2013 Fiscal Year Research-status Report
Project/Area Number |
24540033
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Research Institution | Tohoku University |
Principal Investigator |
花村 昌樹 東北大学, 理学(系)研究科(研究院), 教授 (60189587)
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Keywords | 混合モティーフ / Hodge structure / semi-algebraic set |
Research Abstract |
(Logarithmic integrals associated to mixed Tate motives) We studied integrals which appear as periods of mixed Tate motives. (1) For a semi-algebraic set S in complex n-dimensional space and a differential form having poles along co -ordinate hyperplanes, assuming a certain condition on the dimension of the intersection of S and the pole divisor, we showed that the integral of the form on S absolutely converges. (2) We showed the Cauchy formula for semi-algebraic sets in complex n-space with respect to its intersection with coordinate hyper -planes. (3) We defined a complex of semi-algebraic chains of complex n-space, and showed that it calculates the homology of the n-space. (Relative algebraic correspondences and quasi DG categories) (1) Fix an algebraic variety S as a base. We defined the complex of algebraic correspondences between varieties over S; we showed that the class of varieties over S, together with the complex of algebraic correspondences constitutes a quasi DG category (a generalization of a DG category). (2) Given a quasi DG category C, we gave the construction of another quasi DG category C', whose associated homotopy category has a natural structure of a triangulated category.
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Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
(Log integrals appearing in the theory of mixed motives.)The result that the convergence of a log integral follows from a certain condition on the dimension of intersection of the semi-algebraic set with the pole divisor is noteworthy. The proof turned out to be highly non-trivial, relying on the Hironaka-Spivakovsky method of resolution as well as results on semi-algebraic sets such as Lojasiewicz inequality. (Relative algebraic correspondences and quasi DG categories) The notion of quasi DG category is noteworthy and will turn out to be useful in various contexts. We have documented the theory of producing a triangulated quasi DG category out of a quasi DG category.
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Strategy for Future Research Activity |
(Theory of Hodge realization of mixed Tate motives. ) We intend to achieve: (1) Construction of the mixed Hodge structure associated to a mixed Tate motive. We will do this via Deligne cohomology. (2) Another construction of the mixed Hodge structure associated to a mixed tate motive, using the log integral method we have established (the theory of integration over semi-algebraic sets mentioned in the summary of achievements). (3) Proof that the two methods give the same structure.
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Expenditure Plans for the Next FY Research Funding |
研究集会を2回行う計画であったが,主要な参加者の複数が国内に不在の時期にあたり,それが遂行できなくなった. 研究集会を今年度,2回程度,またワークショップを今年度に数回おこなう.
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Research Products
(2 results)