Let be an algebraically closed field of characteristic . We give a birational characterization of ordinary abelian varieties over : a smooth projective variety is birational to an ordinary abelian variety if and only if and . We also give a similar characterization of abelian varieties as well: a smooth projective variety is birational to an abelian variety if and only if , and the Albanese morphism is generically finite. Along the way, we also show that if (or if and is generically finite), then the Albanese morphism is surjective and in particular .
"Birational characterization of Abelian varieties and ordinary Abelian varieties in characteristic ." Duke Math. J. 168 (9) 1723 - 1736, 15 June 2019. https://doi.org/10.1215/00127094-2019-0008