## Geometry & Topology

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

#### 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 $k∕n$. 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
Accepted: 20 April 2017
First available in Project Euclid: 1 February 2018

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

#### 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