Nakano vanishing in positive characteristic 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:

$(\ast) \quad$ $\mathrm H^0 \big( X, \Omega_X^p \otimes \mathscr L^{-1} \big) = 0$ for all $p < \dim X$ and ample line bundles $\mathscr L$.

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).
 A: 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. 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.
A: A general (logarithmic) Kodaira-Akizuki-Nakano vanishing theorem in characteristic $p$ is proven in $\S$ 11 of
H. Esnault, E. Viehweg: Lectures on vanishing theorems. Notes, grew out of the DMV-seminar on algebraic geometry, held at Reisensburg, October 13-19, 1991, DMV Seminar. 20. Basel: Birkhäuser Verlag. 164 p. (1992). ZBL0779.14003,
see in particular Corollary 11.3. The precise statement is the following

Theorem. Let $k$ be a perfect field, let $X$ be a proper smooth $k$-scheme and $D \subset X$ a normal crossing divisor, both admitting a lifting $\tilde{D} \subset \tilde{X}$ to $W_2(k)$.
For $p+q < \mathrm{min}\{\mathrm{char}(k), \, \dim X\}$ and $\mathscr{L}$ ample and invertible, we have $$H^q(X, \, \Omega^p(\log D) \otimes \mathscr{L}^{-1})=0.$$

So you cannot have a counterexample to $(*)$, unless $k$ is not perfect or the pair $(X, \, D)$ does not lift to the ring of Witt vectors. I guess that counterexamples in the last case could be obtained by using the same strategy of Reynaud's famous counterexample for Kodaira vanishing in positive characteristic (see the reference below), but unfortunately I do not know any of them.
M. Raynaud: Contre-exemple au ''vanishing theorem” en caracteristique $p>0$, C.P. Ramanujam. - A tribute. Collect. Publ. of C.P. Ramanujam and Pap. in his Mem., Tata Inst. fundam. Res., Stud. Math. 8, 273-278 (1978). ZBL0441.14006.
