Hodge standard conjecture

Let $X$ be a smooth projective variety, let $n$ be a codimension parameter, and let $q_Z$ be the quadratic form on algebraic cycles modulo numerical equivalence introduced from the Lefschetz polarization:

$$q_Z:\mathcal{Z}^n(X)_{/\operatorname{num}}\otimes\mathcal{Z}^n(X)_{/\operatorname{num}}\longrightarrow\mathbb{Q}.$$

Hodge standard conjecture. The quadratic form $q_Z$ is positive definite. This is Grothendieck’s Hodge-type standard conjecture; it is known in characteristic zero through Hodge–Riemann positivity, but remains open in positive characteristic in general.