How should the notation in this description of a global section of a line bundle in Huybrechts' complex geometry book be interpreted?

Question

In Huybrechts' book Complex Geometry, An Introduction, in problem 2.4.8 on page 97 the author describes a global meromorphic section of $K_{\mathbb P^n}$ as follows. Let $f \in H^0(\mathbb P^n, \mathcal O(n+1))$. Let $\alpha = \sum_i (-1)^i z_i f^{-1} dz_0 \wedge \dots \wedge \hat{dz_i} \wedge \dots \wedge dz_n$. My question is how this apparently informal description of $\alpha$ ought to be interpreted. If we formally attempt to pull the sum back to some $U_i$ each term with a $dz_i$ wedge now has a $d1$ wedge, hence is zero, so with $h_i$ denoting the appropriate homogenization of $f$ do we have $\alpha = (-1)^i (z_i/z_i) h^{-1} d(z_0/z_i) \wedge \dots \wedge \hat{d(z_i/z_i)} \wedge \dots \wedge d(z_n/z_i)$? In that case, what is the $z_i$ in each term actually doing here? Is it something else?

Answer 1

One way to interpret this form is first to interpret it as a global meromorphic $n$ form on $\mathbb{A}^{n+1}-0$ with the condition that in order for it to descend to $\mathbb{P}^n$ it should be invariant under the $\mathbb{C}^*$ action. So now $f^{-1}(\lambda z)=\frac{1}{\lambda^{n+1}}f^{-1}(z)$ and $d(\lambda z_0) \wedge ... \hat{d (\lambda z_i)} ... \wedge d(\lambda z_n)=\lambda^n dz_0 \wedge \dots \wedge \hat{dz_i} \wedge \dots \wedge dz_n$, so there is one power of $\lambda$ missing and an extra $z_i$ is exactly what compensates for that.

In local coordinates you wanted to use it would still be what you expected, so I think this extra $z_i$ was there so that the global expression in homogeneous coordinates makes sense, as I outlined in the first paragraph.