# Fubini-Study metric/form

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?

• 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. Commented May 18 at 17:30

$$\newcommand\C{\mathbb{C}}$$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.
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.