Effective characterization of ordinary abelian varieties, and beyond

We prove that the Albanese morphism of any normal proper variety $X$ in positive characteristic satisfying $S^0(X,{\omega }_X)\mathrm{\ne }0$ and $P_2(X)=1$ is surjective with connected fibers, adn that $\mathrm{A}\mathrm{l}\mathrm{b}(X)$ is ordinary. We obtain from a variant of the above a purely positive characteristic proof of Chen and Hacon’s effective birational characterization of complex abelian varieties.