The integral Hodge conjecture

Let $X$ be a smooth projective complex variety. Write

$$\operatorname{cl}^i\colon \operatorname{CH}^i(X)\to\mathrm{H}^{2i}(X,\mathbb{Z}(i))$$

for the integral cycle class map, and let

$$\mathrm{Hdg}^{2i}(X,\mathbb{Z}):= \mathrm{H}^{2i}(X,\mathbb{Z})\cap\mathrm{H}^{i,i}(X)$$

be the group of integral Hodge classes. Integral Hodge conjecture. For every such $X$ and every $i$, all integral Hodge classes are integral linear combinations of algebraic cycles, equivalently

$$\operatorname{im}(\operatorname{cl}^i)=\mathrm{Hdg}^{2i}(X,\mathbb{Z}).$$

The conjecture is the integral analogue of the Hodge conjecture. It is false in general: Atiyah and Hirzebruch first established a failure, so this statement is refuted as a universal conjecture.