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

Holomorphic symmetric differentials and a birational characterization of Abelian Varieties