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:

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