Singularities of theta divisors and the Tannakian Schottky problem

With the Tannakian formalism, one can attach to any principally polarized abelian variety a reductive group, along with a representation. We obtain a completely new characterization of Jacobians relying on this Tannakian data and certain characteristic classes for singular varieties. As a corollary, we show that the Tannakian Schottky conjecture holds in dimension up to 5. More generally, we show that the Tannakian Schottky conjecture holds on the bielliptic Prym locus in all dimensions. This provides the first non-trivial cases of this conjecture.