2019 Fiscal Year Final Research Report
Study of DG triangulated categories related to mixed motives
| Project/Area Number |
16K05087
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Tohoku University |
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Principal Investigator |
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| Project Period (FY) |
2016-04-01 – 2020-03-31
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| Keywords | 混合モティーフ / 混合Hodge構造 / 代数的サイクル / 周期積分 |
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| Outline of Final Research Achievements |
For a semi-algebraic set A in a complex affine space and a differential form with logarithmic poles along the coordinate hyperplanes, satisfying a certain condition on the intersection of A and the coordinate hype-rplanes, we showed convergence of the integral of the form on A. When A has dimension m+1 and the form has degree m, we showed that the integral of the residue of the form along H (a coordinate hyperplane) on the intersection of A and H, coincides with the integral of the original form on the topological boundary of A. Using these, we can associate to each mixed Tate motif a mixed Hodge structure. This generalizes and makes precise the construction made by Bloch-Kriz, under certain conditions. (Work with Kenichiro Kimura and Tomohide Terasoma).
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| Free Research Field |
代数幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
混合Tateモティーフ理論はデデキントゼータ値に関するZagier予想や多重ゼータ値などとも関係し，基本的な重要性を持つが，そのHodge構造との関係について厳密かつ理解しやすい理論展開をすることが望まれる．混合Tateモティーフの圏論の一つは代数的サイクルを用いた構成で，BlochとKrizが与えた．Hodge構造との関係についてもBloch-Krizはその先鞭をつけたが，条件付きの部分や正しくない部分もあり，研究者に理解されているとは言いにくい．我々の研究は厳密で理解でき，使うことができる方向を目指し成果を得ている．
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