Let $X$ be a projective complex manifold of dimension $n$. Are torsion cohomology classes in $H^{2n-2}(X,\mathbb{Z})$ algebraic? (We may assume, without loss of generality, that $n=3$, because of the Lefschetz hyperplane theorem.) I know that torsion classes (of even codimension) aren't always algebraic; the first counterexamples were found by Atiyah and Hirzebruch in 1960s. But I do not know any counterexample in this codimension.

Note that by Poincare duality $H^{2n-2}(X,\mathbb{Z})\cong H_2(X)$, so the equivalent question is whether
torsion classes in $H_2(X)$ are generated by algebraic curves in $X$. This looks like a "dual" to the well known fact that torsion classes in $H^{2}(X,\mathbb{Z})$ are generated by divisors (it is a torsion part of the Neron-Severi group).

  • $\begingroup$ Here is something worth considering. Let $Y$ and $Z$ be complex, projective manifolds each of which has $p$-torsion in $H_1$, e.g., $Y$ and $Z$ are both Enriques surfaces so $H_1=\mathbb{Z}/2\mathbb{Z}$. Now let $X$ be the product $Y\times Z$. By Kunneth's theorem, there is torsion in $H_2(X,\mathbb{Z})$. What algebraic cycle gives rise to this torsion? $\endgroup$ Jun 22 '14 at 16:49
  • $\begingroup$ It makes sense. This trick does not work for divisors, because $H^1$ has no torsion. $\endgroup$ Jun 23 '14 at 12:39

This 2013 paper of Totaro says that it is an open question. But the integral Hodge conjecture is known to fail for dimension 1, just not via torsion.


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