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 $N_{Z/X}$?
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 $N_{Z/X}$? 


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

