  Let $X$ be a smooth cubic threefold and $J(X)$ be its intermediate Jacobian.
We show that there exists a codimension 2 cycle $Z$ on $J(X)\times X$ with
$Z_{t}$ homologically trivial for each $t\in J(X)$, such that the morphism
$\phi_{Z}: J(X)\rightarrow J(X)$ induced by the Abel-Jacobi map is the
identity. This answers positively a question of Voisin in the case of the cubic
threefold.
