volume of projective space $\text{Vol}(\mathbb CP^N)$

Question

How can we compute the volume of projective space

$$\text{Vol}(\mathbb CP^N)$$

Answer 1

Let $S^1$ be the set of complex numbers of unit norm. It acts on $\mathbb C^{N+1}$ by isometries by coordinate-wise multiplication. The unit sphere $S^{2N+1}\subset \mathbb C^{N+1}$ is $S^1$-invariant, and the orbits are ``great circles", ie the intersection of (real) 2-dimensional subspaces with the unit sphere, hence are circles of radius 1 and perimeter $2\pi$. The quotient space $S^{2N+1}/S^1$ is the complex projective space $\mathbb C P^N,$ and the Fubini-Study metric on it is the quotient metric, ie the projection $S^{2N+1}\to \mathbb C P^N$ is a riemannian submersion. It follows, basically by Fubiny Theorem of integral calculus, that $vol(S^{2N+1})=vol(S^1)vol(\mathbb C P^N)$, hence $vol(\mathbb C P^N)={1\over 2\pi}vol(S^{2N+1})={\pi^N\over N!}.$ Note that $\mathbb CP^1\cong S^2$ and $vol(\mathbb CP^1)=\pi$, hence the Fubiny-Study metric on $\mathbb C P^1$ is a round 2-sphere of radius 1/2.