Varieties are isomorphism

Question

Let $f:Y\to X$ be a birational morphism, Y is projective. Let $H$ be a very general ample divisor on $Y$. If $f^{*}f_{*}H=H$, is it true that $Y$ is isomorphic to $X$?

Answer 1

By $f^{*}f_{*}H=H$, there is no curve contracted by $f$. Hence $f$ is finite. If assume $X$ is normal, then this birational morphism must be an isomorphism by Zariski Main Theorem.