2021 Fiscal Year Research-status Report
Study of mixed motives by the bar construction
| Project/Area Number |
17K05157
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| Research Institution | University of Tsukuba |
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Principal Investigator |
木村 健一郎 筑波大学, 数理物質系, 講師 (50292496)
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| Project Period (FY) |
2017-04-01 – 2023-03-31
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| Keywords | admissible chains |
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| Outline of Annual Research Achievements |
We tried to find applications of the admissible semi-algebraic chains on smooth complex varieties. We found that the relative cohomology groups of a quasi-projective smooth variety relative to a normal crossing divisor can be described in terms of the admissible chains. We also found that the duality pairing between de Rham cohomology goups and the singular cohomology groups can be described in terms of the integral of logarithmic differential forms on the admissible chains. As an application, we showed that the Abel-Jacobi map on higher Chow cycles can be described as currents constructed from admissible chains. We will try to use this description for some cycles which do not necessarily meet the faces properly.
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| Current Status of Research Progress |
Current Status of Research Progress
3: Progress in research has been slightly delayed.
Reason
We tried to use our result to elliptic polylog cycles, but we face difficulty which arises from the fact that they do not meet the faces properly.
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| Strategy for Future Research Activity |
We will try to generalize the description of the Abel-Jacobi map to some cycles which do not meet the faces properly. As an application we will give a description of the Abel-Jacobi map for elliptic polylog cycles.
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| Causes of Carryover |
コロナ禍で出張ができないため、成果発表の機会が無かった。現在取り組んでいる一般化を完成させて、合わせて成果として発表するために予算が必要である。
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