Rank 2 sheaves on toric 3-folds: classical and virtual counts

TL;DR

This paper studies rank 2 sheaves on toric 3-folds, describing fixed loci, introducing combinatorics for counting invariants, and relating classical and virtual counts via explicit formulas involving the MacMahon function.

Contribution

It introduces double box configurations for computing sheaf moduli counts and connects classical Euler characteristics with virtual invariants on toric Calabi-Yau 3-folds.

Findings

Explicit formulas for generating functions involving the MacMahon function.

Description of fixed loci as products of projective lines.

Connection between classical and virtual counts via localization.

Abstract

Let $\mathcal{M}$ be the moduli space of rank 2 stable torsion free sheaves with Chern classes $c_i$ on a smooth 3-fold $X$. When $X$ is toric with torus $T$, we describe the $T$-fixed locus of the moduli space. Connected components of $\mathcal{M}^T$ with constant reflexive hulls are isomorphic to products of $\mathbb{P}^1$. We mainly consider such connected components, which typically arise for any $c_1$, "low values" of $c_2$, and arbitrary $c_3$. In the classical part of the paper, we introduce a new type of combinatorics called double box configurations, which can be used to compute the generating function $\mathsf{Z}(q)$ of topological Euler characteristics of $\mathcal{M}$ (summing over all $c_3$). The combinatorics is solved using the double dimer model in a companion paper. This leads to explicit formulae for $\mathsf{Z}(q)$ involving the MacMahon function. In the virtual part of the paper, we define Donaldson-Thomas type invariants of toric Calabi-Yau 3-folds by virtual localization. The contribution to the invariant of an individual connected component of the $T$-fixed locus is in general not equal to its signed Euler characteristic due to $T$-fixed obstructions. Nevertheless, the generating function of all invariants is given by $\mathsf{Z}(q)$ up to signs.