| 研究課題/領域番号 |
23K20205
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| 補助金の研究課題番号 |
20H01794 (2020-2023)
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| 研究種目 |
基盤研究(B)
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| 配分区分 | 基金 (2024)
補助金 (2020-2023) |
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| 応募区分 | 一般 |
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| 審査区分 |
小区分11010:代数学関連
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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, 客員上級科学研究員 (00726408)
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| 研究分担者 |
大川 新之介 大阪大学, 大学院理学研究科, 准教授 (60646909)
桑垣 樹 京都大学, 理学研究科, 准教授 (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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| 研究課題ステータス |
交付 (2024年度)
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| 配分額 *注記 |
17,030千円 (直接経費: 13,100千円、間接経費: 3,930千円)
2024年度: 3,250千円 (直接経費: 2,500千円、間接経費: 750千円)
2023年度: 3,380千円 (直接経費: 2,600千円、間接経費: 780千円)
2022年度: 3,250千円 (直接経費: 2,500千円、間接経費: 750千円)
2021年度: 3,380千円 (直接経費: 2,600千円、間接経費: 780千円)
2020年度: 3,770千円 (直接経費: 2,900千円、間接経費: 870千円)
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| キーワード | coherent sheaf / reflexive sheaf / derived category / NCR / schober / resolution / perverse sheaf / Fukaya category / schobers / Derived categories / Floor theory / noncommutative / resolutions / Schober / Derived category / Perverse sheaves / Minimal Model Program / Algebraic varieties / complex manifold / Chern classes / quantization / spherical functor |
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| 研究開始時の研究の概要 |
In the first half of the academic year, we will concentrate on approaches to constructing similar schobers from the perspective of different areas of mathematics: algebraic geometry, representation theory, symplectic geometry, homological algebra.
In the second half of the academic year, we will look on the applications of construction to various problems, such as the classification of algebras of finite global dimensions, derived equivalence of of partial noncommutative resolutions, description of Fukaya categories and their relation to quantization problems.
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| 研究実績の概要 |
A criterion for a finite dimensional algebra to be quasi-hereditary is given in terms of a pair of exceptional collections of modules over the algebra.
Noncommutative resolutions of reduced curves were studied via fibered-cofibered squares of curves. Nocommutative resolutions of some finite length schemes were constructed via null categories of birational morphisms of smooth surfaces. It was investigated how to reconstruct a normal surface from the category of reflexive sheaves on it.
A version of the Riemann-Hilbert correspondence in the presence of the Planck parameter is proven.
A twist-closed dg-enhancement for the category of restricted objects in the derived category of coherent sheaves on noncompact complex-analytic manifolds was constructed via dbar-superconnections.
For noncommutative surfaces which are finite over their centers Artin stacks were constructed which are Morita equivalent to the noncommutative surfaces up to taking direct summands.
A generalization of the concept of spherical functors, which was named N-spherical functor and which describes N-periodic semi-orthogonal decomposition was developed. This allowed us to categorify Euler's continuants in the theory of continued fractions.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
1: 当初の計画以上に進展している
理由
The progress of the work on the research project is good. Noncommutative resolutions for curves were investigated from the point of view of fibered-cofibered squares of curves. A new interesting class of resolutions for 0-dimensional schemes was constructed via the null-categories of birational morphisms of smooth surfaces. It is proven that every restricted object of the directed category of coherent sheaf on noncompact manifold allows a presentation via a dbar-superconnection, thus giving a way to construct moduli spaces of this kind of objects. It was shown that the compactified moduli space of weighted projective lines is endowed with the sheaf of abelian categories of finite global dimension.
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| 今後の研究の推進方策 |
We plan to arrange a workshop in June 2023, where we invite the members of the team, our collaborators and leading experts working on the subject of the research project to give talks on their research and to exchange knowledge within our group and with the experts. We plan to develop the study of noncommutative resolutions and relevant schobers for 0-dimensional schemes, curves and surfaces. We wilI construct microlocal categories over Novikov rings, which should be the sheaf-theoretic counterpart of Fukaya categories over Novikov rings. We will study how to compute Efimov's categorical punctured neighborhood. We will describe how to reconstruct the normal surface from the category of reflexive sheaves on it.
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