Endomorphisms of quasi-projective varieties -- towards Zariski dense orbit and Kawaguchi-Silverman conjectures
Fuente:
arXiv
Guardado en:
| Autores principales: | , , , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2021
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866917836418449408 |
|---|---|
| author | Jia, Jia Shibata, Takahiro Xie, Junyi Zhang, De-Qi |
| author_facet | Jia, Jia Shibata, Takahiro Xie, Junyi Zhang, De-Qi |
| contents | Let $X$ be a quasi-projective variety and $f\colon X\to X$ a finite surjective endomorphism. We consider Zariski Dense Orbit Conjecture (ZDO), and Adelic Zariski Dense Orbit Conjecture (AZO). We consider also Kawaguchi-Silverman Conjecture (KSC) asserting that the (first) dynamical degree $d_1(f)$ of $f$ equals the arithmetic degree $α_f(P)$ at a point $P$ having Zariski dense $f$-forward orbit. Assuming $X$ is a smooth affine surface, such that the log Kodaira dimension $\barκ(X)$ is non-negative (resp. the étale fundamental group $π_1^{\text{ét}}(X)$ is infinite), we confirm AZO, (hence) ZDO, and KSC (when $\operatorname{deg}(f)\geq 2$) (resp. AZO and hence ZDO). We also prove ZDO (resp. AZO and hence ZDO) for every surjective endomorphism on any projective variety with ''larger'' first dynamical degree (resp. every dominant endomorphism of any semiabelian variety). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_05339 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Endomorphisms of quasi-projective varieties -- towards Zariski dense orbit and Kawaguchi-Silverman conjectures Jia, Jia Shibata, Takahiro Xie, Junyi Zhang, De-Qi Algebraic Geometry Dynamical Systems Number Theory 14J50, 08A35, 32H50, 37B40 Let $X$ be a quasi-projective variety and $f\colon X\to X$ a finite surjective endomorphism. We consider Zariski Dense Orbit Conjecture (ZDO), and Adelic Zariski Dense Orbit Conjecture (AZO). We consider also Kawaguchi-Silverman Conjecture (KSC) asserting that the (first) dynamical degree $d_1(f)$ of $f$ equals the arithmetic degree $α_f(P)$ at a point $P$ having Zariski dense $f$-forward orbit. Assuming $X$ is a smooth affine surface, such that the log Kodaira dimension $\barκ(X)$ is non-negative (resp. the étale fundamental group $π_1^{\text{ét}}(X)$ is infinite), we confirm AZO, (hence) ZDO, and KSC (when $\operatorname{deg}(f)\geq 2$) (resp. AZO and hence ZDO). We also prove ZDO (resp. AZO and hence ZDO) for every surjective endomorphism on any projective variety with ''larger'' first dynamical degree (resp. every dominant endomorphism of any semiabelian variety). |
| title | Endomorphisms of quasi-projective varieties -- towards Zariski dense orbit and Kawaguchi-Silverman conjectures |
| topic | Algebraic Geometry Dynamical Systems Number Theory 14J50, 08A35, 32H50, 37B40 |
| url | https://arxiv.org/abs/2104.05339 |