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I wrote out some elementary details at basic complex line bundle on the 2-sphere.
Isn’t it just the Hopf fibration $S^1 \to S^3 \to S^2$?
Sure, it’s the complex line bundle canonically associated to the complex Hopf fibration.
I added in the description that this is the tautological line bundle of the complex projective line, which I think is probably the simplest description. Whether that belongs in the Idea section or not, I’ll let someone else decide.
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