On endomorphisms of projective varieties with numerically trivial canonical divisors
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2019
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| _version_ | 1866913845796143104 |
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| author | Meng, Sheng |
| author_facet | Meng, Sheng |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1901_07089 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | On endomorphisms of projective varieties with numerically trivial canonical divisors Meng, Sheng Algebraic Geometry Dynamical Systems 14E30, 32H50, 08A35, 14J50, 11G10 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). |
| title | On endomorphisms of projective varieties with numerically trivial canonical divisors |
| topic | Algebraic Geometry Dynamical Systems 14E30, 32H50, 08A35, 14J50, 11G10 |
| url | https://arxiv.org/abs/1901.07089 |