2016 Fiscal Year Research-status Report
| Project/Area Number |
16K05087
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| Research Institution | Tohoku University |
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Principal Investigator |
花村 昌樹 東北大学, 理学研究科, 教授 (60189587)
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Keywords | motive / semi-algebraic set / Cauchy formula / Borel-Moore homology |
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| Outline of Annual Research Achievements |
(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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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
(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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| Strategy for Future Research Activity |
(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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| Causes of Carryover |
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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| Expenditure Plan for Carryover Budget |
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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