Grothendieck's standard conjecture of Hodge type

Let $X$ be a smooth projective variety of dimension $d$ over an algebraically closed field, and let $\ell \in A^1_{\operatorname{num}}(X)$ be an ample class.

Grothendieck's standard conjecture of Hodge type. For any $i \le d/2$, the symmetric bilinear form

$$(x, y) \mapsto (-1)^i\deg(\ell^{d - 2i} xy)$$

is positive definite on the kernel of multiplication by $\ell^{d - 2i + 1}$.

This is the standard conjecture of Hodge type for numerical Chow groups. The paper verifies new cases using its Kähler-package theorem, while the conjecture remains open for the projectivization of a toric vector bundle over a field of positive characteristic.