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

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

11 pages, comments are welcome, v3. the exposition is improved; a section on stably birational invariants is added; an appendix on the Roitman theorem for the Walker maps and decomposition of the diagonal is added; to appear in Mathematical Research Letters

Cited by in corpus (2)

• On the coniveau of rationally connected threefolds
• Factorization of the Abel-Jacobi maps