Minimal Cohomology Classes and Jacobians

We show that on the Jacobian $(JC,\theta)$ of a smooth curve $C$ of genus $g$, any effective cycle in $JC$ with cohomology class $\theta^d/d!$ is a translate of $W_{g-d}(C)$ or $-W_{g-d}(C)$. We then use this result to prove that for $1<d<g$, the Jacobian locus (\resp the locus of intermediate Jacobians of cubic threefolds) is an irreducible component of the set of principally polarized abelian varieties of dimension $g$ for which $\theta^d/d!$ (\resp $\theta^3/3!$) is the class of an effective algebraic cycle. Moreover, on the intermediate Jacobian of a generic cubic threefold, $\theta^2/2!$ is not the class of an effective algebraic cycle.