Polarized endomorphisms of normal projective threefolds in arbitrary characteristic
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Mathematische Annalen
Polarized endomorphisms of normal projective threefolds in arbitrary characteristic Paolo Cascini1 · Sheng Meng2 · De-Qi Zhang2 Received: 6 November 2017 / Revised: 18 June 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
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 ∼ q H for some ample Cartier divisor H and integer q > 1. Suppose f is separable and X is QGorenstein 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. Mathematics Subject Classification 14H30 · 32H50 · 14E30 · 11G10 · 08A35
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Communicated by Ngaiming Mok.
B
De-Qi Zhang [email protected]
Extended author information available on the last page of the article
123
P. Cascini et al. 3 Proof of Theorem 1.1 . . . . . . . . . . . . 4 K X pseudo-effective case . . . . . . . . . . 5 Proof of Theorems 1.2 and 1.4 . . . . . . . . 6 Descending of polarized endomorphisms . . 7 Endomorphisms compatible with a fibration 8 Q-abelian case . . . . . . . . . . . . . . . . 9 Global index-1 cover . . . . . . . . . . . . . 10 Surface case and the proof of Theorem 1.5 . 11 Proofs of Theorems 1.6, 1.7 and 1.8 . . . . . References . . . . . . . . . . . . . . . . . . . .
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