Please use this identifier to cite or link to this item: https://doi.org/10.1016/j.aim.2017.11.026
Title: Building blocks of polarized endomorphisms of normal projective varieties
Authors: Meng, Sheng 
Zhang, De-Qi 
Keywords: Science & Technology
Physical Sciences
Mathematics
Polarized endomorphism
Minimal model program
Q-abelian variety
Fano variety
MINIMAL MODELS
SINGULARITIES
VOLUME
Issue Date: 5-Feb-2018
Publisher: ACADEMIC PRESS INC ELSEVIER SCIENCE
Citation: Meng, Sheng, Zhang, De-Qi (2018-02-05). Building blocks of polarized endomorphisms of normal projective varieties. ADVANCES IN MATHEMATICS 325 : 243-273. ScholarBank@NUS Repository. https://doi.org/10.1016/j.aim.2017.11.026
Abstract: An endomorphism f of a projective variety X is polarized (resp. quasi-polarized) if f⁎H∼qH (linear equivalence) for some ample (resp. nef and big) Cartier divisor H and integer q>1. First, we use cone analysis to show that a quasi-polarized endomorphism is always polarized, and the polarized property descends via any equivariant dominant rational map. Next, we show that a suitable maximal rationally connected fibration (MRC) can be made f-equivariant using a construction of N. Nakayama, that f descends to a polarized endomorphism of the base Y of this MRC and that this Y is a Q-abelian variety (quasi-étale quotient of an abelian variety). Finally, we show that we can run the minimal model program (MMP) f-equivariantly for mildly singular X and reach either a Q-abelian variety or a Fano variety of Picard number one. As a consequence, the building blocks of polarized endomorphisms are those of Q-abelian varieties and those of Fano varieties of Picard number one. Along the way, we show that f always descends to a polarized endomorphism of the Albanese variety Alb(X) of X, and that the pullback of a power of f acts as a scalar multiplication on the Néron–Severi group of X (modulo torsion) when X is smooth and rationally connected. Partial answers about X being of Calabi–Yau type, or Fano type are also given with an extra primitivity assumption on f which seems necessary by an example.
Source Title: ADVANCES IN MATHEMATICS
URI: https://scholarbank.nus.edu.sg/handle/10635/234660
ISSN: 0001-8708
1090-2082
DOI: 10.1016/j.aim.2017.11.026
Appears in Collections:Staff Publications
Elements
Staff Publications

Show full item record
Files in This Item:
File Description SizeFormatAccess SettingsVersion 
1606.01345v4.pdf386.1 kBAdobe PDF

OPEN

Pre-printView/Download

SCOPUSTM   
Citations

19
checked on Nov 25, 2022

WEB OF SCIENCETM
Citations

20
checked on Nov 25, 2022

Page view(s)

11
checked on Nov 17, 2022

Download(s)

1
checked on Nov 17, 2022

Google ScholarTM

Check

Altmetric


Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.