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-étale 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.

Minor revision, 33 pages, Mathematische Annalen (to appear)

References in corpus (2)

• Étale fundamental groups of strongly $F$-regular schemes
• Birational characterization of abelian varieties and ordinary abelian varieties in characteristic p>0

Cited by in corpus (16)

• Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism
• Non-density of points of small arithmetic degrees
• Invariant subvarieties with small dynamical degree
• Rigidity of rationally connected smooth projective varieties from dynamical viewpoints
• Kawaguchi-Silverman conjecture for endomorphisms on rationally connected varieties admitting an int-amplified endomorphism
• Non-isomorphic endomorphisms of Fano threefolds
• Surjective endomorphisms of projective surfaces: the existence of infinitely many dense orbits
• Amplified endomorphisms of Fano fourfolds
• Int-amplified endomorphisms of compact Kähler spaces
• Global $F$-splitting of surfaces admitting an int-amplified endomorphim
• Boundedness of finite morphisms onto Fano manifolds with large Fano index
• Totally invariant divisors of int-amplified endomorphisms of normal projective varieties
• Kawaguchi-Silverman conjecture for endomorphisms on several classes of varieties
• Advances in the equivariant minimal model program and their applications in complex and arithmetic dynamics
• Normal projective varieties admitting polarized or int-amplified endomorphisms
• On endomorphisms of projective varieties with numerically trivial canonical divisors