Fourier-Mukai transforms of slope stable torsion-free sheaves and stable 1-dimensional sheaves on Weierstrass elliptic threefolds
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. In the first half of this article, we define the notion of limit tilt stability, which is closely related to Bayer's polynomial stability. We 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 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. In the second half of this article, we define a limit stability for complexes that vanish on the generic fiber of the fibration. We show that one-dimensional stable sheaves with positive twisted third Chern character correspond to such limit stable complexes under a Fourier-Mukai transform. When the elliptic fibration has a numerically $K$-trivial base, we show that these limit stable complexes are the stable objects with respect to a Bridgeland stability on a triangulated subcategory of the derived category of coherent sheaves on the threefold.