A remark on the example of Colliot-Thélène and Voisin

Fumiaki Suzuki

Published 2018-05-03Version 1

A classical question asks whether the Abel-Jacobi map $\psi^{i} \colon A^{i}(X)\rightarrow J^{i}_{a}(X)$ is universal among all regular homomorphisms. Meanwhile, Colliot-Th\'el\`ene and Voisin constructed a smooth projective $3$-fold $Y$ such that the Hodge conjecture is false for degree $4$ integral Hodge classes in a strong sense and $H^{i}(Y, \mathcal{O}_{Y}) = 0$ for all $i>0$. We prove that the Abel-Jacobi map $\psi^{3}\colon A^{3}(Y\times E)\rightarrow J^{3}_{a}(Y\times E)$ is not universal for some smooth elliptic curve $E$ if the generalized Bloch conjecture holds for $Y$.