Exceptional Tannaka groups only arise from cubic threefolds

We show that under mild assumptions, the Fano surfaces of lines on smooth cubic threefolds are the only smooth subvarieties of abelian varieties whose Tannaka group for the convolution of perverse sheaves is an exceptional simple group. This in particular leads to a considerable strengthening of our previous work on the Shafarevich conjecture. A key idea is to control the Hodge decomposition on cohomology by a cocharacter of the Tannaka group of Hodge modules, and to play this off against an improvement of the Hodge number estimates for irregular varieties by Lazarsfeld-Popa and Lombardi.

💡 Research Summary

The paper studies smooth irreducible subvarieties X of an abelian variety A over an algebraically closed field of characteristic zero, focusing on the Tannaka group attached to the convolution algebra of perverse sheaves supported on X. For a smooth subvariety X⊂A of dimension d<g (the dimension of A), the perverse intersection complex δ_X=i_*F_X