Higher rank sheaves on threefolds and functional equations
Amin Gholampour, Martijn Kool
TL;DR
The paper establishes a higher-rank DT/PT framework on smooth projective threefolds by analyzing moduli of torsion-free sheaves with controlled singularities. It proves that the generating function of Euler characteristics for the open locus with empty or 0-dimensional singularities factors as a MacMahon function power times a Laurent polynomial, with duality and, in rank 2, palindromicity properties. The authors build a bridge between Quot schemes of Ext^1 of a higher-rank reflexive sheaf and a locus of Pandharipande-Thomas pairs, using wall-crossing and a Hall-algebra setup to derive a key affine identity linking quotients of M to quotients of Ext^1(M,A). This affine result underpins the global generating function formula and the associated functional equations, extending DT/PT-type correspondences to higher rank and providing a robust structural framework via Hall algebras.
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 $\leq 1$. We consider the open subset of moduli space consisting of sheaves with empty or 0-dimensional singularity set. For fixed Chern classes $c_1,c_2$ and summing over $c_3$, 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 $q \leftrightarrow q^{-1}$ (upon replacing $c_1 \leftrightarrow -c_1$). For some choices of $c_1,c_2$ 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.