  A vector bundle on a smooth projective variety, if it is generically
generated by global sections, yields a rational map to a Grassmannian, called
Kodaira map. We investigate the asymptotic behaviour of the Kodaira maps for
the symmetric powers of a vector bundle, and we show that these maps stabilize
to a map dominating all of them, as it happens for a line bundle via the Iitka
fibration. Through this Iitaka-type construction, applied to the cotangent
bundle, we give a new characterization of Abelian varieties.
