Fourier-Mukai transforms of slope stable torsion-free sheaves on a product elliptic threefold

TL;DR

This paper studies how Fourier-Mukai transforms relate slope stable torsion-free sheaves to limit tilt stable objects on a product elliptic threefold, revealing a correspondence that preserves stability properties under certain conditions.

Contribution

It introduces a notion of limit tilt stability on a product elliptic threefold and demonstrates its preservation under Fourier-Mukai transforms for specific stable sheaves.

Findings

Slope stable torsion-free sheaves are transformed into limit tilt stable objects.

Limit tilt semistable objects are mapped to slope semistable sheaves, up to codimension 2 modifications.

The work establishes stability-preserving properties of Fourier-Mukai transforms on complex threefolds.

Abstract

On the product elliptic threefold $X = C \times S$ where $C$ is an elliptic curve and $S$ is a K3 surface of Picard rank 1, we define a notion of limit tilt stability, which satisfies the Harder-Narasimhan property. We show that under the Fourier-Mukai transform $\Phi$ on $D^b(X)$ induced by the classical Fourier-Mukai transform on $D^b(C)$, a slope stable torsion-free sheaf satisfying a vanishing condition in codimension 2 (e.g. a reflexive sheaf) is taken to a limit tilt stable object. We also show that a limit tilt semistable object on $X$ is taken by $\Phi$ to a slope semistable sheaf, up to modification by the transform of a codimension 2 sheaf.