Homology of Hilbert schemes of points on a locally planar curve

Abstract

Let C be a proper, integral, locally planar curve, and consider its Hilbert schemes of points C [n] . We define 4 creation/annihilation operators acting on the rational homology groups of these Hilbert schemes and show that the operators satisfy the relations of a Weyl algebra. The action of this algebra is similar to that defined by Grojnowski and Nakajima for a smooth surface.As a corollary, we compute the cohomology of C [n] in terms of the cohomology of the compactified Jacobian of C together with an auxiliary grading on the latter. This recovers and slightly strenghtens a formula recently obtained in a different way by Maulik and Yun and independently Migliorini and Shende.Here Q[2] denotes the space Q with homological degree 2. A very similar statement was recently shown by Maulik and Yun [MY14] and Migliorini and Shende [MS13]. See Section 1.5 for a discussion of how these papers relate to this one.

Hecke correspondences for Hilbert schemes of reducible locally planar curves

“…V is naturally a bigraded Q-vector space, graded by the number of points n and homological degree d. We denote by V n,d the (n, d)-graded piece of V . We define the following operators on V , following ideas of Rennemo [34] (and that originally go back to Nakajima and Grojnowski [27,17]). (1) Let c i ∈ C sm i be fixed smooth points and Proof.…”

“…By [34,Lemma 3.4], the intersection of the images of f i and g i is codimension one in X i . Consider a point(Z ⊂ Z ∪ s i , Z) ∈ Im(f i ) ∩ Im(g i ), which can also be written as…”

“…Following Rennemo, we approach the problem of computing the homologies of the Hilbert schemes in question from the point of view of representation theory. In [34], a Weyl algebra in two variables acting on V ∶= ⊕ n≥0 H * (C [n] ) was constructed for integral locally planar curves, and V was described in terms of the representation theory of the Weyl algebra. When C has m irreducible components, we construct an algebra A acting on V , where A is an explicit subalgebra of the Weyl algebra in 2m variables.…”

Supersymmetric Wilson loops in two dimensions and duality

“…Operators of this kind has been considered in relation to theories with boundaries [67,68]. Also, correspondence of such boundary supersymmetric connections under Seiberg-like dualities has been studied in [69,70]. stage of this work was carried out at the Galileo Galilei Institute in Florence during the workshop Supersymmetric Quantum Field Theories in the Non-perturbative Regime.…”

Birational Chow–Künneth decompositions

“…Proposition 4.1 (Rennemo [7]). The correspondences µ ± (x 0 ) and µ ± (C) satisfy the commutation relations…”