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I know that the topic is classical and even "folklore", but many treatments make use of local coordinates and such treatments are rather messy. Could somewhere maybe provide some reference(s)...

Suppose that we have a complex manifold $X$, and a line bundle $L$ over $X$. It is known that the line bundles over $X$ are parametrized by their Chern class, the Chern class being the cohomology ...