2022 Fiscal Year Research-status Report
Categorical Representation Theory on an Algebraic Surface
| Project/Area Number |
22K13889
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| Research Institution | The University of Tokyo |
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Principal Investigator |
Zhao Yu 東京大学, カブリ数物連携宇宙研究機構, 特任研究員 (10928667)
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| Project Period (FY) |
2022-04-01 – 2025-03-31
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| Keywords | Nakajima quiver variety / derived ag / vanishing theorem / quantum loop algebra / quantum toroidal algebra |
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| Outline of Annual Research Achievements |
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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| Current Status of Research Progress |
Current Status of Research Progress
1: Research has progressed more than it was originally planned.
Reason
We revealed the surprising relation between the categorical representation theory and derived algebraic geometry. Moreover, in order to compute the categorical commutators of certain commutators of quantum loop and toroidal algebras, we introduced concepts from birational geometry, and established a Grauert-riemenschneider type vanishing theorem for derived algebraic geometry.Moreover, we introduced the discrepancy concept into derived algebraic geometry, and proved that the derived push-forward behaves very differently when the discrepancy is non-negative or negative. When the discrepancy is non-negative, the derived push-forward has properties similar to canonical singularities in birational algebraic geometry.
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| Strategy for Future Research Activity |
First we will establish a categorical comparison theorem for the derived blow-up of quasi-smooth derived schemes. It will verify a conjecture of myself when studying the birational geometry of nested quiver varieties.
Second we will establish a Beilinson type resolution of the diagonal of derived projectivization of a two term complex over a derived scheme. It will lead to another proof of the semi-orthogonal result of Jiang-Leung.
Finally, we will apply the categorical representation theory of quantum affine and toroidal algebras to the derived categories of certain moduli spaces, like the Kuznetsov component of fourfolds.
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| Causes of Carryover |
Due to the COVID-19 pandemic, many international conferences I supposed to take part in were delayed to 2023.
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