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). 



A special case of the integer Hodge conjecture
This [2013 paper](https://www.math.ucla.edu/~totaro/papers/public_html/integral.pdf) 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.


