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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 |
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