A remark on the Abel–Jacobi morphism for the cubic threefold

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)×X with Zt homologically trivial for each t∈J(X), such that the morphism ϕZ:J(X)→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.