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

掲載種別：研究論文（学術雑誌）  

DOI： 10.11650/tjm/210101

掲載種別：研究論文（学術雑誌）  

Abstract

Let $f:X\to X $ be a dominant self-morphism of an algebraic variety. Consider the set $\Sigma _{f^{\infty } }$ of $f$-periodic subvarieties of small dynamical degree (SDD), the subset $S_{f^{\infty } }$ of maximal elements in $\Sigma _{f^{\infty } }$, and the subset $S_f$ of $f$-invariant elements in $S_{f^{\infty } }$. When $X$ is projective, we prove the finiteness of the set $P_f$ of $f$-invariant prime divisors with SDD and give an optimal upper bound $$\begin{align*} &\sharp P_{f^n}\le d_1(f)^n(1+o(1))\end{align*}$$as $n\to \infty $, where $d_1(f)$ is the 1st dynamic degree. When $X$ is an algebraic group (with $f$ being a translation of an isogeny), or a (not necessarily complete) toric variety, we give an optimal upper bound $$\begin{align*} &\sharp S_{f^n}\le d_1(f)^{n\cdot\dim(X)}(1+o(1))\end{align*}$$as $n \to \infty $, which slightly generalizes a conjecture of S.-W. Zhang for polarized $f$.

DOI： 10.1093/imrn/rnab039

掲載種別：研究論文（学術雑誌）  

掲載種別：研究論文（学術雑誌）  

We prove a conjecture by Kawaguchi–Silverman on arithmetic and dynamical degrees, for self-morphisms of semi-abelian varieties. Moreover, we determine the set of the arithmetic degrees of orbits and the (first) dynamical degrees of self-morphisms of semi-abelian varieties.

DOI： 10.1017/etds.2018.117

掲載種別：研究論文（学術雑誌）  

DOI： 10.1016/j.aim.2020.107086

掲載種別：研究論文（学術雑誌）  

DOI： 10.1353/ajm.2020.0045