Varieties with representable $\mathrm{CH}_0$-group and a question of Colliot-Thélène

We continue our investigation of the geometry of the Albanese morphism on 0-cycles.We provide an example of a smooth projective variety with representable CH 0 -group but with no universal 0-cycle, which answers a question asked by Colliot-Thlne.Our construction relies on a counterexample to the integral Hodge conjecture provided by Benoist and Ottem.