A remark on a $3$-fold constructed by Colliot–Thélène and Voisin

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–Thélène and Voisin in the context of the study of the defect of the integral Hodge conjecture in degree $4$.

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Received 13 June 2018

Accepted 14 November 2018

Published 8 April 2020