2023 Fiscal Year Final Research Report
D-critical birational geometry and categorification of Donaldson-Thomas invariants
| Project/Area Number |
19H01779
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Review Section |
Basic Section 11010:Algebra-related
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| Research Institution | The University of Tokyo |
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Principal Investigator |
Toda Yukinobu 東京大学, カブリ数物連携宇宙研究機構, 教授 (20503882)
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| Project Period (FY) |
2019-04-01 – 2024-03-31
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| Keywords | Donaldson-Thomas不変量 / 連接層の導来圏 / 行列因子化 / 圏論的Donaldson-Thomas理論 / D-臨界的双有理幾何学 / 準BPS圏 / 圏論的壁超え公式 |
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| Outline of Final Research Achievements |
The moduli spaces of stable objects on Calabi-Yau 3-folds admit d-critical structures introduced by Joyce. We introduced the notion of d-critical flips and flops between them, which are regarded as virtual birational maps. Moreover we proved that these virtual birational transformations appear as wall-crossing of stability conditions which are important in enumerative geometry. We also introduced the notion of DT categories for local surfaces which categorify Donaldson-Thomas invariants, and formulated d-critical D/K conjecture which describes the behavior under d-critical flips and flops. We also proved several properties on categorical wall-crossing of DT categories.
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| Free Research Field |
代数幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
3次元カラビヤウ多様体上の曲線や安定層を数え上げるDonaldson-Thomas不変量は数学・物理双方にとって重要な研究課題である。従来の研究によって安定性条件の壁超え公式を用いることでそれらの種々の公式が導かれることが知られていた。今回の研究でそれらの公式の背後にある双有理幾何的・表現論的理解を、Donaldson-Thomas不変量を圏論化する「DT圏」と双有理幾何学の仮想版である「D-臨界的双有理幾何学」を確立することで明らかにした。これらは新しい概念であり、今後の更なる発展も期待される。
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