# Stable reflexive sheaves and localization

Research paper by **Amin Gholampour, Martijn Kool**

Indexed on: **14 Sep '15**Published on: **14 Sep '15**Published in: **Mathematics - Algebraic Geometry**

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#### 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 dense open 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. For $c_1=-1$, we show its
leading term is $12c_2 q^{c_{2}^{2}}$.
Next, we study moduli spaces of rank 2 stable torsion free sheaves on
$\mathbb{P}^3$ and consider the generating function $\mathsf{Z}(q)$ 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.
These Quot schemes and their fixed point loci are studied in a sequel with B.
Young. The components of these fixed point loci are products of
$\mathbb{P}^1$'s and give rise non-trivial combinatorics. For $c_1=-1$ and
$c_2=1$, we obtain $\mathsf{Z}(q) = 4(q+q^{-1}) M(q^{-2})^8$, where $M(q)$ is
the MacMahon function.
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
$\mathbb{P}^2 \times \mathbb{P}^1$.