This question deals with the classic Hodge conjecture on projective non-singular complex varieties, or in other words, projective Kähler manifolds. In Deligne's writeup for the Clay Foundation he says (on page 2):
No counterexample is known to the statement that integral $(p, p)$ classes killed by all $d_r$ are integral linear combinations of classes $cl(Z)$. One has no idea of which classes should be effective, that is, of the form $cl(Z)$, rather than a difference of such.
In the first sentence, the differentials $d_r$ are those appearing in the Atiyah-Hirzebruch spectral sequence for K-theory, and $cl$ is the map associating to an analytic cycle the corresponding cohomology class. Is it still true that there are no such counterexamples? Is there any evidence for this refinement of Hodge, or against (eg in work of Voisin)?
In the second sentence, I'm not sure what Deligne means. It seems like something K-theory-like.