Let $ X$ be a smooth projective curve, and $ E$ an indecomposable vector bundle on $ X$ with $ \mathrm{deg} E<0$ . Is it true that $ H^0(X,E)=0$ ?

This is true if $ E$ is a line bundle, which means it is also true whenever $ X$ is $ \mathbb{P}^1$ , since all vector bundles split here.

It is also true by results of Atiyah if $ X$ is an elliptic curve. What about for curves of higher genus?

The assumption that $ E$ is indecomposable is of course necessary.