Graded character sheaves, HOMFLY-PT homology, and Hilbert schemes of points on ℂ2

Ho, Quoc P¹ ; Li, Penghui²

¹Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
²Yau Mathematical Sciences Center, Tsinghua University, Beijing, China

Geometry & topology, Tome 29 (2025) no. 5, pp. 2463-2546

Résumé

Using a geometric argument building on our new theory of graded sheaves, we compute the categorical trace and Drinfel’d center of the (graded) finite Hecke category HWgr = Chb(SBimW) in terms of the category of (graded) unipotent character sheaves, upgrading results of Ben-Zvi and Nadler, and Bezrukavnikov, Finkelberg, and Ostrik. In type A, we relate the categorical trace to the category of 2-periodic coherent sheaves on the Hilbert schemes Hilbn(ℂ2) of points on ℂ2 (equivariant with respect to the natural ℂ∗× ℂ∗ action), yielding a proof of (a 2-periodized version of) a conjecture of Gorsky, Neguţ, and Rasmussen which relates HOMFLY-PT link homology and the spaces of global sections of certain coherent sheaves on Hilbn(ℂ2). As an important computational input, we also establish a conjecture of Gorsky, Hogancamp, and Wedrich on the formality of the Hochschild homology of HWgr.

Détail

DOI : 10.2140/gt.2025.29.2463

Keywords: categorical trace, Drinfel'd center, character sheaves, Hilbert scheme of points, Hecke categories, Soergel bimodules, Khovanov–Rozansky triply graded link homology

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