On endomorphisms of projective varieties with numerically trivial canonical divisors
Sheng Meng
TL;DR
The paper studies surjective endomorphisms of projective varieties with numerically trivial canonical divisor, introducing amplified, quasi-amplified, and PCD endomorphisms and linking them to entropy and density of periodic points. It proves that a quasi-amplified endomorphism on a klt Calabi–Yau type variety becomes amplified after iterating and performing an $f$-equivariant birational contraction sequence, reducing to an amplified endomorphism on the final model. In the Hyperkähler case, quasi-amplified is equivalent to positive entropy, with consequences for Zariski-dense periodic points, while for abelian varieties it establishes precise cohomological criteria for amplified and PCD endomorphisms and discusses Albanese-type decompositions. The results illuminate how amplification, entropy, and periodic points behave under lifting, descending, products, and Albanese maps, providing structural decompositions and criteria that separate the abelian, Hyperkähler, and Calabi–Yau components. Overall, the work advances the understanding of dynamical properties of endomorphisms on numerically trivial Calabi–Yau type varieties and their special cases.
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 $f^*D-D$ 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).