The standard conjecture of Künneth type

Let $X$ be a smooth projective variety of dimension $n$ over $\mathbf{K}$, and write the Künneth decomposition of the diagonal class as

$$\operatorname{cl}_{X\times X}(\Delta_X)=\sum_{i=0}^{2n}\pi_i,$$

with $\pi_i\in H^i(X)\otimes H^{2n-i}(X)$. Standard conjecture of Künneth type. Each Künneth component $\pi_i$ is algebraic: there are rational algebraic cycles $\Delta_i\in\mathsf{Z}^n(X\times X)_\mathbf{Q}$ such that

$$\pi_i=\operatorname{cl}_{X\times X}(\Delta_i).$$

The conjecture is known for curves, surfaces, abelian varieties, and varieties over finite fields; over $\mathbf{C}$ it follows in general from the Hodge conjecture, but it is open for arbitrary varieties in positive characteristic.