Fourier-Mukai transforms on elliptic fibrations

In their 2002 JAG paper, Bridgeland-Maciocia laid out a few ideas for computing Fourier-Mukai transforms for sheaves on elliptic threefolds, which allowed them to construct an isomorphism between two moduli spaces of stable sheaves. I will explain how their ideas can be developed into a toolchest for computing Fourier-Mukai transforms on elliptic fibrations. On elliptic fibrations of any dimension, this toolchest can be used to embedd a moduli of complexes in a moduli of sheaves, as well as set up a 1-1 correspondence between line bundles and spectral sheaves.