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 ∼ q H 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.