In case of smooth projective scheme $X$ over a field of characterics $0$ I want to prove the isomorphism: $$ L^{n-p-q}: H^{q}(X, \Omega_{X/k}^p) \rightarrow H^{n-p}(X, \Omega_{X/k}^{n-q})$$ where $L$ is an operator defined by $_ \wedge \omega$ for $\omega = c_1(\mathcal{O}_X(1))\in H^{1}(X, \Omega_{X/k})$. We can translate this problem using GAGA to a Kähler manifold $X^{an}$. This approach fails without the assumption of characteristics 0. What can be said about projective scheme over field of finite characteristics, or what conditions do we need to impose for this theorem to still be true?
Thank you very much.
Beyond the basic sheaf cohomology for schemes, there is a notion of a Weil cohomology theory. This is a collection of axioms that mimic the theorems for singular cohomology. Among these axioms are: Poincaré duality, vanishing results $H^i = 0$ for $i < 0$ or $i > 2 \dim X$, and weak and hard Lefschetz.
The four known Weil cohomology theories are:
l-adic cohomology is an instance of étale cohomology which is designed to be more "geometric" than sheaf cohomology (i.e. should have similar properties to singular cohomology i.e. is a Weil cohomology theory). For instance, $$ H^r_{\rm ét}(X,A) \cong H^r_{\rm sing}(X(\mathbf{C}),A) $$ for every finite abelian group $A$. Taking an inverse limit over $\mathbf{Z}/p$ and tensoring with $\mathbf{Q}_\ell$ gives an isomorphism in l-adic cohomology: $$ H^r_{\rm ét}(X,\mathbf{Q}_\ell) \cong H^r_{\rm sing}(X(\mathbf{C}),\mathbf{Q}_\ell).$$ That is, in characteristic 0, l-adic cohomology is just singular cohomology but over the l-adic numbers.
Hard Lefschetz was proven for l-adic cohomology by Deligne in 1980 "La conjecture de Weil. I et II."
If you want to learn more, I would read up on étale and l-adic cohomology, for instance using Milne's notes. Then you can read Deligne's papers if you really want to know the proof of hard Lefschetz for l-adic cohomology.