On the integral Hodge conjecture for varieties with trivial Chow group

TL;DR

This paper provides examples of smooth projective varieties over complex numbers and number fields that violate the integral Hodge conjecture despite having finite rank Chow groups.

Contribution

It constructs explicit examples of varieties with trivial Chow groups that do not satisfy the integral Hodge conjecture, advancing understanding of this conjecture's limitations.

Findings

Existence of varieties over  that violate the integral Hodge conjecture.

Examples of such varieties defined over number fields.

Varieties with finite rank Chow groups that do not satisfy the conjecture.

Abstract

We obtain examples of smooth projective varieties over $\mathbb{C}$ that violate the integral Hodge conjecture and for which the total Chow group is of finite rank. Moreover, we show that there exist such examples defined over number fields.