles and asymptotic behaviors of functions given by the integral over the cycles, we noticed that, if a cycle which give a pure asymptotic behavior suivives in the process of confluence, the cycle after confluence also will give a pure behavior. Using this understanding, we can evaluate several connection coefficients and global behaviors for confluent hypergeometric functions.For local systems defined over multi-dimensional spaces, we have constructed bases of twisted (co) homology groups by using the exterior power structure. We noticed that the asymptotic behavior of a function given by a vanishing cycles can be discribed by functions corresponding to degenerate arrangements of hyperplanes.The study of confluent hypergeometric functions corresponding to degenerate arrangements of hypersurfaces will be substantial in future, and they will be closely related to rigid local systems and GKZ hypergeometric functions.