2014 Fiscal Year Research-status Report
| Project/Area Number |
24540033
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| Research Institution | Tohoku University |
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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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| Project Period (FY) |
2012-04-01 – 2016-03-31
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| Keywords | semi-algebraic set / logarithmic form / Cauchy formula |
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| Outline of Annual Research Achievements |
(Logarithmic integrals related to mixed Tate motives) (1) A semi-algebraic set of complex n-space is said to be admissible if the intersection with any face F has dimension at most n-2cod(F). Using a theorem on resolution of singularities, we proved that the integral of a logarithmic holomorphic n-form on a compact admissible semi-algebraic set is absolutely convergent. (2) We also developed the theory of intersection of a semi-algebraic set in complex n-space and a coordinate hyperplane, with the help of cap product in Borel-Moore homology.
(3) With regard to the above notion of intersection and integration over a
log arithmic form, we have rigorously formulated and proved a Cauchy formula (generalizing the classical Cauchy formula for residue of a holomorphic function) for admissible compact semi-algebraic sets in complex n-space.
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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
主な結果（積分の収束の定理，Cauchyの積分公式）の証明は改善を加えた結果，理解しやすいものになった．
また，交叉理論については，Borel-Moore homologyとcap積を用いると記述が統一的になり，証明も簡略されることが分かった．
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| Strategy for Future Research Activity |
We will apply the results thus obtained as a basic tool for construction the
Hodge realization of a mixed Tate motive (defined via cycle complexes).(1)Construct the universal iterated extensions of Tate objects in the category
of mixed Hodge structures, and related them to the bar complex arising from the Hodge complexes of Tate objects (in the category of mixed Hodge structures).(2) Relate the bar complex arising from the Hodge complexes of Tate objects to the bar complex arising from the cycle complex (this is achieved via the cycle
map).
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| Causes of Carryover |
予定されていた２回の研究集会および共同研究が，外国人参加者の都合でその時期にできなくなったため．
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| Expenditure Plan for Carryover Budget |
研究集会を２回組織し，当該研究の研究者の発表の場とする．
また１月に招待されているインドにおける研究集会の参加の費用にあてる．
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