Stable reflexive sheaves and localization

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 $\mathbb{P}^2 \times \mathbb{P}^1$.Comment: 27 pages. Published version. Typo's correcte

Donaldson-Thomas Invariants of 2-Dimensional sheaves inside threefolds and modular forms

Motivated by the S-duality conjecture, we study the Donaldson-Thomas invariants of the 2 dimensional Gieseker stable sheaves on a threefold. These sheaves are supported on the fibers of a nonsingular threefold X fibered over a nonsingular curve. In the case where X is a K3 fibration, we express these invariants in terms of the Euler characteristic of the Hilbert scheme of points on the K3 fiber and the Noether-Lefschetz numbers of the fibration. We prove that a certain generating function of these invariants is a vector modular form of weight -3/2 as predicted in S-duality.Comment: Some corrections were made and some arguments were extended. Many thanks to the referee's helpful comments. 22 pages, to Appear in Adv. Math. (2018). arXiv admin note: text overlap with arXiv:1305.133

Higher rank sheaves on threefolds and functional equations

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.Comment: 29 pages. Published versio