Let $X \subset \mathbb{P}^N$ be a smooth complete intersection, say over the complex numbers, and let $g$ be a finite order automorphism of $X$.

I would like to prove that $g^\ast$ acts trivially on $H^j(X(\mathbb{C}), \mathbb{Q})$ for all $i \neq \dim X$.

Can someone help me?

I guess what is needed is to relate $H^j(X(\mathbb{C}), \mathbb{Q})$ to $H^i(\mathbb{P}^N(\mathbb{C}), \mathbb{Q})$ which is very simple.

I know that automorphisms of the projective space act trivially on cohomology, but to use that it would be necessary to know that $g$ is induced by an automorphism of $\mathbb{P}^N$. Is that true?

Or maybe the reason is simply that $H^i(X(\mathbb{C}, \mathbb{Q})$ is 0 or 1-dimensional for $i \neq \dim X$?

Thanks for your help!