Categorical Representation Theory on an Algebraic Surface

我们通过最近发展的衍生投影和衍生膨胀理论的新视角，研究了嵌套颤振变种的二元几何。证明了对于任意套颤器（带环或不带环），两类套颤器变体是同构的，在导出对角线鼓化后，是同构的。给出了中岛颤振群上的量子环和环面代数作用的弱分类。在此基础上，我们还研究了拟光滑导出格式的去微分理论。我们证明了对于特征为0的情况，任何拟光滑的导出格式（嵌入到光滑的环境变量中），都允许通过导出的膨胀光滑中心进行去分辨。它允许我们对任何拟光滑的派生格式给出k理论虚基类的一个猜想。本研究揭示了范畴表征理论与两族几何和派生代数几何之间的惊人关系。它允许我们计算变体的更多对角分解（k -理论，Chow群或Fourier-Mukai变换），这对几何表示理论的研究至关重要。

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.