| 研究課題/領域番号 |
16K05087
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| 研究機関 | 東北大学 |
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研究代表者 |
花村 昌樹 東北大学, 理学研究科, 教授 (60189587)
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| 研究期間 (年度) |
2016-04-01 – 2019-03-31
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| キーワード | motive / semi-algebraic set / Cauchy formula / Borel-Moore homology |
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| 研究実績の概要 |
(Borel-Moore homology and cap product operations) (a) We defined cap product operation on locally finite singular homology of a topological space. (b) We established comparison isomorphism between locally finite singular homology and Borel-Moore homology (introduced by Borel and Moore, using sheaf theory), and showed that cap product is compatible with the isomorphism. (c) When the space has a triangulation so that it may be viewed as a simplicial complex, we proved that the locally finite singular homology is isomorphic with (infinite) simplicial homology, and it is compatible with cap product. (d) We developed intersection theory of semi-algebraic sets in affine n-space.
(Hodge realization of mixed Tate motives via period integrals) Previously we have studied integration
of logarithmic forms on semi-algebraic sets; in particular we gave a sufficient geometric condition for absolute convergence, and proved a version of the Cauchy formula. Using these, one can construct a model for the Hodge complex of affine n-spaces. Then we examined the construction of the Hodge realization functor (via the period integrals) from the category of mixed Tate motives. Previously this was done under a certain hypothesis.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
(1) Regarding Borel-Moore homology, we completed a thorough exposition on the theory of cap product operation, comparing three homology theories-- sheaf theoretic Borel-Moore, locally finite
singular homology, and simplicial homology. This not only serves our purpose of intersection theory
of semi-algebraic sets, but also fills gaps in the literature where the assertion is taken for granted.
(2) The Hodge realization functor via integration was considered by Bloch and Kriz only under some
hypothese; we have fully examined this hypothesis.
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| 今後の研究の推進方策 |
(1) We will further generalize the Cauchy formula to the case of a pair of normal crossing divisor on a smooth complex variety. (So far we did it only for affine n-space and with coordinate hyperplanes.) Using this, for any smooth variety construct a model of Hodge complex in which the Betti part is the complex of topological chains.
(2) We will prepare a text on the mixed Tate categories associated with a graded DGA. In this framework, we re-examine the category of mixed Tate motives and mixed Tate Hodge structures.
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| 次年度使用額が生じた理由 |
予定していた研究打ち合わせが，４回，学内の急な業務のため，できなくなったため．
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| 次年度使用額の使用計画 |
適切な時期に，昨年度できなかった研究打ち合わせの分を実施する．
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