Part 1: Holomorphic line bundles as first cohomology classes.

Let (X, O) be a locally ringed space consisting of a compact Riemann surface X and its structure sheaf of holomorphic functions O. We denote O* to be the sheaf of non-zero holomorphic functions.

Line bundles are completely determined by their transition functions in the following sense:

We won't go through the details of these facts for now but they do carry a powerful consequence: they allow us to describe line bundles in terms of cohomology classes. This is mostly due to the fact that the transition functions obey the same cocyle relations from Cech cohomology. Denote the isomorphism classes of holomorphic line bundles as LB(X). We are now ready to prove the main theorem of this post!

Remark: In the proof, we never really used the fact that all of these maps were holomorphic. This statement generally holds for line bundles over locally ringed spaces, such as topological, smooth and algebraic line bundles.

By virtue of this theorem, we can obtain a sheaf-theoretic description of LB(X). In the next post, we will explore this description by considering the classification of topological, smooth and holomorphic line bundles. Through this, I hope to motivate why different viewpoints of LB(X) in the holomorphic settings are useful.