# How to write holomorphic line bundles on compact Riemann surfaces in coordinates

I would like to write down coordinate charts for holomorphic vector bundles on a smooth complex (compact) curve $C$.

For example, if $C = \mathbb C \mathbb P^1 = \{ [x_0 : x_1 ] \}$, let \begin{align*} U_0 &= \{ [x_0 : x_1] \mid x_0 \neq 0 \} = \{ [1 : z] \mid z \in \mathbb C \} \simeq \mathbb C\\ U_1 &= \{ [x_0 : x_1] \mid x_1 \neq 0 \} = \{ [w : 1] \mid w \in \mathbb C \} \simeq \mathbb C \end{align*} be the standard coordinate charts so that $U_0 \cap U_1 \simeq \mathbb C^*$ where we have $w = z^{-1}$.

A complex (holomorphic) line bundle is isomorphic to $\mathcal O_{\mathbb P^1} (n)$ for some $n \in \mathbb Z$. In the above coordinates, this line bundle may be given by the transition function $z^{-n}$ (from $U_0$ to $U_1$) or $(z^{-n})^{-1} = z^n = w^{-n}$ (from $U_1$ to $U_0$).

I would like to do the same for curves of higher genus. Since holomorphic line bundles on punctured Riemann surfaces are trivial, I again should only need two charts, but I don't know how to write the charts in coordinates or how to give the transition functions in such coordinates.

So my questions are

1. Let $C$ be a smooth complex compact curve (i.e. a smooth Riemann surface endowed with a complex structure) of genus $g > 0$. Then $C$ can be covered by two affine open sets (i.e. acyclic for coherent sheaf cohomology) $U_0$ and $U_1$. How to write $U_0$ and $U_1$ in coordinates?

2. Let $E$ be a rank $r$ vector bundle on $C$. Then $E$ is isomorphic to a bundle which is trivial when restricted to $U_0$ and $U_1$ and which is given by a transition matrix $T$ (an $r {\times} r$ matrix) with entries holomorphic in $U_0 \cap U_1$. Even for the case of line bundles ($r = 1$), what sorts of transition functions can appear?

• Since the Picard group of a curve is isomorphic to the integers, “the same thing” will happen, but with powers of the coordinate change functions. Are you looking for something more than that? Jun 2, 2018 at 9:37
• @GunnarÞórMagnússon For one, I am looking for explicit coordinates on the curve, i.e. two charts $U_0, U_1$ with acyclic intersection (acyclic for coherent sheaf cohomology). But you've startled me by saying that "the Picard group of a curve is isomorphic to the integers". As far as I know, for $C$ an elliptic curve, its Jacobian variety (another elliptic curve) parametrizes line bundles of degree $0$ on $C$ — and the Jacobian variety is very far from being isomorphic to the integers. I don't see at all how "the same thing" will happen. Jun 2, 2018 at 9:57
• Yeah, I think I’m confusing something with the degree of bundles. Never mind me. Jun 2, 2018 at 11:08