Iitaka fibrations for vector bundles

TL;DR

This paper studies the asymptotic behavior of Kodaira maps for symmetric powers of vector bundles on smooth projective varieties, revealing stabilization phenomena and new characterizations of Abelian varieties through Iitaka-type fibrations.

Contribution

It introduces an Iitaka-type fibration for vector bundles, generalizing classical results for line bundles, and applies this to characterize Abelian varieties.

Findings

Kodaira maps for symmetric powers stabilize to a dominant map.

The stabilization leads to a new characterization of Abelian varieties.

The results extend Iitaka fibration concepts from line bundles to vector bundles.

Abstract

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.