ag.algebraic geometry - Polarizations on intermediate Jacobians

If enough Hodge numbers vanish so that the Hodge structure $H^{2k+1}(X)$ has level one,
then $J^kX$ should be an abelian variety. This applies to Fano (e.g. cubic) 3-folds for example.

Later that day: Partly in response to Charles Siegel's comment/question, let me
sketch a proof of a slightly more general statement. Suppose X is a projective rather
than just Kaehler (which I forgot to say before), so $H$ has a polarization $Q$. Assume
further that
$$H^{2k+1}(X) = H= H^{pq}oplus H^{qp}$$
has only two terms.
Let $G$ be the same thing as $H$ viewed as a weight one structure. More precisely,
the lattices $G_Z=H_Z$ are the same, and $G^{10}=H^{pq}$.

Then one sees
that $J^kH= G^{01}/G_Z$, and that $pm Q$ gives a polarization on $G$. So this is abelian variety.