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

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 dimX. 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 -> 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 -> A is surjective and in particular dim A