Hodge conjecture for smooth projective complex varieties

Let $X$ be a smooth projective complex variety of complex dimension $n$, and let $p$ be an integer with $0\le p\le n$. Let $Z^p(X)$ be the free abelian group on irreducible closed subvarieties of codimension $p$, with rational equivalence generated by divisors of nonzero rational functions on irreducible codimension-$(p-1)$ subvarieties. Set ${\mathrm{C}\mathrm{H}}^p(X{)}_{\mathbb{Q}}:=(Z^p(X)\mathrm{/}{\sim }_{\mathrm{r}\mathrm{a}\mathrm{t}}){\otimes }_{\mathbb{Z}}\mathbb{Q}$. For each irreducible subvariety, take the Poincaré-dual cohomology class of its fundamental homology class, and extend linearly to the cycle-class map $\mathrm{c}\mathrm{l}:{\mathrm{C}\mathrm{H}}^p(X{)}_{\mathbb{Q}}\to H^{2p}(X;\mathbb{Q})$. Write $H^{2p}(X;\mathbb{C})={⨁}_{a+b=2p}{H}^{a,b}(X)$ for the Hodge decomposition and $H^{p,p}(X{)}_{\mathbb{Q}}:=\{\alpha \in H^{2p}(X;\mathbb{Q}):\alpha \otimes 1\in H^{p,p}(X)\}$. Is every $\alpha \in H^{p,p}(X{)}_{\mathbb{Q}}$ in the image of $\mathrm{c}\mathrm{l}$?

Deligne, §1

Voisin, §2.3.1

As of 2026-11-04 the Clay Mathematics Institute lists the Hodge conjecture as unsolved. The official description records known special cases, including the low-dimensional cases, and states that the general problem remains open. This item is a precise conjectural question, not a proof premise; the proved algebraicity theorem for Hodge loci is a separate claim.

Clay Mathematics Institute, Hodge Conjecture

The open problem is not a supplier for the Hodge-locus theorem on the A page.