Abstract: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.
| Comments: | v2: final version; v1: 68 pages |
| Subjects: | Algebraic Geometry (math.AG); Representation Theory (math.RT) |
| Report number: | MPIM-Bonn-2023 |
| Cite as: | arXiv:2311.13415 [math.AG] |
| (or arXiv:2311.13415v2 [math.AG] for this version) | |
| https://doi.org/10.48550/arXiv.2311.13415 arXiv-issued DOI via DataCite |
From: Alexandre Minets [view email]
[v1]
Wed, 22 Nov 2023 14:25:26 UTC (77 KB)
[v2]
Mon, 25 May 2026 10:29:18 UTC (79 KB)
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