Categorical Representation Theory on an Algebraic Surface

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。

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.