Let $X$ be a smooth projective variety defined over a field $k$.
In characteristic zero, the following is a special case of the (Kodaira-Akizuki-)Nakano vanishing theorem:



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> $(\ast) \quad$ $\mathrm H^0 \big( X, \Omega\_X^p \otimes \mathscr L^{-1} \big) = 0$ for all $p < \dim X$ and ample line bundles $\mathscr L$.
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I am interested in what happens in positive characteristic.


**Question 1:** Is there a counterexample to $(\ast)$ when $\mathrm{char}(k) > 0$?


Since the statement is obviously related to Kodaira vanishing, I would guess that a counterexample is already known.
The following question, however, might be harder:


**Question 2:** Is there a counterexample as above, but additionally with $K\_X \sim\_{\mathbb Q} 0$ or $X$ Fano?


Actually, a log version would also be sufficient, with $\Omega\_X^p$ replaced by $\Omega\_X^p(\log D)$, where $(X, D)$ is an snc pair (and in the second question, $K\_X + D \sim\_{\mathbb Q} 0$ or $(X, D)$ log Fano).



Nakano vanishing in positive characteristic
There exists singular Fano varieties of dimension $n\ge 3$ in characteristic $p$ ($p$ small when compared to $n$) with a non-zero section of $\Omega^{n-1}\_X\otimes \mathcal L^\*$ for an ample $\mathcal L$.
The existence of these examples is established in Kollár’s paper [Nonrational hypersurfaces](https://www.ams.org/journals/jams/1995-08-01/S0894-0347-1995-1273416-8/S0894-0347-1995-1273416-8.pdf). They are constructed as degree $p$ coverings of Fano manifolds ramified over smooth hypersurfaces. It is unclear to me if in any of his examples $X$ is actually smooth.


