2023 Fiscal Year Final Research Report
Representation theory of homotopy algebras and geometry
| Project/Area Number |
18K03293
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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 | Chiba University |
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Principal Investigator |
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| Project Period (FY) |
2018-04-01 – 2024-03-31
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| Keywords | ホモトピー代数 / ミラー対称性 / 導来圏 |
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| Outline of Final Research Achievements |
Homological mirror symmetry conjecture states that the derived category of coherent sheaves on a complex manifold and the derived category of Fukaya category of the mirror dual symplectic manifold are equivalent to each other as triangulated categories. We discuss the case where a complex manifold is a toric manifold and propose a formulation of a version of homological mirror symmetry based on SYZ torus fibrations.
We in particular show explicitly this version of homological mirror symmetry when a toric manifold is a complex projective plane, etc. The derived category of coherent sheaves on a toric manifold is known to have a full exceptional collection, which implies that the derived category is generated by a directed A-infinity category. Thus, our discussions as above are interesting, too, in the sense that we obtain many examples of triangulated categotries generated by directed A-infinity categories from geometry.
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| Free Research Field |
代数的位相幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
ホモロジー的ミラー対称性は，シンプレクティック多様体と複素多様体という異なる２つの幾何の上で定まる三角圏の同値性を主張するものである。この一見異なる幾何学の間に対応があることが興味深く，現在でもホモロジー的ミラー対称性が成り立つような様々な例について議論されている。一方で，なぜそれが成り立つか，という問いに関して決定的な結果は今のところ知られていない．現在この問の解決に一番近いと思われるのがSYZトーラス束によるミラー対の構成に基づく議論であるが，この方向性では解決すべき主張の厳密な証明が難しい状況にある．本研究ではこれを複素側がトーリック多様体に限定した場合に解決する方法を提案している．
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