Non-uniqueness of Fourier-Mukai kernels

We will discuss some properties of the functor sending an object in the bounded derived category of the product of two smooth projective varieties to the Fourier-Mukai functor with that object as kernel. In particular we will show that this functor is not always essentially injective, namely that there exist non-isomorphic kernels defining isomorphic Fourier-Mukai functors. On the other hand, the cohomology sheaves of a kernel are always uniquely determined, up to isomorphism, by the functor. This is a joint work with P. Stellari.