Abstract
We define functors on the derived category of the moduli space ℳ of stable sheaves on a smooth projective surface (under Assumptions A and S below), and prove that these functors satisfy certain commutation relations. These relations allow us to prove that the given functors induce an action of the elliptic Hall algebra on the \(K\)-theory of the moduli space ℳ, thus generalizing the action studied by Nakajima, Grojnowski and Baranovsky in cohomology.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Arinkin, A. Căldăraru and M. Hablicsek, Formality of derived intersections and the orbifold HKR isomorphism, J. Algebra, 540 (2019), 100–120.
V. Baranovsky, Moduli of sheaves on surfaces and action of the oscillator algebra, J. Differ. Geom., 55 (2000), 193–227.
R. Basili, On commuting varieties of upper triangular matrices, Commun. Algebra, 45 (2017), 1533–1541.
I. Burban and O. Schiffmann, On the Hall algebra of an elliptic curve I, Duke Math. J., 161 (2012), 1171–1231.
E. Carlsson, E. Gorsky and A. Mellit, The \({\mathbf {A}}_{q,t}\) algebra and parabolic flag Hilbert schemes, Math. Ann., 376 (2020), 1303–1336.
S. Cautis and A. Licata, Heisenberg categorification and Hilbert schemes, Duke Math. J., 161 (2012), 2469–2547.
I. Cherednik, Double Affine Hecke Algebras, Cambridge University Press, Cambridge, 2005. xii+434 pp. ISBN 0-521-60918-6.
J. Ding and K. Iohara, Generalization of Drinfeld quantum affine algebras, Lett. Math. Phys., 41 (1997), 181–193.
G. Ellingsrud and M. Lehn, Irreducibility of the punctual quotient scheme of a surface, Ark. Mat., 37 (1999), 245–254.
G. Ellingsrud and A. Strømme, On the homology of the Hilbert schemes on points in the plane, Invent. Math., 87 (1987), 343–352.
B. Feigin and A. Tsymbaliuk, Heisenberg action in the equivariant \(K\)-theory of Hilbert schemes via Shuffle Algebra, Kyoto J. Math., 51 (2011), 831–854.
B. Feigin, K. Hashizume, A. Hoshino, J. Shiraishi and S. Yanagida, A commutative algebra on degenerate \({\mathbf {C}}{\mathbf {P}}^{1}\) and MacDonald polynomials, J. Math. Phys., 50, 095215 (2009).
L. Fu and M.-T. Nguyen, Orbifold products for higher K-theory and motivic cohomology, Doc. Math., 24 (2019), 1769–1810.
E. Gorsky and A. Neguţ, The trace of the affine Hecke category, 2201.07144.
E. Gorsky, A. Neguţ and J. Rasmussen, Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology, Adv. Math., 378, 107542 (2021). 115 pp.
L. Göttsche, The Betti numbers of the Hilbert scheme of points on a smooth projective surface, Math. Ann., 286 (1990), 193–207.
I. Grojnowski, Instantons and affine algebras I. The Hilbert scheme and vertex operators, Math. Res. Lett., 3 (1996), 275–291.
D. Huybrechts and M. Lehn, The Geometry of Moduli Spaces of Sheaves, 2nd ed., Cambridge University Press, Cambridge, 2010. ISBN 978-0-521-13420-0.
A. Krug, Symmetric quotient stacks and Heisenberg actions, Math. Z., 288 (2018), 11–22.
D. Maulik and A. Neguţ, Lehn’s formula in Chow and conjectures of Beauville and Voisin, J. Inst. Math. Jussieu, 21 (2022), 933–971. https://doi.org/10.1017/S1474748020000377.
D. Maulik and A. Okounkov, Quantum Groups and Quantum Cohomology, Astérisque, vol. 408, 2019. ix+209 pp. ISBN 978-2-85629-900-5.
K. Miki, A \((q, \gamma )\) analog of the \(W_{1+\infty}\) algebra, J. Math. Phys., 48, 123520 (2007).
H. Nakajima, Heisenberg algebra and Hilbert schemes of points on projective surfaces, Ann. Math., 145 (1997), 379–388.
A. Neguţ, The shuffle algebra revisited, Int. Math. Res. Not., 22 (2014), 6242–6275.
A. Neguţ, Moduli of flags of sheaves and their \(K\)-theory, Algebr. Geom., 2 (2015), 19–43.
A. Neguţ, The \(q\)-AGT-W relations via shuffle algebras, Commun. Math. Phys., 358 (2018), 101–170.
A. Neguţ, Shuffle algebras associated to surfaces, Sel. Math. (N.S.), 25, 36 (2019). 57 pp.
A. Neguţ, \(W\)-algebras associated to surfaces, Proc. Lond. Math. Soc. (2022). https://doi.org/10.1112/plms.12435.
A. Neguţ, AGT relations for sheaves on surfaces, 1711.00390.
O. Schiffmann, Drinfeld realization of the elliptic Hall algebra, J. Algebraic Comb., 35 (2012), 237–262.
O. Schiffmann and E. Vasserot, The elliptic Hall algebra and the equivariant \(K\)-theory of the Hilbert scheme of \({\mathbf {A}}^{2}\), Duke Math. J., 162 (2013), 279–366.
O. Schiffmann and E. Vasserot, Cherednik algebras, \(W\)-algebras and the equivariant cohomology of the moduli space of instantons on \(\mathbf {A}^{2}\), Publ. Math. Inst. Hautes Études Sci., 118 (2013), 213–342.
B. Toën, Proper local complete intersection morphisms preserve perfect complexes, 1210.2827.
C. Voisin, On the Chow ring of certain algebraic hyper-Kahler manifolds, Pure Appl. Math. Q., 4 (2008), 613–649.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
This article is published under an open access license. Please check the 'Copyright Information' section either on this page or in the PDF for details of this license and what re-use is permitted. If your intended use exceeds what is permitted by the license or if you are unable to locate the licence and re-use information, please contact the Rights and Permissions team.
About this article
Cite this article
Neguţ, A. Hecke correspondences for smooth moduli spaces of sheaves. Publ.math.IHES 135, 337–418 (2022). https://doi.org/10.1007/s10240-022-00131-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10240-022-00131-1