Let X be a smooth, projective variety over a field $k \hookrightarrow \mathbb{C}$ and let $g$ be an automorphism of $X$ of finite order. Consider the induced automorphism on the singular cohomology
$g^\ast: H^j(X(\mathbb{C}), \mathbb{Q}) \to H^j(X(\mathbb{C}), \mathbb{Q})$
(or in the De Rham cohomology). Is is true that $g^\ast=id$ when $j \neq \dim X$?
I would really appreciate your help!
Thanks