A surface with representable ${\text{CH}}_0$-group but no universal zero-cycle

Abstract

We introduce a new obstruction to the existence of a universal $0$-cycle on a smooth projective complex variety. As an application, we construct a smooth projective complex surface whose Chow group of $0$-cycles is representable but which does not admit a universal $0$-cycle. This provides a two-dimensional analogue of Voisin's recent threefold counterexample to a question of Colliot-Thélène. As a further consequence, we exhibit the first example of a smooth projective threefold of Kodaira dimension zero carrying a non-torsion Hodge class of degree $4$ that is not algebraic. The construction relies on the geometry of bielliptic surfaces of type 2.

30 pages, minor revisions; several assumptions have been relaxed, examples have been added and the exposition has been improved