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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 $H^0(X(\mathbb{C}), \mathbb{Q})$?

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    I think the problem is to reduce to the case of an automorphism on $H^0(X(\mathbb{C}), \mathbb{Z})$, which can only be $\pm 1$, and eliminate the case $-1$...2012-06-28

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