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

Ho, Quoc P; Li, Penghui

doi:10.2140/gt.2025.29.2463

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Subscriptions

ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

Author index

To appear

Other MSP journals

Graded character sheaves, HOMFLY-PT homology, and Hilbert schemes of points on $\mathbb{C}^2$

Quoc P Ho and Penghui Li

Geometry & Topology 29 (2025) 2463–2546

DOI: 10.2140/gt.2025.29.2463

Abstract

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 $H_{W}^{g r} = {C h}^{b} ({S B i m}_{W})$ 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 ${H i l b}_{n} (ℂ^{2})$ of points on $ℂ^{2}$ (equivariant with respect to the natural $ℂ^{*} \times ℂ^{*}$ 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 ${H i l b}_{n} (ℂ^{2})$ . As an important computational input, we also establish a conjecture of Gorsky, Hogancamp, and Wedrich on the formality of the Hochschild homology of $H_{W}^{g r}$ .

Keywords

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

Mathematical Subject Classification

Primary: 14A30, 14F08, 20C15, 57K18

Secondary: 18N25

References

Publication

Received: 28 August 2023

Revised: 10 September 2024

Accepted: 5 October 2024

Published: 14 August 2025

Proposed: Ciprian Manolescu

Seconded: Benson Farb, David Fisher

Authors

Quoc P Ho
Department of Mathematics The Hong Kong University of Science and Technology Clear Water Bay Hong Kong

Penghui Li
Yau Mathematical Sciences Center Tsinghua University Beijing China

Open Access made possible by participating institutions via Subscribe to Open.

Volume 29, issue 5 (2025)