Abstract
A method of Gabber (Enseign Math (2) 48(1–2):127–146, 2002) produces unramified cohomology classes in the products of certain varieties with an elliptic curve. The connection between third unramified cohomology and integral Hodge conjecture for codimension 2 cycles (Colliot-Thelene et Voisin in Duke Math J 161(5):735–801, 2012) then gives many examples of such a product for which this conjecture fails. The special case of the product with an Enriques surface was established by Benoist and Ottem (Failure of the integral Hodge conjecture for threefolds of Kodaira dimension zero, 2018. arXiv:1802.01845v1 ).
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