Fubini-Study metric induced by submersion

Question

The Fubini-Study metric $g:=g_{FS}$ is the unique $U(n+1)$-invariant Riemannian metric on the complex projective space $\mathbb{CP}^{n}$ the complex projective space which by $U(n+1)$-invariance can be wlog definined on tangent bundle over affine chart $ U_0 :=\{Z \in \mathbb{CP}^n \ \vert z_0 \neq 0 \} \subset \mathbb{CP}^n $ and then it extends uniquely by $U(n+1)$-action to all affine pieces $U_i$.

So we can restict our considerations to tangent space $T_Z \mathbb{CP}^n$ over a point

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

Let $\{\partial _{1},\ldots ,\partial _{n}\}$ the be frame of tangent space
$T_Z \mathbb{CP}^n$ at $Z$, ie the canonical $\mathbb{C}$-basis of $T_Z \mathbb{CP}^n$. Then the Fubini Study metric is defined by

$$\tag{FS} g_{i{\bar {j}}}=h(\partial _{i},{\bar {\partial }}_{j})= {\frac {(1+|\mathbf {z} |^{2})\delta _{i{\bar {j}}}- {\bar {z}}_{i}z_{j}}{(1+|\mathbf {z} |^{2})^{2}}} $$

At first glance one can think that $g$ somehow "has hallen from heaven" but it is well knows that it's not. One way to derive it is using Kähler potentials and canonical choice of a distinguished connection: Chern connection: See Here the excellent answer by Arctic Char.

The concern of this question is to find out if it also possible to derive $g$ on $\mathbb{CP}^{n}$ naturally by recognizing $(\mathbb{CP}^{n},g)$ as Riemannian submersion of $(\mathbb{C}^{n+1} \backslash \{0\},h)$ where $h$ is to restriction of standard Hermitian metric on $\mathbb{C}^{n+1}$ to $\mathbb{C}^{n+1} \backslash \{0\}$.

What do I mean by 'naturally"?

Well, $\mathbb{CP}^{n}$ can be considered as submersion of complex manifold $\mathbb{C}^{n+1} \backslash \{0\}$ via canonical projection

$$\pi: \mathbb{C}^{n+1} \backslash \{0\} \to \mathbb{CP}^n$$

The complex manifold is moreover Hermitian since $ \mathbb{C}^{n+1} \backslash \{0\}$ inherits naturally the standard Hermitian metric $h$ on $ \mathbb{C}^{n+1} $. Futhermore by $\pi$ the $\mathbb{CP}^{n}$ can also be considered as quotient $ \mathbb{C}^{n+1} \backslash \{0\} / \mathbb{C}^*$.

By definition a Riemannian submersion $F: (M,g_M) \to (N, g_N)$ is a submersion from one Riemannian manifold $M$ with Riemannian metric $g^M$ to another Riemannian manifold $N$ with Riemannian metric $g^N$ that respects the metrics, meaning that it is an orthogonal projection on tangent spaces.

Namely that means that for every $x \in M$ and $y=F(X)$ the restriction of the differential map $dF_x$ to $ker(dF_x)^{\perp} \subset T_x M $ gives an isometry $d_x F: ker(dF_x)^{\perp} \to T_y N$ such that the metric $g^N$ is induced by $g^M$ is following sense:

Let $A, B \in T_y N$ and denote by $\overline{A}, \overline{B} \in ker(dF_x)^{\perp}$ their unique preimages with respect the isometry $d_x F: ker(dF_x)^{\perp} \to T_y N$.

Then

$$\tag{RieSub} g^N _y(A,B)= g^M _x(\overline{A}, \overline{B})$$

QUESTION:

How can be proved that the Fubini-Study metric $g$ arises exactly on this way? ie that the canonical projection

$\pi: \mathbb{C}^{n+1} \backslash \{0\} \to \mathbb{CP}^n$

extends naturally to Riemannian submersion $\pi: (\mathbb{C}^{n+1} \backslash \{0\},h) \to (\mathbb{CP}^n, g)$

is the sense above, where $h$ is the restriction of the stadard Hermitian metric on $\mathbb{C}^{n+1}$ and $g$ is the Fubini- Study metric defined above in (FS).

In other words we need to verify that the metrics $h$ and $g$ are related to each other as in (RieSub).

Take as $y= Z =[1:z_1:....z_n]$ and $x =(1,z_1,..., z_n) \in \mathbb{C}^{n+1}$ and $\partial _{i}, \partial _{j} \in T_y \mathbb{CP}^n$ element of the canonical basis of the tangent space at $y$. Firstly we need an explicit identification between the tangent space $T \mathbb{CP}^n$ and $ker(d \pi)^{\perp}$. We need this isometry explicitly since in order to prove (FS) we have to work with unique preimages of elements in $T_y \mathbb{CP}^n$ living in $ker(d \pi_x)^{\perp}$.

But I don't see how can I show that indeed

$$g(\partial _{i},{\bar {\partial }}_{j})= {\frac {(1+|\mathbf {z} |^{2})\delta _{i{\bar {j}}}- {\bar {z}}_{i}z_{j}}{(1+|\mathbf {z} |^{2})^{2}}} $$

holds. On the left side we have already identified $\partial _{i}, \partial _{j} \in T_y \mathbb{CP}^n$ with their unique preimages in $ker(d_x \pi)^{\perp}$.

Answer 1

Let me start from scratch. Note that everything below uses only the definition of complex projective space and the natural Hermitian inner product on $\mathbb{C}^{n+1}$. Also, the construction is coordinate-independent in the sense that everything below can be done with an abstract complex vector space with a Hermitian inner product without using any basis.

Recall that canonical projection map is defined to be \begin{align*} \mathbb{C}^{n+1}\backslash\{0\} &\rightarrow \mathbb{C}P^n\\ z &\mapsto [z], \end{align*} where $[z] = \{ tz\ :\ t \in \mathbb{C} \}$. For each $z \ne 0$, the tangent space of $\mathbb{C}^{n+1}$ at $z$ splits naturally $$ T_z\mathbb{C}^{n+1} = [z] \oplus z^\perp, $$ where $z^\perp = \{ w \in \mathbb{C}^{n+1}\ :\ \langle w,z\rangle = 0 \}$. For each $z$, the pushforward map $\pi_*: T_z\mathbb{C}^{n+1} \rightarrow T_{[z]}\mathbb{C}P^n$ is an isomorphism if restricted to $z^\perp$.

We want to use the Hermitian inner product to define a Kähler metric $g$ on $\mathbb{C}P^n$. Observe that, given such a metric, $\pi^*g$ is an degenerate Hermitian $2$-tensor $h$ on $T_z\mathbb{C}$ that is scale invariant in the sense that if $R_t(z) = tz$, then $(R_t)^*h = h$. Conversely, a Hermitian metric on $\mathbb{C}^{n+1}$ defines one on $\mathbb{C}P^n$ only if its restriction to each $z^\perp$ is scale invariant.

If we want to define the metric using only the standard Hermitian inner product pointwise on $\mathbb{C}^{n+1}$, then the only possibility is one of the form $$ g = f(|z|^2)|dz|^2 $$ restricted to $z^\perp$, for each $z \in \mathbb{C}^{n+1}\backslash\{0\}$. On the other hand, if you work out all the definitions carefully, you find that $$ R_t^*g = f(|t|^2|z|^2)|t|^2|dz|^2, $$ which equals $g$ if and only if there exists a real constant $c$ such that $f(|z|^2) = c|z|^{-2}$. By the observations above, this metric can be pushed down to a metric on $\mathbb{C}P^n$, and it can be verified that it is, up to a scale factor, the Fubini-Study metric.

Note that the explanation above is just the definition of a Riemannian submersion but for the specific situation here.