A surface with representable $\text{CH}_{0}$-group but no universal zero-cycle

TL;DR

This paper constructs a complex surface with a representable Chow group of 0-cycles that lacks a universal 0-cycle, revealing new obstructions and providing examples related to algebraic cycles and Hodge classes.

Contribution

It introduces a novel obstruction to universal 0-cycles and constructs the first example of a surface with these properties, extending Voisin's counterexample to a new dimension.

Findings

Existence of a surface with representable Chow group but no universal 0-cycle

First example of a threefold with non-torsion non-algebraic Hodge class of degree 4

Application of bielliptic surface geometry in cycle obstructions

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-Th\'el\`ene. 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.