| Project/Area Number |
18K03313
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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 11020:Geometry-related
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| Research Institution | Rikkyo University |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2024-03-31
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| Project Status |
Completed (Fiscal Year 2023)
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| Budget Amount *help |
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2022: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2021: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2020: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2019: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
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| Keywords | 変形理論 / 正則曲線 / 代数曲線 / モース理論 / 写像の変形理論 / 複素幾何学 / シンプレクティック幾何学 |
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| Outline of Final Research Achievements |
I conducted research on holomorphic curves on complex manifolds and related objects. In particular, I developed a method to determine the obstructions to deforming singular curves on complex surfaces through local calculations. As an application of this method, I proved a correspondence between holomorphic curves on Abelian surfaces and tropical curves on real 2-dimensional tori, which was a long-standing problem. On the other hand, by investigating gauge theory on 2-dimensional complex tori, I proved that as a limit of Hermitian-Yang-Mills connections on complex tori, a Lagrangian submanifold on the mirror torus naturally correspond, and I partially proved the mirror symmetry conjecture related to D-branes.
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| Academic Significance and Societal Importance of the Research Achievements |
以前知られていた手法では扱いが難しい対象について, 新しい手法を開発することにより研究を可能にした。具体的には, 計算が難しい障害がある場合の変形理論について, 障害の計算を局所的な計算に帰着させることにより, 長年未解決であった問題の解決に役立てた。また, これも扱いが難しい, 横断正則性が成り立たない状況でのゲージ理論について, 新たな手法を開発することで研究を進め, ミラー対称性予想の一部を証明した。
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