When is a Stein manifold a complex affine variety? I had thought that there was a theorem saying that a variety which is Stein and has finitely generated ring of regular functions implies affine, but in the comments to my answer [here][1], Serre's counterexample was brought up.  I'm guessing that the answer is that the ring of regular functions must be nontrivial somehow, like it must separate points, but I'm curious about what the exact condition is.


  [1]: http://mathoverflow.net/questions/2071/non-finitely-generated-ring-of-regular-functions