| 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 / 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 |
The principal investigator A. Bondal developed the theory of noncommutative resolutions in the geometric and algebraic contexts. Algebraic resolutions were constructed via generalized noncommutative differential calculus for a collection of algebras and homomorphisms between them. Noncommutative resolutions for non-normal algebraic varieties were constructed in collaboration with co-Investigator S. Okawa by means of the universal fibered and cofibered squares.
Co-investigator M. Kapranov (in collaboration with V. Schechtman) explicitly described perverse sheaves on the Ran space of the complex line. The categorical interpretations of this construction was explored.
Co-investigator S. Okawa proved that the category of coherent right modules over a smooth noncommutative surface finite over its center is equivalent to a direct summand of the category of coherent sheaves of a smooth tame algebraic stack, which is canonically associated to it, thereby confirming that such nc surfaces are noncommutative geometric schemes in the sense of Orlov. The paper on this results is submitted to the electronic arxive.
As a byproduct of his research on sheaf-theoretic quantization co-investigator T.Kawasaki found a sheaf-theoretic version of the bounding cochain, which was known before in the context of Floer theory.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
COVID restrictions did not allow to implement the travel plans and to invite people collaborators from oversees.
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| 今後の研究の推進方策 |
We will develop the study of noncommutative resolutions via exact categories. We expect to obtain various schobers which govern the families of noncommutative resolutions of a category by means of varying exact structures on the category and considering the corresponding derived categories. We plan to apply this technique to constructing schobers of resolutions for finite dimensional algebras, as well as schobers of singularities of varieties.
We will work on the conjectural relationship between affine Weyl groups and polarizations of nc del Pezzo surfaces, and phantoms and quasi-phantom categories in positive characteristics.
Our new formalism of nonexact sheaf quantization at least enables us to formulate the sheaf theoretic side of the expected correspondence between sheaf quantization and Floer theory. We plan to explore the Floer side and the correspondence in the next year.
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