Application of orbifold signature to singularities and singular fibers
| Project/Area Number |
24540048
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Tohoku Gakuin University |
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Principal Investigator |
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| Co-Investigator(Renkei-kenkyūsha) |
Konno Kazuhiro 大阪大学, 大学院理学研究科, 教授 (10186869)
Yoshikawa Ken-ichi 京都大学, 大学院理学研究科, 教授 (20242810)
Matsumoto Yukio 学習院大学, 理学部, 客員研究員 (20011637)
Ishida Hirotaka 宇部工業高等専門学校, 一般科, 准教授 (30435458)
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| Project Period (FY) |
2012-04-01 – 2016-03-31
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| Project Status |
Completed (Fiscal Year 2015)
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| Budget Amount *help |
¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000)
Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2012: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
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| Keywords | 特異点 / 符号数 / デデキンド和 / トーリック幾何 / 連分数 / オービフォルド / 特異ファイバー / 相互律 / 退化 / 局所不変量 / モジュライ空間 / モノドロミー / トーリック写像 / 商特異点 / 安定曲線 / リーマン面 / モジュライ / singularity / toric resolution / continued fraction / Fourier-Dedekind sum / Todd class / orbifold / 双有理変換 / 符号不足数 / オービフォールド |
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| Outline of Final Research Achievements |
First we generalized Zagier’s reciprocity for higer-dimensional Dedekind sum to the case where the given integers satisfy weaker conditions. Our method is geometrical in the sense that the comparison of the invariants of several cyclic quotient singularities induce the reciprocity. The main tool of the proof is the orbifold signature theorem with non-isolated fixed points locus.
Second, we extended Myerson-Holzapfel formula which expresses the classical Dedekind sum by means of continued fraction as follows: We define the notion of higher-dimensional continued fraction whose algorithm corresponds to the toric resolution process of Fujiki-Oka for cyclic quotient singularities. Then we expressed the three-dimensional Fourier-Dedekind sum with weight zero by means of this continued fraction. As a tool of the proof, we slightly extend the observation of Pommersheim for the Chow ring of simplicial toric varieties.
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Report
(5 results)
Research Products
(17 results)
