Stein Manifold - Non-compact Riemann Surfaces Are Stein

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.

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