2000 Fiscal Year Final Research Report Summary
Algebraic structures and complex structures of Quasi-Abelian varieties
| Project/Area Number |
11640195
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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 | Kyshu Sangyo University |
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Principal Investigator |
UMENO Takashi Faculty of Engineering, Kyshu Sangyo University Professor, 工学部, 教授 (30098769)
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| Project Period (FY) |
1999 – 2000
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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 proved that any n-dimensional Quasi-Abelian variety of type q and kind 0 is a C^<*n-q>-principal bundle over an Abelian variety of dimension q, using our standard form of the period matrix. The proof is found in the paper : Period Matrices of Quasi-Abelian Varieties, Bulletin of the Faculty of Engineering, Kyushu Sangyo University 36(1999), 283-286.
2. Further we showed that for every Quasi-Abelian variety C^n/Γ of type q and kind s, there exists an associated Quasi-Abelian variety C^n/Γ of type q and kind 0 satisfying the following conditions :
(1) C^n/Γ is a covering manifold of C^n/Γ_0 and
(2) The ample Riemann form which defines a Quasi-Abelian structure on C^n/Γ is induced by C^n/Γ_0. Combining this and the result of 1, we get the fibration theorems of Quasi-Abelian varieties which was obtained by A.Andreotti-F.Gherardelli and Y.Abe.
3. The above results were announced in the Third International Workshop on Real and Complex Analysis (June, Ewha Womans University, Korea), Autumn Conference of Mathematical Society of Japan (September, Kyoto University) and the Fourth International Workshop on Real and Complex Analysis (October, Hiroshima University). Papers including these results were published in the Proceedings of the Third International Workshop on Real and Complex Analysis (2000), 75-82 and the Proceedings of the Fourth International Workshop on Real and Complex Analysis (2000), 81-87 .
4. From the result of 2, we can construct every Quasi-Abelian variety from the associated Quasi-Abelian variety of kind 0. We have many examples of them which will be published later, using the computers and the softwares which was brought by the Grant-in-Aid for Scientic Research.
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