Essential $p$-dimension and Chern numbers

Let $X$ be a smooth, projective, connected variety over a field $k$ containing a root of unity of order $p$. If $X$ has a Chern number prime to $p$, we show that every finite $p$-group acting on $X$ acts through a subgroup of $\operatorname{GL}_n(k)$, where $n=\dim X$. This allows one to transfer pr