I have a question about a remark on Fubini Study metric on $\mathbb{CP}^n$ from Notes on canonical Kähler metrics

on page 8 is remarked (Example 2.12 4.):

Fix a Hermitian innerproduct on $\mathbb{C}^{n+1}$. Then $\mathbb{CP}^n$inheritsa unique (up to scale) $U(n+1)$-invariant Riemannian metric, called the Fubini– Study metric. It is Kähler, as can be seen in various ways. [...]

My question is how Fubini–Study metric $f_{FS}$ concretly
arise as *inherited*
metric from standard Hermitian product $\langle-,-\rangle_h$ on $\mathbb{CP}^{n+1}$?
It suggests that to obtain $f_{FS}$ we consider the canonical projection
$\mathbb{C}^{n+1} \backslash \{0\} \to \mathbb{CP}^n$ where
$\mathbb{C}^{n+1} \backslash \{0\}$ is endowed still with restricted metric
$\langle-,-\rangle_h$ and $f_{FS}$ on is induced from $\langle-,-\rangle_h$ in certain canonical
way.

Could somebody give a sketch how this explicit construction of $f_{FS}$ from standard metric $\langle-,-\rangle_h$ works?

More precisely the problem is that in almost
every source
the Fubini Study metric beeing introduced by a
"let *define* the FS as ..." and then one verifiy that the defined metric is
$U(n+1)$-invariant Riemannian and so on (see e.g. Here or the classical literature by Griffith/Harris or R. Wells)

But the quoted sentence above suggest that there is a way to
*construct* the Fubini-Study metric from
$\langle-,-\rangle_h$ on $\mathbb{C}^{n+1}$ explicitly / canonically.
That's what I understand
by "inherited". And I'm asking where I can find source
on Fubini-Study metric where it is *constructed* from $\langle-,-\rangle_h$$
and not simply defined as something "fallen from heaven" &
has desired properties.

The most "natural" approach I found was by defining

$$f_{FS}:= \bar\partial\partial \log <Z,Z>= \bar\partial\partial \log (1+z^2)$$

where $Z =[1:z_1:....z_n] \in U_0 :=\{Z \in \mathbb{CP}^n \ \vert z_0 \neq 0 \} \subset \mathbb{CP}^n$.

Although this construction indeed only depends on $\langle-,-\rangle_h$ this also not persuades me completly. The $\bar\partial\partial \log \langle Z,Z \rangle_h$ shape looks too artificially; i.e. more like a "sophisticated guess" than something "canonical".

I *expected* to find something like a general machinery for it like: when $M$ is
a Riemannian/Hermitian manifold endowed with metric $h$ and suppose
a group acts on $M$ nice enough such that $M/G$ is also a manifold.

Then there is a standard method to obtain a metric $\bar{h}$ on $M/G$ inherited from $h$ on $M$. And in our example the $\bar{h}$ is nothing but the Fubini Study.

Does there exist such procedure in cases for $M$ & group action by $G$ nice enough? Or do I looking for something that not exist and the definition $f_{FS}= \bar\partial\partial \log \langle Z,Z \rangle_h$ is indeed the only meaningful way to introduce Fubini-Study metric?

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