A note on varieties of non-negative Kodaira dimension with polarized self maps

TL;DR

This paper proves that smooth projective varieties with non-negative Kodaira dimension, a rational point, and a polarized self map are finite free quotients of abelian varieties, revealing a structural classification.

Contribution

It establishes a new characterization of such varieties, linking their geometric structure to abelian varieties via polarized self maps.

Findings

Varieties with the given conditions are quotients of abelian varieties.

The result applies to varieties over fields with rational points.

Provides a structural insight into varieties with polarized self maps.

Abstract

In this note we prove that a smooth projective variety (defined over a field $k$) of non-negative Kodaira dimension that has a $k$-rational point and a polarized self map must be a finite free quotient of an abelian variety.