Holomorphic symmetric differentials and a birational characterization of Abelian Varieties

TL;DR

This paper provides a birational characterization of abelian varieties using properties of symmetric differentials and the Kodaira map, linking geometric structures to the minimal model program conjectures.

Contribution

It establishes a new criterion for identifying abelian varieties based on the generically generated symmetric powers of the cotangent sheaf and the Kodaira dimension.

Findings

A birational criterion for abelian varieties under MMP conjectures.

Connection between symmetric differentials and the birational classification.

Answer to a previously open question about Kodaira maps.

Abstract

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.