| We will start from the definitions of various base loci for vector bundles, and see that some of the positivity properties existing in the literature can be recovered from these base loci, as in the well known case of divisors. In particular we will concentrate on semiampleness, and construct an Iitaka fibration. We will see that various pathologies can arise in the case of vector bundles, nevertheless an asymptotical construction similar to the Iitaka fibration can be achieved, both in the semiample case, and in a more general birational setting. As an application we give a characterization of abelian varieties, and some possible birational characterizations. This is a joint work with Stefano Urbinati. |