Holomorphic vector bundles and Swan's theorem Is every holomorphic vector bundle a direct summand of a trivial vector bundle on submanifolds of C^n? What about projective varities? I believe Swan's theorem says something about the first question. But I wanted to make sure.
 A: The statement for Stein manifolds follows indeed from the analogue of the Serre-Swan theorem for Stein manifolds, which was proven first in 1967 in "Zur Theorie der Steinschen Algebren un Moduln" by O. Forster. The situation is a bit more complicated than the affine scheme or manifold case, but the final result relevant for the purposes of the question is the same
The category of locally free sheaves of finite rank is the same as the category of finitely generated projective modules over the global sections $\Gamma(O_X)$. Then one notes that a f.g. projective module is always a direct summand of a finite free module. 
A: Yes, for $\mathbb{C}^n$ itself, since vector bundles are (holomorphically) trivial. See Griffiths and Adams " Topics in Algebraic and
Analytic Geometry" p 209. I would need to think about the case of submanifolds, before giving
an answer. But definitely NO for nontrivial projective varieties: an ample line bundle won't be a summand of a trivial vector bundle. Proof: If it were, then its dual would be generated
by global sections, and this is impossible.
