ag.algebraic geometry - 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?