MOS

Abstract

Since its introduction by Mukai, the theory of integral functors and Fourier-Mukai transforms have been important tools in the study of the geometry of varieties and moduli spaces.

Working with a fibered scheme over a base $T$ it is quite natural to look at the group of $T$-linear autoequivalences. The description of this group seems a hard problem. We will restrict ourselves to the subgroup given by relative Fourier-Mukai transforms. In this talk, I will explain how for a projective fibration the knowledge of the structure of the group of autoequivalences of its fibres and the properties of relative integral functors provide a machinery to study that subgroup. I will work out the case of a Weierstrass fibrations and report about the results for abelian schemes and Fano or anti-Fano fibrations.