Hecke correspondences for smooth moduli spaces of sheaves

Abstract

We define functors on the derived category of the moduli space M of stable sheaves on a smooth projective surface (under Assumptions A and S below), and prove that these functors satisfy certain relations. These relations allow us to prove that the given functors induce an action of the elliptic Hall algebra on the K-theory of the moduli space M, thus generalizing the action studied by Nakajima, Grojnowski and Baranovsky in cohomology.

Moduli Space of Sheaves and Categorified Commutator of Functors

“…In Theorem 7.1, we prove a stronger result by constructing explicit morphisms e i f j → Rι * (f j e i ) when i + j < 0 and morphisms Rι * (f j e i ) → e i f j when i + j > 0 such that the cones are filtered by combinations of symmetric and wedge product of the universal sheaf U k and its derived dual. Combining with Theorem 1.1 of [22], we obtain a weak categorification of the quantum toroidal algebra action on the Grothendieck groups of moduli space of stable sheaves.…”

“…Description of our results. Schiffmann-Vasserot [27,26] (when S = P 2 , M is the moduli space of framed sheaves) and Negut ¸ [20,22] (for all algebraic surfaces S) constructed the quantum toroidal algebra U q1,q2 ( gl 1 ) action on the Grothendieck group of moduli space of stable sheaves over an algebraic surface S. The main result of this paper is a weak categorification of the above action.…”

“…More commutator relations are imposed in this presentation, which we do not know how to categorify right now. See [22] for more details.…”

“…The main reason we assume Assumption S is that: Proposition 1.7 (Proposition 2.8 of [22]). Under Assumption S, the moduli space M k is smooth of dimension const + 2rk, where const only depends on S, H, r, c 1 .…”

“…Strategy of the proof. We adapt the framework of Negut ¸ [20,23,21,22] for the moduli space of nested sheaves. The construction when r = 1 was made in the author's paper [39].…”