掲載種別：研究論文（学術雑誌）  

DOI： 10.1007/s12220-022-01156-y

その他URL： https://link.springer.com/article/10.1007/s12220-022-01156-y/fulltext.html

掲載種別：研究論文（学術雑誌）  

Abstract

Let $X$ be a projective variety and let $f$ be a dominant endomorphism of $X$, both of which are defined over a number field $K$. We consider a question of the 2nd author, Meng, Shibata, and Zhang, who asks whether the tower of $K$-points $Y(K)\subseteq (f^{-1}(Y))(K)\subseteq (f^{-2}(Y))(K)\subseteq \cdots $ eventually stabilizes, where $Y\subset X$ is a subvariety invariant under $f$. We show this question has an affirmative answer when the map $f$ is étale. We also look at a related problem of showing that there is some integer $s_0$, depending only on $X$ and $K$, such that whenever $x, y \in X(K)$ have the property that $f^{s}(x) = f^{s}(y)$ for some $s \geqslant 0$, we necessarily have $f^{s_{0 } }(x) = f^{s_{0 } }(y)$. We prove this holds for étale morphisms of projective varieties, as well as self-morphisms of smooth projective curves. We also prove a more general cancellation theorem for polynomial maps on ${\mathbb {P } }^1$ where we allow for composition by multiple different maps $f_1,\dots ,f_r$.

DOI： 10.1093/imrn/rnac058

掲載種別：研究論文（学術雑誌）  

DOI： 10.1007/s00208-021-02305-4

その他URL： https://link.springer.com/article/10.1007/s00208-021-02305-4/fulltext.html

掲載種別：研究論文（学術雑誌）  

DOI： 10.1016/j.jnt.2021.03.016

掲載種別：研究論文（学術雑誌）  

We collect some results on endomorphisms on projective varieties related to the Kawaguchi–Silverman conjecture. We discuss certain conditions on automorphism groups of projective varieties and positivity conditions on leading real eigendivisors of self-morphisms. We prove Kawaguchi–Silverman conjecture for endomorphisms on projective bundles on a smooth Fano variety of Picard number one. In the last section, we discuss endomorphisms and augmented base loci of their eigendivisors.

DOI： 10.1142/s0129167x21500853