Please use this identifier to cite or link to this item: https://doi.org/10.1007/s00208-019-01877-6
Title: Polarized endomorphisms of normal projective threefolds in arbitrary characteristic
Authors: Cascini, Paolo
Meng, Sheng 
Zhang, De-Qi 
Keywords: Science & Technology
Physical Sciences
Mathematics
14H30
32H50
14E30
11G10
08A35
VARIETIES
3-FOLDS
Issue Date: 1-Oct-2020
Publisher: SPRINGER HEIDELBERG
Citation: Cascini, Paolo, Meng, Sheng, Zhang, De-Qi (2020-10-01). Polarized endomorphisms of normal projective threefolds in arbitrary characteristic. MATHEMATISCHE ANNALEN 378 (1-Feb) : 637-665. ScholarBank@NUS Repository. https://doi.org/10.1007/s00208-019-01877-6
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 - KX 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 KX being pseudo-effective and Q-Cartier means being a torsion Q-divisor. Let fGal: X¯ → X be the Galois closure of f. We show that if p> 5 and co-prime to deg fGal 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.
Source Title: MATHEMATISCHE ANNALEN
URI: https://scholarbank.nus.edu.sg/handle/10635/234646
ISSN: 0025-5831
1432-1807
DOI: 10.1007/s00208-019-01877-6
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