Applications of algebro-analytic method in algebraic geometry
| Project/Area Number |
17540023
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
SAITO Morihiko Kyoto University, Research Institute for Mathematical Sciences, Associate Professor, 数理解析研究所, 助教授 (10186968)
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| Project Period (FY) |
2005 – 2006
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| Project Status |
Completed (Fiscal Year 2006)
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| Budget Amount *help |
¥2,100,000 (Direct Cost: ¥2,100,000)
Fiscal Year 2006: ¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 2005: ¥1,400,000 (Direct Cost: ¥1,400,000)
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| Keywords | Chow-Kunneth decomposition / b-function / hyperplane arrangement / multiplier ideal / Lefschetz-Verdier trace formula / pole order filtration / ホッジ・フィルトレーション / 青本予想 / レフシェッツ公式 / 交叉コホモロジー / 跳躍係数 / 非特性超平面切断 |
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| Research Abstract |
Concerning the injectivity of the refined cycle maps that was mentioned in the research proposal, no good results were obtained so that the target of the research was changed to Murre's conjecture on the Chow- Kunneth decomposition. We proved this in the case the level of the cohomology is at most 1, and generalized it to the case of a morphism of smooth varieties whose singular points are discrete and whose target is a curve.
We found a relatively simple method to compute the b-function of a hyperplane arrangement by combining a generalization of Malgrange's theorem with a solution of Aomoto conjecture due to Esnault, Schechtman, Viehweg. Using this, we can calculate by hand the b-function in the case the dimension is 3 and the degree is at most 8. As for the multiplier ideals of a hyperplane arrangement, we proved a conjecture of Mustata that the jumping coefficients are combinatorial invariants.
We showed with Dimca, Maisonobe and Torrelli that the multiplier ideals of a hypersurface are compatible with the restriction to a noncharacteristic hyperplane section, and deduced some results on the spectrum.
After discussions with Hanamura on the algebraic cycle classes in intersection cohomology, we proved a relation between the intersection cohomology and the weight filtration of the cohomology, generalizing a result of Weber.
We showed with Dimca that the Hodge and pole order filtrations on the local cohomology differ for sufficiently general projective hypersurfaces, and proved a conjecture of Wotzlaw related to this in the case there are not so many singular points.
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Report
(3 results)
Research Products
(3 results)
