Rank 2 Sheaves on Toric 3-Folds: Classical and Virtual Counts

Let 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 M-T with constant reflexive hulls are isomorphic to products of 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 article, we introduce a new type of combinatorics called double box configurations, which can be used to compute the generating function Z(q) of topological Euler characteristics of M (summing over all c(3)). The combinatorics is solved using the double dimer model in a companion article. This leads to explicit formulae for Z(q) involving the MacMahon function. In the virtual part of the article, 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 Z(q) up to signs.