Let $X$ be a projective variety and let $L$ be a line bundle on $X$. Suppose for all coherent sheaves $M$ on $X$,
$
H^i(X,{L^*}^{\otimes r} \otimes M)=0
$ for $i<\dim X$ and $r$ sufficiently big. 

Does it follow that $L$ is an ample line bundle? Here $L^*$ denotes the dual of $L$.

This is of course clear  if $X$ is smooth using Serre duality, but how is it in general?