Stable reflexive sheaves and localization

Abstract

We study moduli spaces $\mathcal{N}$ of rank 2 stable reflexive sheaves on $\mathbb{P}^3$. Fixing Chern classes $c_1$, $c_2$, and summing over $c_3$, we consider the generating function $\mathsf{Z}^{\mathrm{refl}}(q)$ of Euler characteristics of such moduli spaces. The action of the torus $T$ on $\mathbb{P}^3$ lifts to $\mathcal{N}$ and we classify all sheaves in $\mathcal{N}^T$. This leads to an explicit expression for $\mathsf{Z}^{\mathrm{refl}}(q)$. Since $c_3$ is bounded below and above, $\mathsf{Z}^{\mathrm{refl}}(q)$ is a polynomial. We find a simple formula for its leading term when $c_1=-1$. Next, we study moduli spaces of rank 2 stable torsion free sheaves on $\mathbb{P}^3$ and consider the generating function of Euler characteristics of such moduli spaces. We give an expression for this generating function in terms of $\mathsf{Z}^{\mathrm{refl}}(q)$ and Euler characteristics of Quot schemes of certain $T$-equivariant reflexive sheaves, which are studied elsewhere. Many techniques of this paper apply to any toric 3-fold. In general, $\mathsf{Z}^{\mathrm{refl}}(q)$ depends on the choice of polarization which leads to wall-crossing phenomena. We briefly illustrate this in the case of $\mathbb{P}^2 \times \mathbb{P}^1$.