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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.

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    $\begingroup$ The first part was known to be false (by a result of Kollar) even before Deligne wrote this! $\endgroup$
    – naf
    Dec 27, 2016 at 5:49
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    $\begingroup$ To elaborate on ulrich's answer, you can see Voisin's explanation of Kollar's argument here: webusers.imj-prg.fr/~claire.voisin/Articlesweb/takagifinal.pdf. The second sentence concerns the question of which hodge classes are genuinely cycle classes of subvarieties, rather than merely the linear combination of such. It's easy to produce examples of hodge classes which are not "effective" in this sense: for instance, the generator of H^2 for a general hypersurface of high degree in P^3 would have to be represented by a line, but such a hypersurface doesn't contain a line. $\endgroup$
    – user84144
    Dec 27, 2016 at 6:06
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    $\begingroup$ Another example for the last question: let $A$ be an abelian variety of dimension $g$ (say, over $\mathbb{C}$), and $\theta \in H^2(A,\mathbb{Z})$ the class of a principal polarization. Then $\frac{\theta^{g-1}}{(g-1)!} $ is a Hodge class in $H^{2g-2}(A,\mathbb{Z})$. It is representable by an effective cycle if and only if $A$ is a Jacobian (Matsusaka's criterion), a rather subtle property that you cannot detect by K-theory or any spectral sequence. Whether it is representable by an algebraic cycle is a wide open question. $\endgroup$
    – abx
    Dec 27, 2016 at 8:08
  • $\begingroup$ @ulrich the most charitable interpretation then is that the formulation given was known to be false, but no counterexamples were (at that time) known. Is this an accurate reading? $\endgroup$
    – David Roberts
    Jan 2, 2017 at 23:45
  • $\begingroup$ That isn't really correct: Kollar's counterexamples were already known in 1990 (see the reference [28] in the article of Voisin linked to by user84144). $\endgroup$
    – naf
    Jan 3, 2017 at 5:29

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