Deformations of minimal cohomology classes on abelian varieties

We show that the infinitesimal deformations of Brill-Noether loci W-d attached to a smooth non-hyperelliptic curve C are in one-to-one correspondence with the deformations of C. As an application, we prove that if a Jacobian J deforms together with a minimal cohomology class out the Jacobian locus, then J is hyperelliptic. In particular, this provides an evidence toward a conjecture of Debarre on the classification of ppavs carrying a minimal cohomology class.