A characterization of ordinary abelian varieties by the Frobenius push-forward of the structure sheaf

TL;DR

This paper characterizes ordinary abelian varieties by examining the decomposition of their Frobenius push-forward of the structure sheaf, establishing a criterion based on this property and Kodaira dimension.

Contribution

It provides a new characterization of ordinary abelian varieties using Frobenius push-forward decompositions and Kodaira dimension conditions.

Findings

Frobenius push-forward decomposes into line bundles for ordinary abelian varieties.

The property characterizes ordinary abelian varieties among smooth projective varieties.

The characterization applies when the Kodaira dimension is non-negative.

Abstract

For an ordinary abelian variety $X$, $F^e_*\mathcal{O}_X$ is decomposed into line bundles for every positive integer $e$. Conversely, if a smooth projective variety $X$ satisfies this property and its Kodaira dimension is non-negative, then $X$ is an ordinary abelian variety.