"Little Picard" states that if a complex function $f(z)$ is entire and non-constant, then the set of values that f(z) assumes is either the whole complex plane or the plane minus a single point. The original proof used modular functions.


I'm wondering if this has been generalized to other open complex manifolds. I don't expect them to be true for all complex manifolds as the trivial counter example $\mathbb{CP}^1-\text{points}$ indicates maybe that is not possible. 


However, I still want to know what condition/structure on complex manifolds will force functions on it to assume all value expect one or two points.



Little Picard for (open) complex manifolds?
This paper seems to address this question in considerable generality:


Some Picard Theorems for Holomorphic Maps to Algebraic Varieties
by ML Green - 1975 -


(it is also very widely cited, so presumably is not the last word on the subject).


