On endomorphisms of projective varieties with numerically trivial canonical divisors

Abstract

Let $X$ be a klt projective variety with numerically trivial canonical divisor. A surjective endomorphism $f:X\to X$ is amplified (resp.~quasi-amplified) if $f^{\ast }D-D$ is ample (resp.~big) for some Cartier divisor $D$. We show that after iteration and equivariant birational contractions, an quasi-amplified endomorphism will descend to an amplified endomorphism. As an application, when $X$ is Hyperkähler, $f$ is quasi-amplified if and only if it is of positive entropy. In both cases, $f$ has Zariski dense periodic points. When $X$ 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).

arXiv commentTheorem 1.8 revised, 28 pages

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