I heard people mentioned this in one sentence, but don't see the reason.


Why a (smooth) variety of general type, i.e. an algebraic variety X with K\_X big, is hyperbolic, i.e. has no non-constant map from the complex number into it?


I don't know what are the necessary assumption on the variety, do we need properness or smoothness?


Edit: according to David Lehavi's reply, we should certainly put some more condition on it. What's the correct statement of the fact?



Why is a variety of general type hyperbolic?
You must be thinking of Lang's conjecture which predicts that a smooth projective variety is (Brody) hyperbolic if and only if all of its irreducible subvarieties are of general type.


This is still not known in general but there are many special cases that are known. A good example is McQuillan's theorem - a smooth surface of general type which satisfies $c\_{1}^{2} > c\_{2}$ and does not contain any rational or elliptic curves is Brody hyperbolic. 


