| Résume | Abstract: Any subvariety of an abelian variety defines a reductive group by taking the Tannaka dual of the tensor category generated by the perverse intersection complex on the subvariety. These Tannaka groups play a fundamental role in the geometry and arithmetic of irregular varieties. I will discuss recent work with Christian Lehn and Marco Maculan: Under mild assumptions, the only smooth subvarieties of abelian varieties with exceptional Tannaka group are the Fano surfaces of lines on smooth cubic threefolds (for which the group is known to be of type E6). 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 play this off against Hodge number estimates by Lazarsfeld-Popa and Lombardi.
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