Question

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!

Answer 1

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.