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Let $X \hookrightarrow \mathbb{P}$ be a smooth varietyhypersurface inside some projective space $\mathbb{P}$ and let $H$ be a smooth hyperplane section of $X$. Now let $\varphi$ be an automorphism of $X$.

Why is it true that $\varphi^\ast$ acts trivially on the cohomology class of $H$?

Let $X \hookrightarrow \mathbb{P}$ be a smooth variety inside some projective space $\mathbb{P}$ and let $H$ be a smooth hyperplane section of $X$. Now let $\varphi$ be an automorphism of $X$.

Why is it true that $\varphi^\ast$ acts trivially on the cohomology class of $H$?

Let $X \hookrightarrow \mathbb{P}$ be a smooth hypersurface inside some projective space $\mathbb{P}$ and let $H$ be a smooth hyperplane section of $X$. Now let $\varphi$ be an automorphism of $X$.

Why is it true that $\varphi^\ast$ acts trivially on the cohomology class of $H$?

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why do automorphisms preserve ample divisors?

Let $X \hookrightarrow \mathbb{P}$ be a smooth variety inside some projective space $\mathbb{P}$ and let $H$ be a smooth hyperplane section of $X$. Now let $\varphi$ be an automorphism of $X$.

Why is it true that $\varphi^\ast$ acts trivially on the cohomology class of $H$?