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

TL;DR

This paper characterizes ordinary abelian varieties in positive characteristic by conditions on the Frobenius push-forward of the structure sheaf and the pseudo-effectiveness of the canonical bundle.

Contribution

It provides a new criterion for identifying ordinary abelian varieties using Frobenius push-forwards and the pseudo-effectiveness of the canonical bundle.

Findings

$F^e_*\mathcal{O}_X$ splits into line bundles for some $e$ with $p^e>2$

$K_X$ is pseudo-effective for such varieties

Characterization is both necessary and sufficient

Abstract

In this paper, we prove that a smooth projective variety $X$ of characteristic $p>0$ is an ordinary abelian variety if and only if $K_X$ is pseudo-effective and $F^e_*\mathcal O_X$ splits into a direct sum of line bundles for an integer $e$ with $p^e>2$.