Categorical Representation Theory on an Algebraic Surface
Categorical Representation Theory on an Algebraic Surface
批准号:
22K13889
负责人:
Zhao Yu
金额:
$1.83万
依托单位:
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
项目状态:
未结题
起止时间:
2022-04-01 至 2025-03-31
中文摘要
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。
英文摘要
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.
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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]
通讯作者:
Yu Zhao