Automorphism that preserves Kahler class The following statement is a Lemma from the paper  "Kahler Manifolds with trivial canonical class" by F. A. Bogomolov:
Let $F:M\mapsto M$ be an automorphism of algebraic manifold $M$, which preserves Kahler class $[w]$ and Volume form $w^n$. Then there exists a Kahler form $w'$, s.t. $[w']=[w]$ and $F^*w'=w'.$
How could we see that the statement is trivial when $M=\mathbb{C}P^n?$

There is only one Kahler class on the projective space (up to positive constant muliples); the on of the Fubini-Study metric $\omega$. If $F$ is an automorphism of the projective space, then $\omega' := F^*\omega$ is again a Kahler metric. The hypothesis that $F$ preserves the volume form implies that the Ricci-form of $\omega'$ is equal to the Ricci-form of the Fubini-Study metric. By uniqueness of the Ricci-positive Kahler metric on the projective space, $\omega' = \omega$. Scaling by positive reals then treats the case when we consider a multiple of the Fubini-Study class.
