Non-compact Riemann Surfaces Are Stein
Let X be a connected non-compact Riemann surface. A deep theorem of Behnke and Stein (1948) asserts that X is a Stein manifold.
Another result, attributed to Grauert and Röhrl (1956), states moreover that every holomorphic vector bundle on X is trivial.
In particular, every line bundle is trivial, so . The exponential sheaf sequence leads to the following exact sequence:
Now Cartan's theorem B shows that, therefore .
This is related to the solution of the Cousin problems, and more precisely to the second Cousin problem.
Read more about this topic: Stein Manifold
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