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 ≥ 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 [5, 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.

Building blocks of amplified endomorphisms of normal projective varieties

“…Then f is quasi-étale by the ramification divisor formula. By [7,Corollary 8.2], there exist a quasi-étale cover π : A → X and a surjective endomorphism f A : A → A such that f • π = π • f A . By Lemma 3.5, f A is int-amplified.…”

“…Since E 1 · E 2 > 0, f s | E 1 and f s | E 2 are both n 2s -polarized (cf. [7,Introduction]). Let f := f s | E 1 × f s | E 2 .…”

“…In the papers of [19] and [7], for a polarized f : For the int-amplified case, we are only able to give a "limit" version in Lemma 3.8. Due to these difficulties, all the generalizations as shown in the previous main theorems are required to adjust the old proofs in [19] and [7] accordingly based on the new methods in Section 3. Finally, we highlight that the int-amplified criteria in Theorem 1.1 by the cone analysis are the keys to making all the subsequent methods and results possible.…”

Totally Invariant Divisors of Int-Amplified Endomorphisms of Normal Projective Varieties

“…Lemma 2.6 and Equation (7)) and f isétale outside f −1 (SingX) ∪ ( V j ) by the purity of branch loci. As a result, we complete the proof of Theorem 1.1 (4). Now, the only thing we need to do is to prove Claim 3.9.…”

“…Suppose X is a normal projective variety with only Q-factorial Kawamata log terminal singularities, and f : X → X is an int-amplified endomorphism of X. If V j ⊆ X(1 ≤ j ≤ c) are all the prime divisors with f −1 (V j ) = V j , and c = ρ(X) + dim X (achieving the upper bound) as in Theorem 1.1 (4), then (X, V j ) is a toric pair. Indeed, it follows immediately from [3, Theorem 1.2] and our main theorem, in which case, the complexity (cf.…”