Abstract
In this note, we consider unramified cohomology with $${{\mathbb {Z}}}/2$$ coefficients for some (degree two) quotient varieties and describe a method that allows one to prove the non-vanishing of these groups under certain conditions. We apply this method to prove a non-vanishing statement in the case of Kummer varieties. Combining this with work of Colliot-Thelene and Voisin, we obtain a new type of three-dimensional counterexample to the integral Hodge conjecture.
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