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

Alexei Oblomkov, Jacob Rasmussen and Vivek Shende

Appendix: Eugene Gorsky

Geometry & Topology 22 (2018) 645–691

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.

plane curve, Hilbert scheme, Khovanov homology
Mathematical Subject Classification 2010
Primary: 14H20, 14N35
Secondary: 57M27
Received: 14 September 2012
Accepted: 20 April 2017
Published: 16 January 2018
Proposed: Peter Ozsváth
Seconded: Lothar Göttsche, Gang Tian
Alexei Oblomkov
Department of Mathematics and Statistics
University of Massachusetts
Amherst, MA
United States
Jacob Rasmussen
University of Cambridge
United Kingdom
Vivek Shende
Department of Mathematics
University of California
Berkeley, CA
United States
Eugene Gorsky
Department of Mathematics
University of California
Davis, CA
United States