Let $X$ be a Kahler variety of dimension $n$. For each odd number $2k-1 \leq n$ one can consider the $k$-th intermediate Jacobian, that is, the complex torus $$J^{k}X := \frac{H^{2k+1}(X, \mathbb{C})}{F^k H^{2k+1}(X, \mathbb{C}) + H^{2k+1}(X, \mathbb{Z})},$$ where $F^{\cdot}$ is the Hodge filtration. In general $J^k X$ is not an abelian variety, except for the extreme cases of the Picard and Albanese varieties (when $X$ itself is algebraic).

Are there any criteria to determine whether $J^K X$ is polarized? Or have some nontrivial cases where $J^k X$ is polarized been worked out?