First sheaf Cohomology on $\mathbb C$

Question

If E is a holomorphic vector bundle on $X= \mathbb C$

is it true that $H^1(X,E)=0$?

I know it's true for $E=O_X$ the sheaf of holomorphic functions by Dolbeaut theorem.

Answer 1

Yes, in general if $E$ is a holomorphic vector bundle over a Stein manifold $X$, then $H^p(X,\mathcal{O}(E))=0$ for all $p\geq 1$, by Cartan's theorem B.