Generic vanishing and minimal cohomology classes on abelian varieties

We establish a-and conjecture further-relationship between the existence of subvarieties representing minimal cohomology classes on principally polarized abelian varieties, and the generic vanishing of the cohomology of twisted ideal sheaves. The main ingredient is the Generic Vanishing criterion established in Pareschi G. and Popa M. (GV-sheaves, Fourier-Mukai transform, and Generic Vanishing. Preprint math.AG/0608127), based on the Fourier-Mukai transform.