  We introduce a new approach to the geometric Bombieri--Lang conjecture for
hyperbolic varieties in characteristic 0. The main idea is to construct an
entire curve on a special fiber of a variety over a complex function field from
an infinite sequence of rational points of the variety. The construction relies
on the classical Brody lemma in complex geometry and is conditional on a
non-degeneracy conjecture of partial heights. In particular, we prove the
geometric Bombieri-Lang conjecture for hyperbolic projective varieties which
have finite morphisms to abelian varieties over function fields of
characteristic 0.
