Please use this identifier to cite or link to this item: https://doi.org/10.1016/j.aim.2017.11.026
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dc.titleBuilding blocks of polarized endomorphisms of normal projective varieties
dc.contributor.authorMeng, Sheng
dc.contributor.authorZhang, De-Qi
dc.date.accessioned2022-11-17T05:26:06Z
dc.date.available2022-11-17T05:26:06Z
dc.date.issued2018-02-05
dc.identifier.citationMeng, 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
dc.identifier.issn0001-8708
dc.identifier.issn1090-2082
dc.identifier.urihttps://scholarbank.nus.edu.sg/handle/10635/234660
dc.description.abstractAn 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.
dc.language.isoen
dc.publisherACADEMIC PRESS INC ELSEVIER SCIENCE
dc.sourceElements
dc.subjectScience & Technology
dc.subjectPhysical Sciences
dc.subjectMathematics
dc.subjectPolarized endomorphism
dc.subjectMinimal model program
dc.subjectQ-abelian variety
dc.subjectFano variety
dc.subjectMINIMAL MODELS
dc.subjectSINGULARITIES
dc.subjectVOLUME
dc.typeArticle
dc.date.updated2022-11-16T08:17:21Z
dc.contributor.departmentDEPT OF MATHEMATICS
dc.description.doi10.1016/j.aim.2017.11.026
dc.description.sourcetitleADVANCES IN MATHEMATICS
dc.description.volume325
dc.description.page243-273
dc.published.statePublished
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