Categorical Representation Theory on an Algebraic Surface

我们研究双有理几何的嵌套类簇，通过一个新的角度非常最近发展的理论，派生的投影和派生爆破。证明了对任意的环（有环或无环），两种套环簇同构，导出对角爆破后同构。在此基础上，我们还研究了拟光滑导方案的去奇异性理论。我们证明了，对于特征0的情况下，任何准光滑的衍生计划（嵌入到一个光滑的环境品种），允许一个程序的desingulization衍生爆破光滑中心。这一研究揭示了范畴表示理论、双有理几何和导出代数几何之间令人惊讶的联系。它允许我们计算更多的对角分解（K理论，周群或傅立叶-向井变换）的品种，这是至关重要的几何表示理论的研究。

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.