Polarized endomorphisms of normal projective threefolds in arbitrary characteristic

Abstract

Let $X$ be a projective variety over an algebraically closed field $k$ of arbitrary characteristic $p \ge 0$. A surjective endomorphism $f$ of $X$ is $q$-polarized if $f^\ast H \sim qH$ for some ample Cartier divisor $H$ and integer $q > 1$. Suppose $f$ is separable and $X$ is $\mathbb{Q}$-Gorenstein and normal. We show that the anti-canonical divisor $-K_X$ is numerically equivalent to an effective $\mathbb{Q}$-Cartier divisor, strengthening slightly the conclusion of Boucksom, de Fernex and Favre (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 $\mathbb{Q}$-Cartier means being a torsion $\mathbb{Q}$-divisor. Let $f^{Gal}:\overline{X}\to X$ 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-\'etale quotient of an abelian variety when $\dim(Y)\le 2$). 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.