One says that a holomorphic line bundle $L$ is globally generated by the sections $s_{0}, \ldots, s_{N}$ if $\operatorname{Bs}\left(L, s_{0}, \ldots, s_{N}\right)=\emptyset$, i.e. those sections have no common zero.
It is well known that for a positive (=ample) line bundle $L$ on a compact complex manifold, $mL$ is globally generated for some big $m$.
Let $X$ be a noncompact complex manifold. Let $L$ be a positive (i.e. admits a smooth Hermitian metric with strictly positive curvature form) line bundle on $X$.
Question: Is $mL$ globally generated for some big $m$?
Clues: Take $X:=\mathbb{C}^n$, $L=\mathcal{O}_X$, then the curvature of $(L, h_L:=e^{-|z|^2})$ is strictly positive. Now a global section of $L$ is just a holomorphic function. It is easy to know that $f=z_1$ and $f=z_1-1$ have no common zero. That is to say, under this setting, the positive bundle $L$ is globally generated.