Oblomkov–Rasmussen–Shende (ORS) conjecture relating Hilbert schemes and link homology

Establish the Oblomkov–Rasmussen–Shende conjecture asserting that for a planar curve singularity C at the origin with associated algebraic link L, the a=0 part of the Khovanov–Rozansky homology HHH(L) equals the direct sum over k and n of the cohomology groups of the punctual Hilbert schemes of C, namely HHH^{a=0}(L) = ⊕_{k,n≥0} H^k(Hilb^n(C,0)).

Background

Given a singular plane curve C⊂C^2 with a planar singularity at the origin, intersecting C with a small S^3 defines an algebraic link L. The punctual Hilbert schemes Hilb^n(C,0) capture local geometric data of the singularity, while the Khovanov–Rozansky homology HHH(L) is a topological invariant of L.

The ORS conjecture proposes a precise equivalence between these two realms by identifying the a=0 part of HHH(L) with the direct sum over n and k of H^k(Hilb^n(C,0)). While verified in several special cases, the conjecture remains unproved in general and is a central open problem at the interface of algebraic geometry and low-dimensional topology.