# Vanishing theorems in positive characteristic

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?

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.