Categorical Representation Theory on an Algebraic Surface

代数曲面上的分类表示论

• 批准号: 22K13889
• 负责人: Zhao Yu
• 金额: $ 1.83万
• 依托单位: The University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Early-Career Scientists
• 财政年份: 2022
• 资助国家: 日本
• 起止时间: 2022-04-01 至 2025-03-31
• 项目状态: 未结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-22K13889/
• 关键词: Nakajima quiver variety; derived ag; vanishing theorem; quantum loop algebra; quantum toroidal algebra; Representation Theory

We study the birational geometry of nested quiver varieties, through a new perspective of very recent developed theories of derived projectivizations and derived blow-ups. We prove that for any quiver (with or without loops), two kinds of nested quiver varieties are isomorphic, after derived blowing-up the diagonals, are isomorphic. It leads to a weak categorification of the quantum loop and toroidal algebras action on the Grothendieck group of Nakajima quiver varieties.After that, we also studied the desingulization theory of quasi-smooth derived schemes. We proved that for the characteristic 0 case, any quasi-smooth derived schemes (with an embedding into a smooth ambient variety), allows a procedure of desingulization by derived blowing up smooth centers. It allowed us to formulate a conjecture of K-theoretic virtual fundmanetal classes for any quasi-smooth derived schemes.This study revealed the surprising relation among the categorical representation theory, birational geometry and derived algebraic geometry. It allows us to compute more diagonal decomposition (K-theoretic, Chow groups or Fourier-Mukai transforms) of varieties, which is crucial to the study of geometry representation theory.

通过最近发展起来的派生射影和派生爆破理论的新视角，我们研究了嵌套箭图变种的双生几何。证明了对于任一箭图(有环或无环)，两类嵌套箭图簇同构，导出爆破对角线后，两类嵌套箭图簇同构。在此基础上，对Nakajima箭图的Grothendieck群上的量子环和环状代数作用进行了弱分类，并研究了拟光滑衍生格式的去奇化理论。我们证明了对于特征为0的情形，任何拟光滑的导出格式(嵌入到光滑的环境簇中)都允许通过爆破光滑中心来去奇异。这项研究揭示了范畴表示理论、二元几何和派生代数几何之间惊人的联系。它允许我们计算更多的簇的对角分解(K-理论、Chow群或Fourier-Mukai变换)，这对于几何表示理论的研究是至关重要的。

项目成果

A CATEGORICAL QUANTUM TOROIDAL ACTION ON THE HILBERT SCHEMES

• DOI: 10.1017/s1474748022000585
• 发表时间: 2020-09
• 期刊: Journal of the Institute of Mathematics of Jussieu
• 影响因子: 0.9
• 作者: Yu Zhao