Let $M=\mathbb{CP}^n$ be our manifold, and $U_j$ be the standard coordinate charts, i.e. $$U_j=\{[z_0:z_1:\cdots:z_n] : z_j\neq 0\}$$ with coordinates $w^{(j)}_i=z_i/z_j$ for $i\neq j.$
Consider the section of $\mathcal{O}(1)$ which is $s=z_0$ in homogeneous coordinates. In local cordinates: over $U_0$ is equal to $1$ and over $U_1$ is equal to $w_1$.
Question: How can I compute the integral $\int_M |s|_h \omega^n_{FS}$? Do I integrate over $U_0$ or over $U_1$?
I thought this shouldn't matter but it does. I am confused! What am I mising?
Any help/idea is appreciated!
You can integrate over only $U_0$. The points that are in $\mathbb{C}P^n$ but missing from $U_0$ are measure zero.
Your section $s=z_0$, it chooses the zeroth coordinate. And on $U_0$, where the zeroth coordinate is guaranteed not to vanish we can scale it to 1 $U_0=\{[1:w_1:\dotsb:w_n]\}$, so on this patch the section $s$ takes the constant value 1, right?
No. The tautological bundle $\mathcal{O}(-1)$ is constructed as a subbundle of $\mathbb{CP}^n\times \mathbb{C}^{n+1}$, and its dual bundle $\mathcal{O}(1)$ is linear functionals on $\mathbb{C}^{n+1}$, restricted to lines. So the section $s$ takes different values on the vectors $(1,0,\dotsc,0)$ and $(2,0,\dotsc,0)$, even though those points are identified in the projective space base. That just means these vectors live in the same fiber of $\mathcal{O}(-1)$. But since one vector is a scalar multiple of the other, the value of the section must also be. Namely $s((1,0,\dotsc,0))=1\neq 2 = s((2,0,\dotsc,0)).$
The upshot is that even though you are integrating over a patch where it seems your first inhomogenous "coordinate" is 1 everywhere (note: this is not actually a coordinate), you may not assume that $s$ takes that value everywhere. All sections of this bundle must have a zero, and so the norm of $s$ must go to zero.
Do you have the formula for $\lvert s \rvert_h$? It should not ever give you a constant.
Right, a natural inner product on $\mathcal{O}(1)$ is $h_i=\frac{\lvert z_i\rvert^2}{\sum\lvert z_j\rvert^2} = \frac{1}{1+\sum \lvert w\rvert^2}$. So to compute $\int \lvert s\rvert \omega_{FS}$ that is what you integrate in local coordinates.