Why are holomorphic vector bundles over $\mathbb{C}^n$ trivial?

Question

Let $E$ be a holomorphic vector bundle over $\mathbb{C}^n$.

How do I show that $E$ is trivial?

I know this to be true for vector bundles over affine varieties but I don't know how to extend the proofto holomorphic vector bundles.

Answer 1

Yes, indeed, one way to argue is that ${\mathbb C}^n$ is contractible, which implies that all complex vector bundles over ${\mathbb C}^n$ are smoothly trivial. The second piece of information is the Oka-Grauert Principle, see for instance Theorem 5.3.1 in

F. Forstneric, Stein Manifolds and Holomorphic Mappings, Springer Verlag, 2011.

According to the Oka-Grauert Principle, the classification of holomorphic vector bundles over Stein manifolds (and ${\mathbb C}^n$ is Stein) is equivalent to their classification as smooth bundles. Putting these two things together, you conclude that every holomorphic vector bundle over ${\mathbb C}^n$ is (holomorphically) trivial.