In the Wikipedia article on the Hodge Standard Conjecture it is written (note: it has since been fixed):
In characteristic zero the Hodge standard conjecture holds, being a consequence of Hodge theory. In positive characteristic the Hodge standard conjecture is known only for surfaces and abelian varieties.
I have three questions:
(1) Is the characteristic 0 version the Hodge Index Theorem?
(2) If so, what is a good reference for an algebraic geometry proof? I know it can be proved for a surface using the Riemann-Roch Theorem. Does this continue to be true for higher dimensions?
(3) Is the conjecture really only known in positive characteristic for surfaces and abelian varieties? Surely it should be possible to at least compute a result for projective $n$-space.