Moduli Spaces of Sheaves in Higher Dimensions

Little is known about the structure of the moduli spaces of sheaves above dimension 2 except for a few basiccases. For abelian varieties, we have two powerful techniques: the Fourier-Mukai transform and Bridgelandstability conditions. These are interlinked and between them we can describe the moduli spaces of sheaves indimension 2. The aim of this project is to attempt to use these techniques in higher dimensions to explore thestructure of the moduli spaces. We can then use this toy model to describe moduli spaces of sheaves on othervarieties.