# Higher rank sheaves on threefolds and functional equations

## TL;DR

This paper studies the moduli space of stable torsion free sheaves on threefolds, deriving generating functions for their Euler characteristics and revealing symmetries and invariance properties through wall-crossing techniques.

## Contribution

It introduces a new approach to compute generating functions for moduli spaces of sheaves with controlled singularities using wall-crossing from Quot schemes to Pandharipande-Thomas pairs.

## Key findings

- Generating functions relate to the MacMahon function and Laurent polynomials.
- Invariance under certain variable transformations is established.
- In some cases, the open subsets cover the entire moduli space.

## 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.