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 $\mathbb{CP}^n$ inherits a 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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    $\begingroup$ Yes, this metric is well-known. If $G$ acts isometrically and freely on a Riemannian manifold $M$, then the quotient space $M/G$ has the so-called Riemannian submersion metric. Points of $M/G$ are in a bijection with $G$-orbits. A tangent plane at a point of $p\in M/G$ is represented by a normal subspace to the corresponding $G$-orbit. The $G$-action isometrically identifies the normal spaces to a $G$-orbit, and the thus gives an inner product on $T_p(M/G)$. $\endgroup$ – Igor Belegradek Jul 15 '20 at 22:54
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    $\begingroup$ Perhaps one can simply say that the Fubini-Study metric is the one induced on the quotient of $S^{2n+1}\subset\mathbb{C}^{n+1}$ (i.e., the unit vectors with respect to the Hermitian metric) by the $S^1$-isometric action of multiplication by $\{\mathrm{e}^{it}\ |\ t\in\mathbb{R}\ \}$. If one wants the other scale factors, then one just takes, instead of $S^{2n+1}$, any other nonzero level set of the Hermitian length function. Since this $S^1$ commutes with $\mathrm{SU}(n{+}1)$ (which acts by isometries on $S^{2n+1}$), one has that $\mathrm{SU}(n{+}1)$ acts via isometries on $\mathbb{CP}^n$. $\endgroup$ – Robert Bryant Jul 15 '20 at 23:06
  • $\begingroup$ Could you give reference where the construction of this Riemannian (Hermitian) submersion metric $\bar{h}$ with respect $ M \to M/G$ and Riemannian (Hermitian) mertic $h$ on $M$ is explicitly described (preferably as "cooking recipe"? If this is exactly the constuction I'm looking for then this precisely gives the Fubini Study metric $f_{FS}$ on $\mathbb{C}^{n+1} / \mathbb{C}^*$ which is (accoring to en.wikipedia.org/wiki/…) on affine chart $U_0$ given by $\endgroup$ – user7391733 Jul 15 '20 at 23:53
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    $\begingroup$ Not sure what you mean by a "cooking recipe". Petersen's "Riemannian geometry" book covers the material; just look up "Riemannian submersion" and "projective space" in the index. $\endgroup$ – Igor Belegradek Jul 16 '20 at 0:36
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    $\begingroup$ The $\partial\bar\partial \log \langle Z,Z\rangle$ is not really just a clever guess and is in fact canonical. You want a Kahler potential that is defined as a function of the Hermitian product (and nothing else) but that defines a Kahler form that is homogeneous of degree $0$, so that it can be pushed down to $\mathbb{C}P^n$. The logarithm is the only function that will do this. $\endgroup$ – Deane Yang Jul 16 '20 at 4:20

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