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"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.

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For starters loook at Kobayashi,Hyperbolic Complex Spaces –  Mohan Ramachandran Dec 5 '11 at 20:04

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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).

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