# Explicit construction of Fubini Study Metric

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 = \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? • 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$T_p(M/G)$. – Igor Belegradek Jul 15 '20 at 22:54 • 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$\mathbb{CP}^n$. – Robert Bryant Jul 15 '20 at 23:06 • 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$given by – user7391733 Jul 15 '20 at 23:53 • Not sure what you mean by a "cooking recipe". Petersen's "Riemannian geometry" book covers the material; just look up "Riemannian submersion" and "projective space" in the index. – Igor Belegradek Jul 16 '20 at 0:36 • The$\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. – Deane Yang Jul 16 '20 at 4:20