TITLE: Equivalent descriptions of Hodge conjecture?
QUESTION [33 upvotes]: I would like to know equivalent descriptions of the Hodge conjecture (with references).

Dan Freed's Version:


Consider a topological cycle (boundary less chains that are free to deform) on a projective manifold. The topological cycle is homologous to a rational combination of algebraic cycles, if and only if the topological cycle has rotation number zero.


Deligne's version (Clay's official description):


On a projective non-singular algebraic variety over $\mathbb{C}$ , any Hodge class is a rational combination of classes $\rm{Cl(Z)}$ of algebraic cycles.


nLab ((Pure)Motivic description):


Let $SmProj^{cor}_\mathbf{C}$ denote the category of algebraic correspondences of smooth projective algebraic varieties over the complex numbers.
  Then the canonical functor
$$ SmProj^{cor} \to HS^{pure} $$
to the category of rational pure Hodge structures, given by taking rational Betti cohomology, is full.

Equivalence between statements of Hodge conjecture

REPLY [4 votes]: A possibly less known reformulation of the Hodge conjecture was given by Richard Thomas. He relates to a problem about finding nodal hypersurfaces with sufficient homology.
https://arxiv.org/abs/math/0212216