Why is a variety of general type hyperbolic? 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?
 A: 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. 
A: you can blow up a point on a general type surface, and get a general type surface containing a copy of C. 
A: Surely you need some assumptions on  your variety, for example for every $n$ there is a smooth surface of degree $n$ in $CP^3$ that contains a line, for $n>3$ such surfaces are minimal and of general type. So for large classes of varieties of general type you need an additional assumption that the variety should be generic.
For hypersufaces in $CP^n$ the most optimistically you can hope that a generic hypersurface of degree $2n+1$ is hyperbolic (hypersurfaces of degree $2n-1$ and less always contain lines). This is related to a conjecture of Kobayshi.
There is a very nice review of Claire Vosin on different aspects of hyperbolicity of complex projective manifold that you can find here http://people.math.jussieu.fr/~voisin/Articlesweb/harvard.pdf
(this also contains the result mentioned by Tony Pantev)
Recently there was a genuing progress in proving  of Kobayashi conjecture
http://arxiv.org/PS_cache/arxiv/pdf/0811/0811.2346v4.pdf
though the obtained bound is very far fron optimal, worse than a triple exponent of n. At the conference due to 80 birthday of Atiyah Kirwan annonced that she can get a much more realistic bound.
