2023 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 | Categorification / Quantum groups / Derived geometry |
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| Outline of Annual Research Achievements |
The quantum toroidal algebra is an affinization of the q-Heisenberg algebra.
Schiffmann-Vasserot proved that the equivariant K-theory of the instanton moduli space over the affine plane is the Fock representation of the quantum toroidal algebra. It was geometrized and generalized by Negut to moduli space of sheaves over an algebraic surface.
The main contribution of this work depends on the rank of the moduli space: when the rank is 1, we adapt the singularity theory from minimal model program and proved that certain nested moduli spaces have rational singularities; for the case that r>1, we applied the derived algebraic geometry and considered a new forms of blow-up of varieties, which both have clear geometric picture and easy formula in computing the derived push-forward of exceptional divisors.
We lifted the generators and relations of Negut operators to the derived category of coherent sheaves and obtained a weak categorification of the action by computing the categorical commutator of the positive and negative part of this algebra. We applied the derived algebraic geometry and considered a new forms of blow-up of varieties, which both have clear geometric picture and easy formula in computing the derived push-forward of exceptional divisors.
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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
We observed that the new blow-up theory of the nested quiver variety is the classical shadow of the derived blow-up theory of Hekking. Particularly, we studied two kinds of (derived) fiber products and showed that their derived blow-up along the diagonal is isomorphic.
While we laid down the foundation for the categorification of Smirnov-Okounkov Yangians actions, the real and striking contribution is that many important but very different phenomenon in algebraic geometry and representation theory, including Thomason's localization and excess intersection theorems, Vakil-Zinger's desingularization, Kuznetsov's homological projective duality and Khovanov's diagrammatic approach in categorification are unified under the framework of quasi-smooth derived schemes and derived blow-up theory.
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| Strategy for Future Research Activity |
We will generalize the weak categorification for the Smirnov-Okounkov Yangian actions on the equivariant K-theory of any quiver varieties. It includes many more examples including quantum loop, toroidal, and moreover Borcherd algebras. Strong categorification needs more work. Unlike the categorification of quantum groups by Khovanov-Lauda-Rouqier, the categorification of quantum loop algebras can only exist in dg or infty-categories.
In the joint work with Qingyuan Jiang, we will reveal a surprising relation between the moduli space of stable objects of the Kuznetsov component and representation theory. Particularly, we will construct an action of the Hall algebra of the category of coherent sheaves on a higher genus curve on the cohomology and Chow group of the moduli space.
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| Causes of Carryover |
As a result of proper execution in accordance with the research plan. I will visit Beijing for the conferences of geometric represenation theory in July.
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