Fourier-Mukai functors in the supported setting

Fourier-Mukai functors play a distinct role in algebraic geometry. Nevertheless two questions remained open: are all exact functors between the bounded derived categories of smooth projective varieties of Fourier-Mukai type? Is the Fourier-Mukai kernel unique? We will answer positively to these questions under some assumptions on the exact functor. This extends previous results by Lunts, Orlov and Ballard. Along the way, we will show that, in geometric contexts, full functors are faithful as well. This is a joint work in collaboration with A. Canonaco and, partly, with D. Orlov.