*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.