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I'm looking to find a way to derive the Fubini-Study metric on $\Bbb CP^n$ and the corresponding Kähler form, but I cannot find a proper derivation for this. In most references I found they just state that the Kähler form is defined either as $$\omega_{FS} = i\partial\bar{\partial}\log \|s\|^2$$ for a section $s$ or something like

$$ \omega_{FS} ={\frac {i}{2}}\partial {\bar {\partial }}\log |\mathbf {Z} |^{2} $$

as wikipedia does. How are these things derived they seem very mysterious?

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    $\begingroup$ Here are a few bits of intuition: You want a $U(n+1)$-invariant metric on $\Bbb CP^n = U(n+1)/(U(n)\times U(1))$. Secondly, in your formula, $s$ is a nowhere-zero (local) holomorphic section of $\mathscr O(-1)$, the tautological line bundle; any two sections differ by multiplication by a nonzero holomorphic function $f$. Note that $\log\|fs\|^2 = \log |f|^2 + \log \|s\|^2$, and $\partial\bar\partial\log|f|^2 = 0$. If you're comfortable with Maurer-Cartan forms on Lie groups, I can give you a different derivation. $\endgroup$ Commented May 18 at 17:30

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$\newcommand\C{\mathbb{C}}$Here is a brief explanation but you'll need to fill in all of the details: First, denote \begin{align*} \pi: \C^{n+1}\backslash\{0\} &\rightarrow \C P^n\\ Z &\mapsto [Z] \end{align*} and for each $Z \in \C^{n+1}\backslash\{0\}$, let $$ \pi_Z: T_z\C^{n+1} \rightarrow T_{[Z]}\C P^n $$ be the pushforward map of $\pi$ at $Z$.

First, observe that given any metric $g$ on $\C^{n+1}\backslash\{0\}$, the restriction of $\pi_Z$, $$ \pi_Z: T^\perp_Z \rightarrow T_{[Z]}\C P^n, $$ where $T^\perp_Z \subset T_Z\C^{n+1}$ is the subspace orthogonal to $\pi^{-1}([Z])$, is an isomorphism. So the idea is to define a metric $\hat{g}$ on $\C^{n+1}\backslash\{0\}$ that is invariant under rescaling by a complex scalar and define the metric on $T_{[Z]}\C P^n$ to be $\hat{g}$ restricted to $T^\perp_Z$.

The Euclidean metric $dZ|^2$ on $\C^{n+1}\backslash\{0\}$ is not invariant under scaling. However, the metric $$ \frac{|dZ|^2}{|Z|^2} $$ is. This metric restricted to $T^\perp_Z$ and pushed forward to $T_{[z]}\C P^n$ is the Fubini-Study metric.

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I find it helps to zoom out a bit. Let $V$ be a complex vector space of dimension at least one and let $P(V)$ be the projective space of lines in it. We have the trivial vector space $V \to P(V)$, and the tautological line bundle $\mathcal O(-1) \subset V$. These fit into a short exact sequence $$ 0 \to \mathcal O(-1) \to V \to V / \mathcal O(-1) \to 0. $$ Now pick a Hermitian inner product $h$ on $V$. This defines a flat Hermitian metric on the trivial bundle $V$, and thus a Hermitian metric on the line bundle by restriction. That metric will have negative curvature by the Codazzi—Griffiths equations, and its negative is the Fubini—Study metric.

If you pick a basis of $V$ and the standard inner product this unpacks to the unmotivated definition in Wikipedia and other places.

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