TITLE: Rational or elliptic curves on Calabi-Yau threefolds
QUESTION [20 upvotes]: Let $X$ be a Calabi-Yau threefold. From a complex analytic point of view, it is widely believed that it should not be Kobayashi hyperbolic, that is it should always admit some non-constant entire map from the complex plane $f\colon\mathbb C\to X$. 
One could be even more ambitious and ask whether a Calabi-Yau threefold always contains a rational or an elliptic curve (or, more generally a non-constant image of a complex torus).
Mostly string theorists have produced lots of examples of such manifolds, mainly by adjunction or crepant resolution of singularities. So my question is:
Is it true that in all known examples of Calabi-Yau threefold one can always find a rational or an elliptic curve (or, more generally a non-constant image of a complex torus)?
Thanks in advance!

REPLY [9 votes]: Let me give a partial answer.
Most of the known examples of Calabi-Yau threefolds contain rational curves. However, there exist examples of Calabi-Yau threefolds without rational curves.
You can find some of them in the paper by Oguiso and Sakurai Calabi-Yau threefolds of quotient type, Asian Journal of Mathematics 5 (2001).
These threefolds, that the authors  call "of Type A", are constructed as the quotient af an Abelian threefold $A$ by a suitable fixed-point free finite group of automorphisms. 
Moreover, a Calabi-Yau threefold $X$ is  of type A if and only if $c_2(X)=0$, and in this case the Picard number $\rho(X)$ is either $2$ or $3$. 
In fact, the authors ask as an open question whether every Calabi-Yau threefold of Picard number   $\rho \neq 2,3$ contains rational curves.
I do not know whether Calabi-Yau threefolds of type A contain elliptic curves, but one can  probably check this directly, since the construction is very explicit.