When are the Smooth Sections of a Bundle Generated as a Module (over Smooth Functions) by the Holomorohic Sections

Question

For a holomorphic vector bundle $E$ over a complex manifold $M$, we denote its space of smooth sections by $\Gamma^{\infty}(E)$, and its space of holomorphic sections by $\Gamma^{hol}(E)$. Now I've been looking at the line bundles $L_k$ over the complex projective spaces ${\bf C} P^N$, and I have managed to show that $\Gamma^{\infty}(L_k)$ is generated as a $C^{\infty}({\bf C} P^N)$-module by $\Gamma^{hol}(E)$, which is to say that every element $\Gamma^{\infty}(E)$ is a sum of elements of the form $ef$, where $e \in \Gamma^{hol}(E)$, and $f \in C^{\infty}({\bf C} P^N)$.

I am guessing that this result is extremely well known, and an example of a well understood general phenomenon. So I would like to ask if there is a characterization of the manifolds for which this result holds, for both the case of line bundles alone, and holomorphic vector bundles of general dimension?

Answer 1

Swan has proved that taking global section gives an anti-equivalence between finitely generate projective $\Gamma^{\infty}(M)$-modules and $C^{\infty}$ vector bundles on $M$; this correspondence is functorial in $M$. Hence a set of section of a $C^{\infty}$ vector bundle $E$ on $M$ generated $\Gamma^{\infty}(E)$ if and only if it generates each fiber of $E$. So if $E$ is holomorphic, the holomorphic sections generate if and only if $E$ is globally generated. In particular, this is always true if $M$ is Stein. In the case of $L_k$ on $\mathbb{CP}^N$, this is true if and only if $k ≥ 0$.