In the paper


Deligne, Pierre; Illusie, Luc (1987), "*Relèvements modulo $p^{2}$ et décomposition du complexe de De Rham*", Inventiones Mathematicae **89** (2): 247–270, doi:10.1007/BF01389078


I found the following



> 
> **Theorem.** Let $X$ be a smooth and projective scheme over a field $k$ with $\textrm{char}(k) = p > 0.$ Assume that $X$ admits a lift to $W\_{2}(k)$. If $L$ is an ample line bundle on $X$ then
>  $$H^{i}(X,L^{-1}) = 0 \: \: \textrm{for} \: \: i < \min(p, \, \dim X).$$
> 
> 
> 


I just wanted to ask if anyone knows a theorem like this under weaker hyphoteses. More precisely I have the following



> 
> **Question.** Does such result hold for $X$ normal and $L$ big and nef?
> 
> 
> 



Vanishing theorems in positive characteristic
In the book *Lectures on Vanishing Theorems* by Esnault and Viehweg (Birkhäuser 1992) this is stated as an open problem. See in particular Problem 11.7 page 132.


However, they are able to prove the result when $\dim X=2$, since in this case one can perform the embedded resolution of singularities for curves on surfaces. See in particular Proposition 11.5 p. 129 and the subsequent remarks.


