Hodge conjecture

Let $X$ be a non-singular complex projective variety of complex dimension $n$ (equivalently, a compact complex manifold admitting a holomorphic embedding into some complex projective space), and let $H^{m}(X,\mathbb{Q})$, $H^{m}(X,\mathbb{C})=H^{m}(X,\mathbb{Q})\otimes_{\mathbb{Q}}\mathbb{C}$ denote its singular cohomology. For each $m$ there is the Hodge decomposition

$$H^{m}(X,\mathbb{C})=\bigoplus_{p+q=m}H^{p,q}(X),\qquad \overline{H^{p,q}(X)}=H^{q,p}(X),$$

where $H^{p,q}(X)$ is the subspace of classes represented by harmonic forms of type $(p,q)$, i.e. forms which in local holomorphic coordinates $z_{1},\dots,z_{n}$ are sums of terms

$$f\, dz_{i_{1}}\wedge\cdots\wedge dz_{i_{p}}\wedge d\bar{z}_{j_{1}}\wedge\cdots\wedge d\bar{z}_{j_{q}}$$

with $f$ a smooth function. For an integer $k$ with $0\le k\le n$ set

$$\operatorname{Hdg}^{k}(X)=H^{2k}(X,\mathbb{Q})\cap H^{k,k}(X),$$

the group of Hodge classes of degree $2k$ on $X$, the intersection being taken inside $H^{2k}(X,\mathbb{C})$.

Each irreducible closed complex algebraic subvariety $Z\subseteq X$ of codimension $k$ determines a class $[Z]\in H^{2k}(X,\mathbb{Q})$, the Poincaré dual of the image of the fundamental homology class of $Z$; such classes lie in $\operatorname{Hdg}^{k}(X)$. For an algebraic cycle $\sum_{i}a_{i}Z_{i}$ with $a_{i}\in\mathbb{Q}$ and each $Z_{i}\subseteq X$ closed irreducible of codimension $k$, put $\left[\sum_{i}a_{i}Z_{i}\right]=\sum_{i}a_{i}[Z_{i}]$, and let

$$\mathrm{Alg}^{k}(X)=\Big\{\textstyle\sum_{i}a_{i}[Z_{i}]\;:\;a_{i}\in\mathbb{Q},\ Z_{i}\subseteq X\ \text{closed irreducible of codimension }k\Big\}\subseteq \operatorname{Hdg}^{k}(X).$$

Then for every such $X$ and every $k$,

$$\operatorname{Hdg}^{k}(X)=\mathrm{Alg}^{k}(X);$$

that is, every rational cohomology class of type $(k,k)$ on $X$ is a linear combination with rational coefficients of cohomology classes of closed complex algebraic subvarieties of $X$ of codimension $k$.