A pencil of Enriques surfaces with non-algebraic integral Hodge classes
The integral Hodge conjecture for a smooth complex projective variety says that every class \(\alpha \in H^{2i}(X, \mathbb{Z})\) whose image in \(H^{2i}(X, \mathbb{C})\) is of type \((i,i)\) is in fact represented by an algebraic cycle of codimension \(i\). The integral Hodge conjecture is known to fail in general. The first counterexample was given in [\textit{M. F. Atiyah} and \textit{F. Hirzebruch}, Topology 1, 25--45 (1962; Zbl 0108.36401)]. More recently many authors have found other examples of varieties admitting non-algebraic integral Hodge classes, both of torsion type [\textit{O. Benoist} and \textit{J. C. Ottem}, Comment. Math. Helv. 95, No. 1, 27--35 (2020; Zbl 1442.14032)] and of non-torsion type [\textit{J.-L. Colliot-Thélène} and \textit{C. Voisin}, Duke Math. J. 161, No. 5, 735--801 (2012; Zbl 1244.14010)]. The very interesting paper under review provides the first example of a threefold with the trivial Chow group of zero-cycles on which the integral Hodge conjecture fails. For this purpose the authors prove the existence of a pencil of Enriques surfaces defined over \(\mathbb{Q}\) admitting non-algebraic integral Hodge classes of non-torsion type. As an application, the authors construct a fourfold which gives the negative answer to a classical question of \textit{J. P. Murre} [Topology 24, 361--367 (1985; Zbl 0585.14009)] about the universality of the Abel-Jacobi maps in codimension three.
scientific article