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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!

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Nice question Simone. Do you know if it is settled for an arbitrary smooth quintic hypersurface in $\mathbb P^4$? – Jorge Vitório Pereira Jul 8 '11 at 0:52
Thanks Jorge! It is very frustrating, but unfortunately no, I don't know... – diverietti Jul 8 '11 at 9:43
Arbitrary I don't know, generic yes. But probably, this you already know! – diverietti Aug 16 '13 at 9:07

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.

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Thank you for your hints. Anyway, a manifold with trivial canonical class and $c_2(X)=0$ is a finite étale quotient of a complex torus, so that it cannot certainly contain rational curves... – diverietti Jul 7 '11 at 17:35
Actually, Oguiso and Sakurai say so themselves right before posing that question "Secondly, our statements (II) and (III) show that there certainly exist smooth Calabi-Yau threefolds containing no rational curves if $\rho(X) = 2$ and $3$, but, on the other hand, suggest some hope to ask the following:" then comes the question quoted by Francesco. Obviously they meant some variation of the question. – Sándor Kovács Jul 7 '11 at 17:44
On the other hand, coming back to my original question, these CY threefold with $c_2(X)=0$ certainly admit a non-constant map from a complex torus. – diverietti Jul 7 '11 at 17:47
Right. Probably Oguiso-Sakurai meant something like that: Is it true that all CY threefolds admit a rational curve or a non-trivial map from an abelian variety. Perhaps even asking that that map from the abelian variety be dominant. – Sándor Kovács Jul 7 '11 at 17:53
@LiYutong, this is because trivial canonical class implies existence of Ricci flat Kähler metrics (this is Yau), then Kobayashi-Lübke inequalities give you that $c_2(X)\ge 0$ and you have equality if and only if the Ricci flat metric is indeed flat itself. Now you conclude by the classical Bieberbach theorem. – diverietti Feb 23 '14 at 11:01

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