The Demailly--Peternell--Schneider conjecture is true in positive characteristic

We prove the Demailly--Peternell--Schneider conjecture in positive characteristic: if $X$ is a smooth projective variety over an algebraically closed field of characteristic $p>0$ with $-K_X$ is nef, then the Albanese morphism $a: X \to A$ is surjective. We also show strengthenings either allowing mild singularities for $X$, or proving more special properties of $a$. The above statement for compact K\"ahler manifolds was conjectured originally by Demailly, Peternell and Schneider in 1993, and for smooth projective varieties of characteristic zero it was shown by Zhang in 1996. In positive characteristic, all earlier results involved tameness assumptions either on cohomology or on the singularities of the general fibers of $a$. The main feature of the present article is the development of a technology to avoid such assumptions.