Exceptional Tannaka groups only arise from cubic threefolds

TL;DR

This paper demonstrates that the Tannaka groups associated with Fano surfaces of lines on smooth cubic threefolds are uniquely exceptional, linking geometric properties to specific algebraic group structures and advancing understanding of the Shafarevich conjecture.

Contribution

It establishes that only Fano surfaces of lines on smooth cubic threefolds have Tannaka groups that are exceptional simple groups, under mild assumptions.

Findings

Fano surfaces are the only smooth subvarieties with exceptional Tannaka groups in this context.

The work strengthens previous results related to the Shafarevich conjecture.

Control of Hodge decomposition via Tannaka groups is a key technique.

Abstract

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.