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

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    $\begingroup$ 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. $\endgroup$ – Donu Arapura Dec 20 '18 at 13:53
  • $\begingroup$ @DonuArapura Thank you for the comment. The abelian variety I have in mind is the product of Jacobians of a curve. $\endgroup$ – Chen Dec 20 '18 at 13:57
  • $\begingroup$ Even so, I would expect that when $X$ is a Jacobian of a general curve, it would fail for $p$ as above. $\endgroup$ – Donu Arapura Dec 20 '18 at 14:09

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