It is known that [Hodge standard conjecture](https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/StandardConjs.pdf) is true for étale cohomology for a field $k$ of characteristic zero. It means that the following pairing 
$$
(x,y)\mapsto (-1)^{i}\langle L^{r-2i}(x),y\rangle
$$
is positive definite, where $\langle -,-\rangle$ is the cup product, $L$ is the Lefschetz operator for this cohomology and $x,y\in C^i(X)\cap P^{2i}(X)$.
In above formula $X$ is a smooth projective scheme over $k$, $C^i(X)$ is a free abelian group generated by closed irreducible subschemes of $X$ of codimension $i$ and $P^i(X)$ is a kernel of $L^{r-i-1}$.



> 
> My question is : what is the main problem with proving the similar theorem for fields of arbitrary characteristic?
> 
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Hodge standard conjecture for étale cohomology
I'm very interested myself on a better answer to this question, but let me point out the obvious: the main problem is that there is no Hodge theory on positive characteristic.


The proof in characteristic zero simply says that it is enough to consider $\mathbb{C}$ (via Lefschetz principle), and that there you can use Hodge theory, the Hodge index theorem in particular (via the comparison theorem).


Note that in positive characteristic the conjecture is known for surfaces, as it is a simple application of Riemann-Roch (see chapter V of Hartshorne for example). A proper answer should adress why no generalization of Riemann-Roch yields the result for other varieties.


