Some Computations of Line Bundles on Riemann Sphere

This is a joint work with Ziyang Zhu.

1. The Transition Functions

It is well known that the complex line bundles over the Riemann sphere $\mathbb{P}^1$ are classified by the 1st Chern Classes $c_1\in H^2(\mathbb{P}^1,\mathbb{Z})\cong \mathbb{Z}$ , for $c_1=n$ , the corresponding line bundle is denoted by $\mathcal{O}(n)$ , a classical prototype is $\mathcal{O}(-1)$ , which is so called the tautological line bundle over $\mathbb{P}^1$ , defined by $\mathcal{O}(-1):=\left\{([z],v)\in\mathbb{P}^1\times\mathbb{C}^2|v\in[z]\right\}$ , and let its dual bundle be $\mathcal{O}(1)$ , and $\mathcal{O}(n)=\mathcal{O}(1)\otimes...\otimes \mathcal{O}(1)$ , where tensors $n$ times.

In this section, we will compute the transition functions of these line bundles.

Recall that the $\mathbb{P}^1$ is constructed by $\left(\mathbb{C}^2\setminus\{0\}\right)/\sim$ , the standard open cover of $\mathbb{P}^1$ is by the upper semi-sphere and the lower semi-sphere, namely $U_1=\{[1,z]|z\in\mathbb{C}\}$ and $U_2=\{[w,1]|w\in\mathbb{C}\}$ , let’s first compute on $\pi:\mathcal{O}(-1)\longrightarrow \mathbb{P}^1$ .

A local trivialization of $\mathcal{O}(-1)$ on $U_1$ is $\phi_1: \pi^{-1}(U_1)\longrightarrow U_1\times \mathbb{C}$ , $([1,z], (t,zt))\mapsto ([1,z],t)$ , the same definition for the case of $\phi_2$ , now for some $([x,y],t)\in U_1\cap U_2\times\mathbb{C}=\mathbb{C}^*\times\mathbb{C}^*\times\mathbb{C}$ , we have

$\begin{aligned}\phi_2\circ\phi_1^{-1}([x,y],t)&=\phi_2\left(\left[1,\frac{y}{x}\right], \left(t,\frac{y}{x}t\right)\right)\\&=\phi_2\left(\left[\frac{x}{y},1\right], \left(\frac{x}{y}t,t\right)\right)\\&=\left([x,y], \frac{y}{x}t\right)\end{aligned}$

Hence the transition function on $\mathcal{O}(-1)$ is $f([x,y])=\frac{y}{x}: \mathbb{C}^*\times\mathbb{C}^*\longrightarrow \mathbb{C}$

Since the transition function of the dual is its inverse, thus we have:

Theorem 1: The transition function on  is .

So we can see that, the cotangent bundle is $\mathcal{O}(-2)$ , thus the tangent bundle is $\mathcal{O}(2)$ .

We call $\mathcal{O}(0)$ the trivial bundle, $\mathcal{O}(1)$ the Serre twist bundle, and $\mathcal{O}(-2)$ the canonical bundle.

2. The Global Sections

Next, let’s compute the global sections on $\mathcal{O}(n)$ , let us start from $\mathcal{O}(1)$ .

Suppose $s(z)=f(z)e_z$ is an arbitrary local holomorphic section on $U_1$ , i.e. $f(z)$ is a holomorphic function on $\mathbb{C}$ , $e_z$ is a basis of fiber on $z$ , let $\sigma(w)=g(w)e_w$ is an arbitrary local holomorphic section on $U_2$ , if two local sections can be glued into a global one, then they will coincide on the intersection part, let’s do the change of the variables from $U_1$ to $U_2$ , that is $f(z)\longrightarrow wf\left(\frac{1}{w}\right)$ , if $g(w)$ is gluable with $f$ , then $g(w)=wf\left(\frac{1}{w}\right)$ , note that $g(w)$ is holomorphic on $\mathbb{C}$ , thus

$\Gamma(\mathcal{O}(1))\cong\left\{f\in\mathbb{C}[[z]]|wf\left(\frac{1}{w}\right)\in\mathcal{O}(\mathbb{C})\right\}$

The RHS is generated by two functions $1, z$ .

Similarly we have

$\Gamma(\mathcal{O}(n))\cong\left\{f\in\mathbb{C}[[z]]|w^nf\left(\frac{1}{w}\right)\in\mathcal{O}(\mathbb{C})\right\}$

Where the RHS is generated by $1, z,..., z^{n}$ .

Theorem 2:

One can easily compute this fact via Riemann-Roch Formula.

3. The View of Algebraic Geometry

It is known that the line bundles correspond with the invertible sheaves, so our goal is to describe the invertible sheaves on the projective scheme $S=\mathrm{Proj}(\mathbb{C}[X,Y])$ over $\mathbb{C}$ , where we can regard $\mathbb{C}[X,Y]=\mathbb{C}\oplus(\mathbb{C}X+\mathbb{C}Y)\oplus...$ as a polynomial ring. Indeed, the invertible sheaves $\mathcal{O}(n)$ are definded by shifting the degree of this polynomial ring. With the natural $\mathcal{O}_S$ -module structure, if we cover it by affine open subsets $D_+(X)$ and $D_+(Y)$ , one can easily find the global sections of $\mathcal{O}(n)$ by the symmetric part of the shiftted ring. For example, consider the affine open subset $D_+(X)$ , the localization of $\mathcal{O}(1)$ on it is $(\mathbb{C}X+\mathbb{C}Y)\oplus(\mathbb{C}X+\mathbb{C}Y^2/X+\mathbb{C}Y)\oplus...$ , when change $X$ to $Y$ , only the invariant part of this is just $\mathbb{C}X+\mathbb{C}Y$ , actually it has dimension 2.