The Tannaka group E6 only arises from cubic threefolds

To any subvariety of an abelian variety one may attach a reductive group via the convolution of perverse sheaves. I will give a brief introduction to these Tannaka groups and then discuss a recent work with Christian Lehn and Marco Maculan: Under mild assumptions, the Fano surfaces of lines on smooth cubic threefolds are the only subvarieties with exceptional Tannaka group. This significantly enlarges the scope of our previous results on the Shafarevich conjecture. The key idea is to control the Hodge decomposition on cohomology by a cocharacter of the Tannaka group, and to compare this with an improvement of the Hodge number estimates by Lazarsfeld-Popa and Lombardi.