2007 Fiscal Year Final Research Report Summary
Finite dimensionality of Motives
| Project/Area Number |
18540033
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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 |
Algebra
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| Research Institution | Hiroshima University |
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Principal Investigator |
KIMURA Shun-ichi Hiroshima University, Graduate school of Science, Associate Professor (10284150)
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| Co-Investigator(Kenkyū-buntansha) |
KIMURA Kenichiro Tsukuba University, Graduate School of Pure and Applied Science, Research Assistant (50292496)
YAMAZAKI Takao Tohoku University, Graduate school of science, Associate Professor (00312794)
IKEDA Atsushi Osaka University, Graduate School of Science, COE Lecturer (40397617)
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| Project Period (FY) |
2006 – 2007
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| Keywords | Motives / Chow group / tensor category / Schur dimension / Bloch's conjecture |
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| Research Abstract |
There are 6 major results during this period.
1. Consider a rigid tensor category, and let V be a Schur finite object. Then there exists a smallest Young diagram which kills V, say λ. namely μ kills V, then μ contains λ. In this way, we can define the Schur dimension of V to be λ.
2. Generalizing the notion of finite dimensionality of an object in a tensor category, we defined the notion of finite dimensional morphism. An object V is finite dimensional if and only if the identity morphism of V is finite dimensional. In this way, we obtain a new and convenient criterion for finite dimensionality.
3. When the motive of X is finite dimensional, then its motivic zeta, formal sum of the Chow motives of the symmetric products of X, becomes rational. In this joint work with Professor Javier Elizondo, we studied what happens if we replace the symmetric products by Chow varieties. To our surprise, the formal sum is not rational, even when X is the projective plane.
4. In a joint work with Prefessor Jacob Murre, we proved the uniqueness of the Picard Functor, assuming that the Chow motive Pic(X) is finite dimensional.
5. Professor Uwe Jannsen prove that if the cycle map is injective, then it is bijective. We generalized this result to Chow motives.
6. In this joint work with Professor Nobuyoshi Takahashi and Professor Kenichiro Kimura, we studied, if the motivic zeta of Schur finite object V becomes rational. The answer depends on the Schur dimension of V. If the Schur dimension is a hook, then the motivic zeta is rational. In general, the motivic zeta is always determinantally ration], but not necessarily uniformly rational.
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Research Products
(72 results)
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[Presentation] On the second Abel-Jacobi map2008
Author(s)
Kenichiro Kimura
Organizer
Cycles, motives and Shimura varieties
Place of Presentation
Tata Institute of Fundamental Research, Bombay.
Year and Date
2008-01-10
Description
「研究成果報告書概要(和文)」より
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[Presentation] モチーフの有限次元性2006
Author(s)
木村俊一
Organizer
日大特異点セミナー
Place of Presentation
日本大学理工学部
Year and Date
2006-05-15
Description
「研究成果報告書概要(和文)」より
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