ag.algebraic geometry - Hodge Standard Conjecture in Positive Characteristic

Probably too late, but...

(1) Not quite. The Hodge index theorem only works for $mathbb{C}$. To extend it to all characteristic $0$ ground fields, you need the Lefschetz principle and the comparison theorem.

(2) As J. S. Milne mentioned, the wikipedia article was mistaken (it has since been fixed). It isn't known how to extend Segre-Grothendieck to higher dimension varieties.

(3) It seems that only surfaces are known so far. The Lefschetz conjecture for projective spaces and Grassmannians follows from intersection theory, but I'm not aware of such results for the Hodge standard conjecture.