The complex-curve criterion for Endo–Pajitnov manifolds

Let $T_M$ be an Endo–Pajitnov manifold with real eigenvalue $b1$, and let $a$ be the corresponding eigenvector.

Complex-curve criterion. $T_M$ admits complex curves if and only if the components of $a$ are not linearly independent over $\mathbb{Z}$.

The preceding argument proves that linear independence over $\mathbb{Z}$ prevents compact complex curves, while the sufficiency of the condition is explicitly left unproved.