| Project/Area Number |
18K13405
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 11020:Geometry-related
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| Research Institution | Kyoto University (2022)
Osaka University (2020-2021)
The University of Tokyo (2018-2019) |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2023-03-31
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| Project Status |
Completed (Fiscal Year 2022)
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| Budget Amount *help |
¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2021: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2020: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2019: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
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| Keywords | 深谷圏 / シンプレクティック幾何 / 超局所層理論 / ミラー対称性 / RH対応 / WKB解析 / Riemann--Hilbert対応 / ホモロジー的ミラー対称性 / 完全WKB解析 / 超局所層 / リーマン・ヒルベルト対応 / 変形量子化 / 層量子化 / 超局所圏 / Riemann-Hilbert対応 / 超局所幾何 |
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| Outline of Final Research Achievements |
Homological mirror symmetry is a mysterious duality between symplectic geometry and algebraic geometry. Traditionally, the symplectic side is studied via Floer theory, which is a geometric analytic theory. Recent advances of algebraic analysis enables us to study the symplectic side through sheaf theory. In this study, we used the technology to prove homological mirror symmetry of toric varieties. Also, we also studied the technology itself, and found a relationship between WKB analysis and microlocal sheaf theory.
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| Academic Significance and Societal Importance of the Research Achievements |
トーリック多様体のミラー対称性を層理論を用いた証明は、（私自身のものとは限らない）さまざまな派生研究（偏屈圏のミラー対称性、トーリック退化・因子のミラー対称性など）を生み、学術的意義は大きかったと考える。また、超局所層理論をもちいたシンプレクティック幾何の研究は、量子化とFloer理論の関係を明確にする上での端緒となると考えられ、今後の発展への期待が大きい。
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