Let $X$ be an abelian variety. In this question, that will mean that $X$ is a complex torus $\mathbb C^n / \Lambda$, where $\Lambda \subset \mathbb C^n$ is a lattice, and that $X$ also a projective variety. Does $X$ contain an elliptic curve?
The question comes from thinking about Demailly's definition of algebraic hyperbolicity. He said a projective variety $X$ is algebraically hyperbolic if for every Kahler class $\omega$ on $X$ there exists an $\varepsilon > 0$ such that if $C \subset X$ is a curve and $\widetilde C$ is its normalization we have $$ \varepsilon \leq \frac{g(\widetilde C)}{\deg_\omega C}. $$ Furthermore he proved that an abelian variety is not algebraically hyperbolic (because it admits a nonconstant map from an abelian variety, namely itself).
An idea for proving that a variety is not algebraically hyperbolic is to try to come up with a sequence of curves $(C_k)$ on $X$ such that $g(\widetilde C_k) / \deg_\omega C_k \to 0$. I'm wondering whether this works at all, or whether this ratio is generally bounded away from $0$ for curves that aren't simply rational or elliptic. If there exists an abelian variety that doesn't contain an elliptic curve, this idea must work sometimes.
