2023 Fiscal Year Final Research Report
Mordell-Weil Groups of elliptically-fibered Calabi-Yau manifolds
| Project/Area Number |
19K03427
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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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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | Chuo University |
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Principal Investigator |
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| Project Period (FY) |
2019-04-01 – 2024-03-31
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| Keywords | 楕円ファイブレーション / Mordell-Weil格子 / 楕円曲面 / Calabi-Yau多様体 / K3曲面 |
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| Outline of Final Research Achievements |
We attempted to extend Mordell-Weil lattice theory for elliptic surfaces to higher-dimensional elliptically fibered varieties. We defined a lattice structure for the Mordell-Weil group of certain elliptic threefolds by introducing an inner product using the notion of height, and determined the structure in some cases. In particular, we have introduced a lattice structure to the rank 6 Mordell-Weil group of certain rational elliptic threefolds, and have shown that it is isomorphic to the root lattice of type E6. We also studied some Calabi-Yau threefolds with Mordell-Weil groups of rank 9 or 10.
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| Free Research Field |
数論的代数幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
楕円曲面の理論は代数幾何，解析学，数論など数学の様々な分野だけでなく，理論物理学まで密接に関連して発展してきた大変興味深い研究対象である．楕円曲面の切断のなす群は，交点形式に由来する内積によりMordell-Weil格子と呼ばれる格子の構造を持ち，その構造は楕円曲面の幾何学的性質を調べるなかで重要な役割を果たし，様々な応用を持つ．理論物理学では，高次元のCalabi-Yau楕円多様体のMordell-Weil群には物理的な意味あり，高次元の楕円多様体のMordell-Weil群に格子の構造を導入する試みは，広い分野に波及する可能性をもつ意義ある研究である．
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