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Let $X$ be a smooth projective algebraic variety and $Z$ a smooth closed subvariety of $X$. Let $f: X \to X$ be an automorphism of $X$ such that $f(Z)\subset Z$.

How does $f$ act on the normal bundle $N_{Z/X}$?

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$N_{Z/X} = TX|_Z/TZ$. By your assumption, $Tf:TX\to TX$ maps the restriction $TX|_Z$ to itself, and also $TZ\to TZ$, with the same foot-point mapping $Z\to Z$. Thus it induces $TX|_Z/TZ\to TX|_Z/TZ$.

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