On endomorphisms of projective varieties with numerically trivial canonical divisors

Author:

Meng Sheng12ORCID

Affiliation:

1. School of Mathematical Sciences, Shanghai Key Laboratory of PMMP East China Normal University, 500 Dongchuan Road, Shanghai 200241, P. R. China

2. Max-Planck-Institut für Mathematik, Vivatsgasse 7, Bonn 53111, Germany

Abstract

Let [Formula: see text] be a klt projective variety with numerically trivial canonical divisor. A surjective endomorphism [Formula: see text] is amplified (respectively, quasi-amplified) if [Formula: see text] is ample (respectively, big) for some Cartier divisor [Formula: see text]. We show that after iteration and equivariant birational contractions, a quasi-amplified endomorphism will descend to an amplified endomorphism. As an application, when [Formula: see text] is Hyperkähler, [Formula: see text] is quasi-amplified if and only if it is of positive entropy. In both cases, [Formula: see text] has Zariski dense periodic points. When [Formula: see text] is an abelian variety, we give and compare several cohomological and geometric criteria of amplified endomorphisms and endomorphisms with countable and Zariski dense periodic points (after an uncountable field extension).

Funder

Science and Technology Commission of Shanghai Municipality

Publisher

World Scientific Pub Co Pte Ltd

Subject

General Mathematics

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