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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?

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  • $\begingroup$ 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? $\endgroup$ – Gunnar Þór Magnússon Jun 2 '18 at 9:37
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    $\begingroup$ @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. $\endgroup$ – Earthliŋ Jun 2 '18 at 9:57
  • $\begingroup$ Yeah, I think I’m confusing something with the degree of bundles. Never mind me. $\endgroup$ – Gunnar Þór Magnússon Jun 2 '18 at 11:08

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