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Unramified cohomology in a product with an elliptic curve. (Cohomologie non ramifiée dans le produit avec une courbe elliptique.) (French. English summary) Zbl 1423.14058

Summary: A method of J.-L. Colliot-Thélène [Enseign. Math. (2) 48, No. 1–2, 127–146 (2002; Zbl 1047.16007)] 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 [J.-L. Colliot-Thélène and C. Voisin, Duke Math. J. 161, No. 5, 735–801 (2012; Zbl 1244.14010)] 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 O. Benoist and J. C. Ottem [“Failure of the integral Hodge conjecture for threefolds of Kodaira dimension zero”, Preprint, arXiv:1802.01845].

MSC:

14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14C25 Algebraic cycles
14C35 Applications of methods of algebraic \(K\)-theory in algebraic geometry
19E15 Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects)
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References:

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