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?

Let $X$ be a projective variety and let $L$ be a line bundle on $X$. Suppose for all locally free 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?

After reading Laurent Moret-Bailly and Karl Schwede's comments, below I changed the condition '$M$ coherent' to '$M$ locally free'.