2002 Fiscal Year Final Research Report Summary
Complex analysis on quasi-Abelian varieties
| Project/Area Number |
13640199
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Kyushu Sangyo University |
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Principal Investigator |
UMENO Takashi Kyushu Sangyo University, Faculty of Engineering, Professor, 工学部, 教授 (30098769)
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| Co-Investigator(Kenkyū-buntansha) |
FURUSHIMA Mikio Kumamoto University, Faculty of Science, Professor, 理学部, 教授 (00165482)
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| Project Period (FY) |
2001 – 2002
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| Keywords | quasi-Abelian variety / Abelian variety / period matrix / toroidal group / de Rham cohomology / ^^-__∂-cohomology |
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| Research Abstract |
1. We obtained all standard forms of period matrices for quasi-Abelian varieties. Using these forms, we got a brief proof of fibration theorems for quasi-Abelian varieties. Further, we have an another proof, which does not depend on the theory of harmonic integrals, of classical Riemann conditions for Abelian varieties.
2. The author gave a talk about the above results at the Fifth International Workshop on Real and Complex Analysis (October 2001, Hiroshima. University). He wrote the paper : Period matrices for quasi-Abelian Varieties, which includes de Rham cohomology of toroidal groups, period matrices and fibration theorems for quasi-Abelian varieties. This will be published in Japanese Journal of Mathematics vol.29-1(2003).
3. From the theorems of period matrices, we found the existence of quasi-Abelian varieties which have principal bundle structures over non-algebraic complex tori. Further we could construct many examples of toroidal groups which have no non-constant meromorphic fu
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nctions on them. This is an extension of a method of Siegel who constructed examples of complex tori which have no non-constant meromorphic functions on them. We could not construct these examples without the computers and the softwares which was brought by the Grant-in-Aid for Scientic Research. Because we needed an enormous amount of calculations of matrices including polynomials.
4. The author gave an lecture about the theory of period matrices as an invited talk at the conference of the mathematical society of Japan, March 2002. Further, he gave a talk about these examples obtained by the theorems of period matrices at the Sixth International Workshop on Real and Complex Analysis (December 2002, Hiroshima University). By these results, We think that the main aims of the research was obtained.
5. As above, we use computers as important tools for a study of mathematics. The method of using computers for mathematics which we have developed will be applicable for mathematical educations. The author gave a talk about a computerization of mathematical education at the conference of information processing education (October 2002, Tokyo University). We think these methods were also obtained by the research. Less
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Research Products
(10 results)
