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.

Read more about this topic:  Stein Manifold

Famous quotes containing the words surfaces and/or stein:

    “Footnotes are the finer-suckered surfaces that allow tentacular paragraphs to hold fast to the wider reality of the library.”
    —Nicholson Baker (b. 1957)

    “As always art is the pulse of a nation.”
    —Gertrude Stein (1874–1946)