Weakly Special Manifolds with no rational curves

Abstract

Assuming the abundance conjecture and the existence of a Zariski dense set of rational curves on terminal Calabi--Yau varieties, we show that a complex projective weakly special manifold $X$ with no rational curves is an \'etale quotient of an Abelian variety. The same conclusion holds true if $X$ contains a Zariski dense entire curve, assuming Lang's conjecture. This implies that any non-hyperbolic complex projective manifold contains the image of an Abelian variety, according to another conjecture of Lang. We illustrate this last conjecture by producing examples of canonically polarised submanifolds of abelian varieties containing no subvariety of general type, except for a finite number of disjoint copies of some simple abelian variety, which can be chosen arbitrarily. We also show, more generally, that any projective manifold containing a Zariski dense entire curve appears as the `exceptional set' in Lang's sense of some general type manifold.