# Intermediate Jacobian of abelian varieties

Is the intermediate Jacobian of an abelian variety again an abelian variety?

• If $H^{2p-1}(X)$ has nonzero Hodge numbers only of type $(p-1,p), (p,p-1)$, then the corresponding intermediate Jacobian is an abelian variety, but otherwise it is not usually true. In particular, if $X$ is a general abelian variety, and $1<p<\dim X$, I would expect that the answer is no. – Donu Arapura Dec 20 '18 at 13:53
• @DonuArapura Thank you for the comment. The abelian variety I have in mind is the product of Jacobians of a curve. – Chen Dec 20 '18 at 13:57
• Even so, I would expect that when $X$ is a Jacobian of a general curve, it would fail for $p$ as above. – Donu Arapura Dec 20 '18 at 14:09