I am trying to solve a problem in Introduction to Complex Geometry by D. Huybrechts, question 3.1.6 which is the following: let $A\in GL(n+1, \mathbb{C})$ be a $\mathbb{C}$-linear transformation $\mathbb{C}^{n+1}\rightarrow\mathbb{C}^{n+1}$, inducing the automorphism $F_{A}:\mathbb{P}^{n}\rightarrow\mathbb{P}^{n}$. Let $\omega_{FS}$ denote the Fubini-Study metric on $\mathbb{P}^{n}$. The question asks
Show that $F^{*}_{A}\omega_{FS}=\omega_{FS}$ if and only if $A\in U(n+1)$.
I could do one direction: if $A\in U(n+1)$ is unitary, then the local coordinates for $F^{*}_{A}\omega_{FS}$ is something like
$\displaystyle F^{*}_{A}\omega_{FS}=\frac{1}{(1+\|Az\|^{2})^{2}}\left((1+\|Az\|^{2})A^{t}\overline{A}-A^{t}\overline{A}\overline{z}z^{t}A^{t}\overline{A}\right)$
which turns out to be $\omega_{FS}$ (unless I am mistaken in my calculations...)
How should I prove the converse?
I mean, if we see this as a metric and apply to $z$ itself, I get something like
$\displaystyle z^{t}F^{*}_{A}\omega_{FS}\overline{z}=\frac{\|Az\|^{2}}{(1+\|Az\|^{2})^2}=\frac{\|z\|^{2}}{(1+\|z\|^{2})^{2}}=z^{t}\omega_{FS}\overline{z},$
which doesn't say anything. I have tried taking the trace and the determinant and it seems like I can't get anything close to proving that it is unitary. Can anyone point references (with proofs!) or give some hints to this question? Thanks!
