An n-fold cup-product Bockstein on products of Enriques surfaces produces non-algebraic 2-torsion integral Hodge classes in dimension 2n under the Brauer-separation hypothesis.
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From Diaz's Enriques Product to an $n$-Fold Cup-Product Bockstein Family of Integral Hodge Counterexamples
An n-fold cup-product Bockstein on products of Enriques surfaces produces non-algebraic 2-torsion integral Hodge classes in dimension 2n under the Brauer-separation hypothesis.