On endomorphisms of projective varieties with numerically trivial canonical divisors

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).