Birational characterization of Abelian varieties and ordinary Abelian varieties in characteristic p>0

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 $b_1(X)=2dimX$. 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 $dimA\le dimX$.