Polarized endomorphisms of normal projective threefolds in arbitrary characteristic

Let X be a projective variety over an algebraically closed field k of arbitrary characteristic p≥ 0. A surjective endomorphism f of X is q-polarized if f^∗H∼ qH for some ample Cartier divisor H and integer q> 1. Suppose f is separable and X is Q-Gorenstein and normal. We show that the anti-canonical divisor - K_X is numerically equivalent to an effective Q-Cartier divisor, strengthening slightly the conclusion of Boucksom, de Fernex and Favre (Duke Math J 161(8):1455–1520, 2012, Theorem C) and also covering singular varieties over an algebraically closed field of arbitrary characteristic. Suppose f is separable and X is normal. We show that the Albanese morphism of X is an algebraic fibre space and f induces polarized endomorphisms on the Albanese and also the Picard variety of X, and K_X being pseudo-effective and Q-Cartier means being a torsion Q-divisor. Let f^Gal: X¯ → X be the Galois closure of f. We show that if p> 5 and co-prime to deg f^Gal then one can run the minimal model program (MMP) f-equivariantly, after replacing f by a positive power, for a mildly singular threefold X and reach a variety Y with torsion canonical divisor (and also with Y being a quasi-étale quotient of an abelian variety when dim (Y) ≤ 2). Along the way, we show that a power of f acts as a scalar multiplication on the Neron-Severi group of X (modulo torsion) when X is a smooth and rationally chain connected projective variety of dimension at most three.