Stable pairs and the HOMFLY polynomial

Abstract

Given a planar curve singularity, we prove a conjecture of Oblomkov-Shende, relating the geometry of its Hilbert scheme of points to the HOMFLY polynomial of the associated algebraic link. More generally, we prove an extension of this conjecture, due to Diaconescu-Hua-Soibelman, relating stable pair invariants on the conifold to the colored HOMFLY polynomial of the algebraic link. Our proof uses wall-crossing techniques to prove a blowup identity on the algebrogeometric side. We prove a matching identity for the colored HOM-FLY polynomials of a link using skein-theoretic techniques.

Euler characteristics of Hilbert schemes of points on simple surface singularities

“…. , P k } of planar singularities however, we have the beautiful conjecture of Oblomkov and Shende [39], proved by Maulik [33], which takes the form (2) Z C (q) = (1 − q) −χ(C) k j=1 Z (Pi,C) (q).…”

Moduli of map germs, Thom polynomials and the Green–Griffiths conjecture

“…If G k = J Hilbert schemes of points on surfaces form a central object of geometry and representation theory and have a huge literature (see for example [39,9]). Recently many interesting connections between Hilbert schemes of points on planar curve singularities and the topology of their links have been discovered [45,46,47,32]. However, much less is known about Hilbert schemes or punctual Hilbert schemes on higher dimensional manifolds.…”

BPS States, Torus Links and Wild Character Varieties

“…In string theory these conjectures have been shown to follow from large N duality for conifold transitions in [24,23]. The physical derivation leads to a colored refined generalization of these conjectures formulated in [23] and proven by Maulik in [59] for the unrefined case. In the present context, these conjectures relate the refined stable pair theory of Y ξ to refined colored invariants of (ℓ, (n − 2)ℓ)-torus links.…”