Effective characterization of ordinary abelian varieties, and beyond

TL;DR

This paper proves properties of Albanese morphisms and abelian varieties in positive characteristic, providing new proofs and characterizations that extend classical results to a broader setting.

Contribution

It offers a new proof of the effective birational characterization of complex abelian varieties in positive characteristic, and establishes surjectivity and connectedness of Albanese morphisms under certain conditions.

Findings

Albanese morphism is surjective with connected fibers for certain varieties.

The Albanese variety is shown to be ordinary in these cases.

Provides a positive characteristic proof of a known complex case result.

Abstract

We prove that the Albanese morphism of any normal proper variety $X$ in positive characteristic satisfying $S^0(X, \omega_X) \neq 0$ and $P_2(X) = 1$ is surjective with connected fibers, adn that $\mathrm{Alb}(X)$ is ordinary. We obtain from a variant of the above a purely positive characteristic proof of Chen and Hacon's effective birational characterization of complex abelian varieties.