I wish someone well initiated in the area , told me , why the $ \mathbb{Z} $ - Hodge conjecture is wrong, which allowed to change its state by tensoring by $ \mathbb{ Q } $ to become as it is known today, as : rational Hodge conjecture. I recently learned that there are two reasons for that, which allowed such a change. First, because of the presence of torsion elements , and the second reason is that a mathematician in the $ 90 $s , managed to produce a counter-example based on $ K_3 $ - surfaces. I do not know what that means.
Is someone can explain to me all this?
Thank you very much .
Atiyah and Hirzebruch observed in Analytic cycles on complex manifolds, Topology, 1 (1962), that the group $H^{2p}(X,\mathbf{Z})$ may contain torsion elements and that a torsion element vanishes in the $H^{2p}(X,\mathbf{C})$, and therefore satisfies obviously the Hodge condition. Then you could ask whether the integral Hodge conjecture is true modulo torsion, in particular as soon as the $H^{2p}(X,\mathbf{Z})$ is torsion free. But Kollár showed in Trento examples. Classification of irregular varieties (Trento, 1990), 134–135. Lecture Notes in Math. 1515, Springer-Verlag, Berlin, 1992, that this is not the case for a general hypersurface in $\mathbf{P}^4$ of large enough degree. There is a very nice exposé of Arnaud Beauville on the Hodge conjecture at the séminaire Bourbaki, which deals also with the integral Hodge conjecture and gives examples about it, that you can find online here :
http://math.unice.fr/~beauvill/pubs/Hodge.pdf