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

Abstract

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 ${\varphi }_Z:J(X)\to J(X)$ induced by the Abel-Jacobi map is the identity. This answers positively a question of Voisin in the case of the cubic threefold.

4 pages