Higher rank sheaves on threefolds and functional equations

Abstract

We consider the moduli space of stable torsion free sheaves of any rank on a smooth projective threefold. The singularity set of a torsion free sheaf is the locus where the sheaf is not locally free. On a threefold it has dimension ≤1. We consider the open subset of moduli space consisting of sheaves with empty or 0-dimensional singularity set. For fixed Chern classes c1,c2 and summing over c3, we show that the generating function of topological Euler characteristics of these open subsets equals a power of the MacMahon function times a Laurent polynomial. This Laurent polynomial is invariant under q↔q−1 (upon replacing c1↔−c1). For some choices of c1,c2 these open subsets equal the entire moduli space. The proof involves wall-crossing from Quot schemes of a higher rank reflexive sheaf to a sublocus of the space of Pandharipande-Thomas pairs. We interpret this sublocus in terms of the singularities of the reflexive sheaf.