Coherent sheaves on surfaces, COHAs and deformed $W_{1+\infty}$-algebras
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866916043598856192 |
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| author | Mellit, Anton Minets, Alexandre Schiffmann, Olivier Vasserot, Eric |
| author_facet | Mellit, Anton Minets, Alexandre Schiffmann, Olivier Vasserot, Eric |
| contents | We compute the cohomological Hall algebra of zero-dimensional sheaves on an arbitrary smooth quasi-projective surface $S$ with pure cohomology, deriving an explicit presentation by generators and relations. When $S$ has trivial canonical bundle, this COHA is isomorphic to the enveloping algebra of deformed trigonometric $W_{1+\infty}$-algebra associated to the ring $H^*(S,\mathbb{Q})$. We also define a double of this COHA, show that it acts on the homology of various moduli stacks of sheaves on $S$ and explicitly describe this action on the products of tautological classes. Examples include Hilbert schemes of points on surfaces, the moduli stack of Higgs bundles on a smooth projective curve and the moduli stack of $1$-dimensional sheaves on a $K3$ surface in an ample class. The double COHA is shown to contain Nakajima's Heisenberg algebra, as well as a copy of the Virasoro algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_13415 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Coherent sheaves on surfaces, COHAs and deformed $W_{1+\infty}$-algebras Mellit, Anton Minets, Alexandre Schiffmann, Olivier Vasserot, Eric Algebraic Geometry Representation Theory We compute the cohomological Hall algebra of zero-dimensional sheaves on an arbitrary smooth quasi-projective surface $S$ with pure cohomology, deriving an explicit presentation by generators and relations. When $S$ has trivial canonical bundle, this COHA is isomorphic to the enveloping algebra of deformed trigonometric $W_{1+\infty}$-algebra associated to the ring $H^*(S,\mathbb{Q})$. We also define a double of this COHA, show that it acts on the homology of various moduli stacks of sheaves on $S$ and explicitly describe this action on the products of tautological classes. Examples include Hilbert schemes of points on surfaces, the moduli stack of Higgs bundles on a smooth projective curve and the moduli stack of $1$-dimensional sheaves on a $K3$ surface in an ample class. The double COHA is shown to contain Nakajima's Heisenberg algebra, as well as a copy of the Virasoro algebra. |
| title | Coherent sheaves on surfaces, COHAs and deformed $W_{1+\infty}$-algebras |
| topic | Algebraic Geometry Representation Theory |
| url | https://arxiv.org/abs/2311.13415 |