Fourier–Mukai transform

Summary

A Fourier–Mukai transform is an exact functor between the derived categories of coherent sheaves of two varieties $X$ and $Y$. It is given by an object on $X \times Y$ called the kernel. Mukai showed that the Poincaré bundle gives an equivalence between the derived categories of an abelian variety and of its dual. By a theorem of Orlov, all fully faithful exact functors between such derived categories of smooth projective varieties are of this form.