On the integral Hodge and Tate conjectures over a number field (Q2849816)

The integral Hodge conjecture for a smooth complex projective variety says that every class \(\alpha \in H^{2i}(X,{\mathbb Z})\) whose image in \(H^{2i}(X,{\mathbb C})\) is of type \((i,i)\) is in fact represented by an algebraic cycle of codimension \(i\). Integral Hodge conjecture is known to be false in general. Hassett and Tschinkel (cf. [\textit{J. L. Colliot-Thélène} and \textit{C. Voisin}, Duke Math. J. 161, No. 5, 735--801 (2012; Zbl 1244.14010)]) gave counterexamples involving \(3\)-folds over number fields. The author studies the integral Hodge conjecture for varieties defined over a number field. By examining and extending the method of Hassett and Tschinkel he concludes that all known counterexamples to the integral Hodge conjecture over complex numbers can be made to work over a number field. The author also discusses the Hassett and Tschinkel method in the context of counterexamples to the integral Tate conjecture.