On the Chow groups of certain geometrically rational 5-folds (Q2017204)

The author constructs explicit smooth projective varieties for which the integral Tate conjectures fails. The examples are constructed as quadric surface fibrations over \(\mathbb{P}^2\). For codimension \(2\) cycles on such \(X\), the author shows that the map \[ \mathrm{CH}^2(X)\otimes \mathbb{Z}_2 \to H^4_{\text{ét}}(X, \mathbb{Z}_2(2)) \] is not surjective. The author also shows that the group \(H^4_{\text{ét}}(X, \mathbb{Z}_2(2))\) is not torsion. So the failure of the integral Tate conjecture is not caused by torsion classes.