2002 Fiscal Year Final Research Report Summary
Theory of motives and algebraic cycles
| Project/Area Number |
12440009
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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 |
Algebra
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| Research Institution | Kyushu University |
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Principal Investigator |
HANAMURA Masaki Kyushu University, Graduate School of Mathematics, Ass. Prof., 大学院・数理学研究院, 助教授 (60189587)
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| Co-Investigator(Kenkyū-buntansha) |
SATO Eiichi Kyushu University, Graduate School of Mathematics, Prof., 大学院・数理学研究院, 教授 (10112278)
YOSHIDA Masaaki Kyushu University, Graduate School of Mathematics, Prof., 大学院・数理学研究院, 教授 (30030787)
KANEKO Masanobu Kyushu University, Graduate School of Mathematics, Prof., 大学院・数理学研究院, 教授 (70202017)
KIMURA Shun-ichi Hiroshima University, Facility of Science, Lecturer, 理学部, 講師 (10284150)
SAITO Shuji Nagoya University, School of Polymathematics, Prof., 大学院・多元数理科学研究科, 教授 (50153804)
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| Project Period (FY) |
2000 – 2002
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| Keywords | motif / decomposition theorem / modular varieties |
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| Research Abstract |
1. Motives of varieties: Let D(k) be the category of mixed motives over a field k. We produced a functor from the category of quasi-projective varieties into D(k). The construction uses the method of cubical hyperresolution.
2. Motivic decomposition theorem: It is of interest to formulate and prove the motivic analogue of the topological decomposition theorem (of Beilinson, Bernstein and Deligne). In the case of the universal family of abelian varieties over the Hilbert modular variety, we showed the existence of the expected motivic decomposition, and deduced from it the Grothendieck-Murre conjecture for the fiber variety.
3. Homology correspondence at chain level: It is well-known to consider homological correspondences and their compositions. We considered this at the chain level. Namely we gave a complex of abelian groups which gives cohomology, and produced the composition map as a map of complexes. This construction is applied to produce the cohomology realization functor from mixed motives.
4. Mixed motivic sheaves: We sketched the construction of the triangulated category of mixed motives over a quasi-projective variety.
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