# Global sections of vector bundles

There are a few posts that are related to what I'm asking (e.g. this one) but aren't precisely what I had in mind. The question is fairly basic but is one that confuses me:

Does a holomorphic vector bundle $V$ over a complex manifold $X$ admit a zero section? If so, why is $H^0(X, V) = 0$ (for bundles in which the structure group acts non-trivially)? I'm sure there's a fairly straightforward answer to this but I'm not so well-informed on the subject.

• All vector bundles have zero sections. This of course does not necessarily mean $H^0(X,V)=0$, it may have non-zero sections. – Mohan May 11 '18 at 0:28

Yes, there's always a zero section. $H^0(X,V)$ is the vector space of sections. Writing $H^0(X,V) = 0$ means that it is the zero vector space, which contains exactly one element, namely the zero section.
• I think I understand now - because it is a single value (0) instead of, say, $\mathbb{R}$, it is zero-dimensional instead of being one-dimensional, correct? That is, $H^0(X, V) = 0 \rightarrow h^0(X, V) = 0$ but $H^0(X, \mathcal{O}_X) = \mathbb{R} \rightarrow h^0(X, \mathcal{O}_X)) =1$? – nonreligious May 11 '18 at 4:19