Geometry & Topology

The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link

Alexei Oblomkov, Jacob Rasmussen, and Vivek Shende

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Abstract

We conjecture an expression for the dimensions of the Khovanov–Rozansky HOMFLY homology groups of the link of a plane curve singularity in terms of the weight polynomials of Hilbert schemes of points scheme-theoretically supported on the singularity. The conjecture specializes to our previous conjecture (2012) relating the HOMFLY polynomial to the Euler numbers of the same spaces upon setting t=1. By generalizing results of Piontkowski on the structure of compactified Jacobians to the case of Hilbert schemes of points, we give an explicit prediction of the HOMFLY homology of a (k,n) torus knot as a certain sum over diagrams.

The Hilbert scheme series corresponding to the summand of the HOMFLY homology with minimal “a” grading can be recovered from the perverse filtration on the cohomology of the compactified Jacobian. In the case of (k,n) torus knots, this space furnishes the unique finite-dimensional simple representation of the rational spherical Cherednik algebra with central character kn. Up to a conjectural identification of the perverse filtration with a previously introduced filtration, the work of Haiman and Gordon and Stafford gives formulas for the Hilbert scheme series when k=mn+1.

Article information

Source
Geom. Topol., Volume 22, Number 2 (2018), 645-691.

Dates
Received: 14 September 2012
Accepted: 20 April 2017
First available in Project Euclid: 1 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.gt/1517454106

Digital Object Identifier
doi:10.2140/gt.2018.22.645

Mathematical Reviews number (MathSciNet)
MR3748677

Zentralblatt MATH identifier
06828597

Subjects
Primary: 14H20: Singularities, local rings [See also 13Hxx, 14B05] 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 57M27: Invariants of knots and 3-manifolds

Keywords
plane curve Hilbert scheme Khovanov homology

Citation

Oblomkov, Alexei; Rasmussen, Jacob; Shende, Vivek. The Hilbert scheme of a plane curve singularity and the HOMFLY homology of its link. Geom. Topol. 22 (2018), no. 2, 645--691. doi:10.2140/gt.2018.22.645. https://projecteuclid.org/euclid.gt/1517454106


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