2000 Fiscal Year Final Research Report Summary
Topological theory of local systems having irregular singularities
| Project/Area Number |
09440065
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| Research Category |
Grant-in-Aid for Scientific Research (B).
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
解析学
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| Research Institution | Kumamoto University |
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Principal Investigator |
HARAOKA Yoshishige Faculty of Science, Kumamoto University Associate Professor, 理学部, 助教授 (30208665)
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| Co-Investigator(Kenkyū-buntansha) |
OHWAKI Shin-ichi Faculty of Science, Kumamoto University Professor, 理学部, 教授 (50040506)
KOHNO Mitsuhiko Faculty of Science, Kumamoto University Professor, 理学部, 教授 (30027370)
KIMURA Hironobu Graduate School of Science and Technology, Kumamoto University Professor, 大学院・自然科学研究科, 教授 (40161575)
KAMIMURA Yutaka Tokyo University of Fisheries Faculty of Fisheries Associate Professor, 水産学部, 助教授 (50134854)
TAKADA Yoshikazu Faculty of Engineering, Kumamoto University Professor, 工学部, 教授 (70114098)
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| Project Period (FY) |
1997 – 2000
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| Keywords | hypergeometric function / confluent hypergeometric function / irrgular singularity / twisted homology / twisted cohomology / rigid local system / arrangement of hyperplanes / accessory parameter |
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| Research Abstract |
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.
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