Let $X$ be a smooth, projective, connected algebraic variety defined over a subfield of $\mathbb{C}$. Assume $X$ is equipped with an automorphism $g: X \to X$.
By functoriality we get morphisms $g^\ast: H^k(X(\mathbb{C}), \mathbb{Q}) \to H^k(X(\mathbb{C}), \mathbb{Q})$ for any $k$.
How can one prove that $g^\ast$ acts trivially on $H^0(X(\mathbb{C}), \mathbb{Q})$?