| 研究課題/領域番号 |
20H01794
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| 配分区分 | 補助金 |
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| 研究機関 | 東京大学 |
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研究代表者 |
Bondal Alexey 東京大学, カブリ数物連携宇宙研究機構, 主任研究員 (00726408)
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| 研究分担者 |
大川 新之介 大阪大学, 理学研究科, 准教授 (60646909)
桑垣 樹 大阪大学, 理学研究科, 助教 (60814621)
KAPRANOV MIKHAIL 東京大学, カブリ数物連携宇宙研究機構, 教授 (90746017)
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| 研究期間 (年度) |
2020-04-01 – 2025-03-31
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| キーワード | coherent sheaf / derived category / complex manifold / Chern classes / quantization / spherical functor |
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| 研究実績の概要 |
The PI A. Bondal in collaboration with A. Rosly developed the theory of dbar-resolutions for objects of the derived category of coherent sheaves. It was proven that the homotopy category of dbar-superconnections is equivalent to the bounded derived category of coherent sheaves for smooth compact complex manifolds. This theory is applied to constructing Chern classes for coherent sheaves, proving that Chern classes have the expected Hodge type, and constructing Bott-Chern classes for sheaves. The paper on these result was submitted and is already accepted for publication. Co-I M. Kapranov (with T. Dyckerhoff and V. Schechtman) developed a generalization of the concept of spherical functors, called N-spherical, which describe N-periodic semi-orthogonal decomposition. This implies the categorification of Euler's continuants in the theory of continued fractions. Co-I S. Okawa proved a relative version of his previous theorem about the relations between semiorthogonal decompositions and canonical linear systems. As an application he proved that minimal surfaces of positive irregularity are semiorthogonally indecomposable, thereby giving further evidence to his conjecture that minimal surfaces are semiorthogonally decomposable if and only if the structure sheaf is acyclic. A paper with these results is posted on the electronic arXiv. Co-I T. Kuwagaki studied sheaf quantization and its application to symplectic geometry and Riemann-Hilbert correspondence. For sheaf quantization, T. Kuwagaki found a formalism of sheaf-theoretic study of non-exact Lagrangian submanifolds.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
3: やや遅れている
理由
The delay was due to COVID restrictions. It was not possible to travel and to invite collaborators. a certain amount of money from the research grant had to be carried over to the next fiscal year.
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| 今後の研究の推進方策 |
We plan to develop the theory of interrelation of the relative singularity categories and the null-categories for Auslander non-commutative resolutions of simple singularities and more general for singularities of finite Cohen-Macaulay type. As an application we expect to prove the Knoerrer periodicity for the null-categories of these singularities, and provide the description of the null-categories in any dimension. We plan to relate the formalism of N-spherical functors to the theory of cluster algebras and cluster categories, as Euler's continuants appear as multplicative moment maps in construction of cluster varieties parametrizing Stokes data for irregular local systems. We will work on projective geometry of noncommutative del Pezzo surfaces in noncommutative (weighted) projective spaces, a key point of which would be to understand the monodromy group of lines on nc surfaces. We will also work on the foundation of nc del Pezzo surfaces and their classifications from the point of view of helices and dg categories, starting with revisiting the case of nc projective planes. We will continue to study relationship between sheaf quantization and Floer theory. In the exact symplectic geometry, such results are obtained by Ganatra-Pardon-Shende recently. We would like to enhance such results in the realm of nonexact symplectic geometry. We plan to arrange a workshop where collaborators of the research supported by the grant will exchange their knowledge and current achievements.
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