  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
$\dim A\leq \dim X$.
