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.