the simplest non-trivial line bundle over Riemann sphere

Question

We define Riemann sphere as $S=\mathbb{C}^2-\{0\}/\sim$. Given a point $p$ over $S$, I have seen somewhere there exists a line bundle $L_p$ associated to $p$, and $L_p$ has a non-zero holomorphic section with only one zero at $p$.

I think this construction is well-known to most people, I want to know how to construct the line bundle associated to $p$, by definition of the line bundle, how to define the space $E$ and the map $E\rightarrow S$ such that each fiber is a complex vector space of dimension $1$?

Thanks!

Answer 1

For simplicity let's take $P = [0: 1]$.

Cover $S$ with the two opens $U_0 = S \setminus [1: 0]$ and $U_\infty = S \setminus [0:1]$.

Now take the disjoint union of two copies of the trivial bundle: $M = (U_0 \times \mathbb{C}) \sqcup(U_\infty \times \mathbb{C})$

We impose an equivalence relation on $M$ by the rule $([a:b], z)_0 \sim ([a:b], bz/a)_\infty$ when neither $a$ nor $b$ are zero, and set $L = M/\sim$.

Since $\sim$ respects the projection to $S$, there is a natural map $L \to S$. The fiber over a point is $\mathbb{C}$, and local triviality is clear (every point is in $U_0$ or $U_\infty$).

To give a section of $L$ is to give a map $f_0: U_0 \to \mathbb{C}$ and a map $f_\infty: U_\infty \to \mathbb{C}$ such that $$ bf_0([a: b]) = af_\infty([a:b]) $$ whenever neither $a$ nor $b$ are zero.

There is such a section: $f_0([a: b]) = a/b$ and $f_\infty([a: b]) = 1$. This is locally holomorphic, so it's holomorphic, and it has a unique zero of order one at $P$ as desired.

Answer 2

This falls into the realm of the correspondence between divisors and holomorphic line bundles on a complex manifold. From this point of view, the line bundle you are describing is denoted $\mathcal{O}_{\mathbb{CP}^1}([p])$. However, this is a particularly special situation which allows for many different descriptions.

It turns out that the isomorphism type of $\mathcal{O}_{\mathbb{CP}^1}([p])$ doesn't depend on the choice of point $p$ (in the language of divisors, for any other point $q$ we have $[p] = [q]$), and is often denoted by $\mathcal{O}_{\mathbb{CP}^1}(1)$. One explicit model for the total space of $\mathcal{O}_{\mathbb{CP}^1}(1)$ is $\mathbb{CP}^2\setminus\{[0, 0, 1]\}$, where the projection map $\pi$ is given by $\pi([z_0, z_1, z_2]) \mapsto [z_0, z_1]$; geometrically, $\pi$ is the projection from the point $[0, 0, 1] \in \mathbb{CP}^2$ to the hyperplane given by $z_2 = 0$. For any choice of point $p = [a, b] \in \mathbb{CP}^1$, the bundle admits a section $\sigma$ which vanishes only at $p$, namely $\sigma([z_0, z_1]) = [z_0, z_1, bz_0 - az_1]$.

All of this generalises to $\mathbb{CP}^n$. The answers to this MathOverflow question give several other interpretations of the line bundle $\mathcal{O}_{\mathbb{CP}^n}(1)$.