# Aggressive geometry – Can a holomorphic vector beam on a compact Riemann surface be defined by a single transition function?

It is known that any holomorphic beam of any rank on a non-compact Riemann surface is trivial. Evidence of this is given in Forster's "Conferences on Riemann Surfaces", section 30.

Let $$E$$ to be a holomorphic vector beam on a compact surface of Riemann $$X$$ with gauge group $$G$$. A consequence of the above theorem is the restriction $$E | _ {X – {p }}$$ for any point $$p in X$$ is a trivial package. So $$E$$ can be recovered by specifying the transition function $$g: D cap (X – {p }) rightarrow G$$ or $$D$$ is a small disk containing $$p$$.

Is it correct? If no, could you give a counterexample? I am mainly interested in learning the space of holomorphic beam modules $$X$$ concretely, for example by using transition functions.