# Birational characterization of abelian varieties and ordinary abelian varieties in characteristic p>0

## TL;DR

This paper characterizes when a smooth projective variety over an algebraically closed field of characteristic p>0 is birational to an (ordinary) abelian variety based on invariants like Kodaira dimension and Albanese morphism properties.

## Contribution

It provides new birational criteria for identifying (ordinary) abelian varieties in positive characteristic, linking invariants like Kodaira dimension and Albanese morphism behavior.

## Key findings

- A variety is birational to an ordinary abelian variety iff $
abla_S(X)=0$ and $b_1(X)=2 ext{dim} X$.
- A variety is birational to an abelian variety iff $
abla(X)=0$ and the Albanese morphism is generically finite.
- If $
abla_S(X)=0$ or $
abla(X)=0$ with a generically finite Albanese morphism, then the Albanese map is surjective and $ ext{dim} A extless= ext{dim} X$.

## Abstract

Let $k$ be an algebraically closed field of characteristic $p>0$. We give a birational characterization of ordinary abelian varieties over $k$: a smooth projective variety $X$ is birational to an ordinary abelian variety if and only if $\kappa_S(X)=0$ and $b_1(X)=2 \dim X$. We also give a similar characterization of abelian varieties as well: a smooth projective variety $X$ is birational to an abelian variety if and only if $\kappa(X)=0$, and the Albanese morphism $a: X \to A$ is generically finite. Along the way, we also show that if $\kappa _S (X)=0$ (or if $\kappa(X)=0$ and $a$ is generically finite) then the Albanese morphism $a:X\to A$ is surjective and in particular $\dim A\leq \dim X$.