Generalized Fourier-Mukai Transforms

The paper sets out a generalized framework for Fourier-Mukai transforms and illustrates their use via vector bundle transforms. A Fourier-Mukai transform is, roughly, an isomorphism of derived categories of (sheaves) on smooth varieties X and Y. We show that these can only exist if the first Chern class of the varieties vanishes and, in the case of vector bundle transforms, will exist if and only if there is a bi-universal bundle on XxY which is "strongly simple" in a suitable sense. Some applications are given to abelian varieties extending the work of Mukai.