The Hodge conjecture for complex projective manifolds

Let $X$ be a complex projective manifold of dimension $n$. For $0\leq p\leq n$, define

$$H^{p,p}(X;\Q):=PD^{-1}(H_{2(n-p)}(X;\Q))\cap H^{p,p}(X).$$

An element $\alpha\in H^{p,p}(X;\Q)$ is representable by algebraic cycles with rational coefficients if there is an algebraic $p$-cycle $R$ with rational coefficients such that

$$PD^{-1}([R])=\alpha.$$

The Hodge conjecture. Every element of $H^{p,p}(X;\Q)$ is representable by some algebraic $(n-p)$-cycles with $\Q$-coefficients.

This is one of the central questions in algebraic geometry. It concerns whether rational Hodge classes on complex projective manifolds arise from algebraic cycles, and remains open in general.