| Building on previous conjectures of Beauville and Ran, Debarre famously conjectured that Jacobians of curves and intermediate Jacobians of smooth cubic threefolds are the only principally polarized abelian varieties which contain a subvariety with minimal cohomology class. In this talk we discuss a sheaf theoretic analogue of that conjecture, due to Pareschi and Popa, and explain how to prove Pareschi--Popa's conjecture in dimension five. Our result is implied by a more general statement about subvarieties of minimal cohomology class whose sum is a theta divisor. This is joint work with Casalaina-Martin and Popa. |