| 研究課題/領域番号 |
21K03197
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| 研究種目 |
基盤研究(C)
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| 配分区分 | 基金 |
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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 |
Sala Francesco 東京大学, カブリ数物連携宇宙研究機構, 客員准科学研究員 (60800555)
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| 研究期間 (年度) |
2021-04-01 – 2026-03-31
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| 研究課題ステータス |
交付 (2023年度)
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| 配分額 *注記 |
4,290千円 (直接経費: 3,300千円、間接経費: 990千円)
2025年度: 520千円 (直接経費: 400千円、間接経費: 120千円)
2024年度: 2,210千円 (直接経費: 1,700千円、間接経費: 510千円)
2023年度: 520千円 (直接経費: 400千円、間接経費: 120千円)
2022年度: 520千円 (直接経費: 400千円、間接経費: 120千円)
2021年度: 520千円 (直接経費: 400千円、間接経費: 120千円)
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| キーワード | Hall algebras / COHAs / Categorification / t-structures / Stability conditions / Calabi-Yau completions / Yangians / Quantum groups / Motivic Hall algebras / PT stable pairs / Space curve singularity / Quantum Groups / Hall Algebras / Higgs bundles / Flat bundles / stable pairs |
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| 研究開始時の研究の概要 |
The project stands at the crossroads between representation theory and geometry. It aims at discovering new algebraic structures (Lie algebras associated with curves) which should encode geometric properties of (moduli) spaces appearing in the so-called Riemann-Hilbert and the non-abelian Hodge correspondences for curves. We expect that this theory of Lie algebras associated with curves will unlock a new striking connection between geometry, algebra, and physics, which needs to be investigated further.
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| 研究実績の概要 |
During FY2023, the preprint [arXiv:2207.08926] underwent a significant revision. The revision includes the construction of the categorified, K-theoretical, and cohomological Hall algebra associated to a t-structure on a finite type dg-category, satisfying certain natural conditions. This new framework yields a multitude of novel examples. For instance, we can now construct categorified, K-theoretical, and cohomological Hall algebras associated to the underlying t-structure of a stability condition on a Kuznetsov component (of a smooth cubic 4fold, etc.).
Furthermore, Diaconescu, Porta, Schiffmann, Vasserot, and I are in the final stages of a project concerning the cohomological Hall algebra of coherent sheaves on a surface supported set-theoretically on a fixed curve. We are establishing a PBW-type theorem for such algebras. Additionally, when we restrict ourselves to minimal resolutions of a type ADE singularity, we can establish a connection between the corresponding COHA and the affine Yangian of the same type.
Finally, I started a project with Luis Alvarez-Consul about moduli spaces of objects on a 2-Calabi-Yau completion of a category of framed quiver representations with valued in a fixed dg-category. When the quiver is A_2 and the target category is the category of coherent sheaves on a curve, the corresponding category admits a torsion pair whose torsion object are Higgs sheaves on the curve. We anticipate that the moduli space of stable torsion-free objects of this category will serve as the curve analog of Nakajima quiver varieties.
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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
I have established a framework enabling the construction of categorified Hall algebras associated with t-structures arising from stability conditions on 2CY categories, advancing the part about categorified Yangians.
One of the anticipated outcomes of the nearly completed project with Diaconescu, Porta, Schiffmann, and Vasserot is a description of the Dolbeault COHA of the projective line in terms of the Yangian of affine type A_1.
During a visit to Kavli IPMU in March 2024, I initiated a project with Diaconescu, Porta, and Yu Zhao aimed at studing Hecke operators associated to modifications along divisors. This is an initial step toward understanding the Dolbeault COHA of a curve of positive genus.
Finally, I have found the right candidate for the curve analog of Nakajima quiver varieties.
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| 今後の研究の推進方策 |
During FY 2024, I am at completing the revision of [arXiv:2207.08926] and the project with Diaconescu, Porta, Schiffmann, and Vasserot.
I plan to complete the project with Diaconescu, Porta, and Yu Zhao about Hecke operators associated to modifications along effective divisors. More precisely, our operators act on the K-theory of certain moduli stacks of vectors bundles on a smooth projective surface satisfying certain conditions. When the divisor is an affine ADE configuration of (-2)-rational curves, we obtain a geometric realization of the corresponding quantum toroidal algebra. I will compute relations between these operators when the divisor is a smooth projective curve of positive genus.
Finally, I will study the moduli stacks and spaces arising from the project with Alvarez-Consul.
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