Varieties with representable CH_0-group and a question of Colliot-Th\'{e}l\`{e}ne

TL;DR

This paper explores the geometry of the Albanese morphism on 0-cycles, providing a counterexample to a question by Colliot-Thélène using a variety with a representable CH_0-group but lacking a universal 0-cycle.

Contribution

It constructs a smooth projective variety with a representable CH_0-group but no universal 0-cycle, answering an open question in the field.

Findings

Counterexample to the integral Hodge conjecture by Benoist and Ottem used in construction.

Existence of a variety with representable CH_0-group but no universal 0-cycle.

Clarification of the relationship between the Albanese morphism and 0-cycle geometry.

Abstract

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-Th\'el\`ene. Our construction relies on a counterexample to the integral Hodge conjecture provided by Benoist and Ottem.