On the unramified cohomology of certain quotient varieties
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Mathematische Zeitschrift
On the unramified cohomology of certain quotient varieties Humberto Diaz1 Received: 24 June 2019 / Accepted: 5 October 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract In this note, we consider unramified cohomology with 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-Thélène and Voisin, we obtain a new type of three-dimensional counterexample to the integral Hodge conjecture. Let X be a smooth projective variety over C. We will consider the unramified cohomology i (X , A) = (X , Hi (A)), where Hi (A) denotes the Zariski sheaf with coefficients in A, Hnr X X over X associated to the presheaf U → H i (U , A). When A = Z/n, it has been known since [11] that these groups are (stably) birational invariants, and this been used quite often (and by many different authors) to prove the existence of unirational varieties are not (stably) rational. Another application of these groups is to the integral Hodge conjecture, which asserts that every α ∈ H p, p (X , Z) is algebraic. When p = 0, dim(X ), this is trivially true and when p = 1 this is true by the Lefschetz (1, 1)-theorem. For all other p, it has been known since the counterexamples of Atiyah and Hirzebruch [2] that the conjecture is false. Of particular interest is the case when X is a threefold (necessarily for p = 2). In this direction, various results have been produced, both positive and negative. On the positive side, Voisin [22] proved that the integral Hodge conjecture holds for uniruled threefolds and for Calabi-Yau threefolds. Grabowski [14] also proved the conjecture holds for Abelian threefolds. Moreover, these results were recently generalized by Totaro in [21], where it is shown that the integral Hodge conjecture holds for threefolds of Kodaira dimension 0 for which H 0 (X , K X ) = 0. On the negative side, Kollár [16] produced the first counterexamples for threefolds, in the form of non-algebraic (2, 2) cohomology classes on general hypersurfaces in P4 of sufficiently large degree. Other counterexamples have been found with Kodaira dimension 1 (Colliot-Thélène and Voisin [12]; Totaro [20]). More recently, Benoist and Ottem [4] produced threefolds of Kodaira dimension zero that fail the Hodge conjecture, the counterexamples they give being products of Enriques surfaces with very general elliptic curves. Their counterexample was generalized by Shen [17] to products of Enriques surfaces with very general odd degree hypersurfaces of higher dimension. Colliot-Thélène [10] reinterpreted the result of Benoist
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Humberto Diaz [email protected] Department of Mathematics, Washington University in St. Louis, St. Louis, MO 63130, USA
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and Ottem using unramified cohomology (together with a degeneration technique of Gabber [13]) and gave furthe
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