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

TL;DR

This paper proves that for a smooth cubic threefold, there exists a specific cycle on the product of its intermediate Jacobian and itself that induces the identity morphism via the Abel-Jacobi map, confirming a question posed by Voisin.

Contribution

It establishes the existence of a codimension 2 cycle on the product of the intermediate Jacobian and the threefold that induces the identity morphism, answering Voisin's question.

Findings

Existence of a special cycle inducing the identity morphism

Positive answer to Voisin's question for cubic threefolds

Advancement in understanding Abel-Jacobi morphisms

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