Fourier-Mukai transforms of slope stable torsion-free sheaves on Weierstrass elliptic threefolds

TL;DR

This paper investigates how Fourier-Mukai transforms relate to slope stable torsion-free sheaves on Weierstrass elliptic threefolds, introducing limit tilt stability and establishing stability preservation under the transform.

Contribution

It defines limit tilt stability on elliptic threefolds and proves the Fourier-Mukai transform preserves stability properties for certain sheaves.

Findings

Fourier-Mukai transform of a slope stable sheaf is limit tilt stable.

Inverse transform of a limit tilt semistable object yields a slope semistable sheaf.

Introduces limit tilt stability in the context of elliptic threefolds.

Abstract

We focus on a class of Weierstrass elliptic threefolds that allows the base of the fibration to be a Fano surface or a numerically $K$-trivial surface. We define the notion of limit tilt stability, and show that the Fourier-Mukai transform of a slope stable torsion-free sheaf satisfying a vanishing condition in codimension 2 (e.g. a reflexive sheaf) is a limit tilt stable object. We also show that the inverse Fourier-Mukai transform of a limit tilt semistable object of nonzero fiber degree is a slope semistable torsion-free sheaf, up to modification in codimension 2.