A remark on a $3$-fold constructed by Colliot-Th\'el\`ene and Voisin

TL;DR

This paper constructs a specific 4-fold to demonstrate that the Abel-Jacobi map is not universal among regular homomorphisms in codimension 3, contingent on the generalized Bloch conjecture for a related 3-fold.

Contribution

It provides a counterexample in codimension 3, showing the non-universality of the Abel-Jacobi map under certain conjectural assumptions.

Findings

Constructs a 4-fold counterexample under the generalized Bloch conjecture.

Shows the Abel-Jacobi map is not universal in codimension 3.

Links the result to the defect of the integral Hodge conjecture in degree 4.

Abstract

A classical question asks whether the Abel-Jacobi map is universal among all regular homomorphisms. In this paper, we prove that we can construct a $4$-fold which gives the negative answer in codimension $3$ if the generalized Bloch conjecture holds for a $3$-fold constructed by Colliot-Th\'el\`ene and Voisin in the context of the study of the defect of the integral Hodge conjecture in degree $4$.