I have a basic confusion. The following is drawn from Huybrechts' Complex Geometry text:
"Let $L$ be a holomorphic line bundle on a complex manifold $X$ and suppose that $s_0,\dots, s_n \in H^0(X, L)$ is a basis." (Prop. 2.3.26)
Here $H^0(X,L)$ is the space of (holomorphic) sections of the line bundle $L$.
What does it mean to have a basis for such a space? Indeed, the existence of a global frame makes any vector bundle trivial, so I presume he means something else.
A basis of sections does not provide a global frame. The sections permit zeros. Morally, one can think of the basis of sections as a basis in a function space. On the other hand a frame is a basis in the bundle fibers (as opposed to the space of sections). A priori, the space of sections could be infinite dimensional, but it is a nice fact that for $X$ compact, $H^0(X,L)$ is finite-dimensional. On the other hand, a frame always has $\mathrm{rank} L=1$ vector at each fiber. That is, a global frame for $L$ is provided by a single nonvanishing section.