  A generically generated vector bundle on a smooth projective variety yields a
rational map to a Grassmannian, called Kodaira map. We answer a previous
question, raised by the asymptotic behaviour of such maps, giving rise to a
birational characterization of abelian varieties.
  In particular we prove that, under the conjectures of the Minimal Model
Program, a smooth projective variety is birational to an abelian variety if and
only if it has Kodaira dimension 0 and some symmetric power of its cotangent
sheaf is generically generated by its global sections.
