Effective Eigendivisors and the Kawaguchi-Silverman Conjecture

Kawaguchi-Silverman conjecture for endomorphisms on several classes of varieties

We prove Kawaguchi-Silverman conjecture (KSC) and Shibata's conjecture on ample canonical heights for endomorphisms on several classes of algebraic varieties including varieties of Fano type and projective toric varieties. We also prove KSC for group endomorphisms of linear algebraic groups. We also propose a possible approach to the conjecture using equivariant MMP.

Kawaguchi-Silverman猜想对某些满射自同态

Kawaguchi-Silverman conjecture for certain surjective endomorphisms

本文证明Kawaguchi-Silverman猜想对任意射影曲面及若干高维簇（如有理连通光滑三维簇）的满射自同态成立，核心方法结合等变极小模型纲领、反典范除子有效性与环面偶对刻画。

拟射影簇的自同态——走向Zariski稠密轨道与Kawaguchi-Silverman猜想

Endomorphisms of quasi-projective varieties -- towards Zariski dense orbit and Kawaguchi-Silverman conjectures

本文研究了拟射影簇上有限满射自同态的Zariski稠密轨道猜想（ZDO）、Adelic Zariski稠密轨道猜想（AZO）及Kawaguchi-Silverman猜想（KSC），并在光滑仿射曲面、射影簇和半阿贝尔簇等特定条件下证明了这些猜想成立。

Ueno型簇的自同态与Kawaguchi-Silverman猜想

Endomorphisms of a variety of Ueno type and Kawaguchi-Silverman Conjecture

本文证明Ueno型簇的可分满自态幺半群等同其自同构群并给出显式描述，进而证实了该型簇及部分Calabi-Yau三维簇的自同构满足Kawaguchi-Silverman猜想。

关于不规则簇双有理自同构的Kawaguchi--Silverman猜想

On the Kawaguchi--Silverman Conjecture for birational automorphisms of irregular varieties

针对Kawaguchi--Silverman猜想，证明当光滑射影簇$X$的Kodaira维数为零且不规则度$q(X) \ge \dim X -1$或为不规则三维簇（除一例外）时，双有理自映射的算术度与第一动力度重合，并研究了Zariski稠密轨道的存在性。

Dynamical Degree and Arithmetic Degree of Endomorphisms on Product Varieties

For a dominant rational self-map on a smooth projective variety defined over a number field, Shu Kawaguchi and Joseph H. Silverman conjectured that the dynamical degree is equal to the arithmetic degree at a rational point whose forward orbit is well-defined and Zariski dense. We give some examples of self-maps on product varieties and rational points on them for which the Kawaguchi-Silverman conjecture holds.

On the realizability of arithmetic degrees of morphisms

The Kawaguchi-Silverman conjecture relates two different invariants of a surjective endomorphism, the dynamical and arithmetic degrees. As the Kawaguchi-Silverman conjecture is only meaningful when a morphism has a Zariski dense orbit, it has no content for varieties with positive Kodaira dimension. A generalization of the Kawaguchi-Silverman conjecture which is meaningful in positive Kodaira dimension is the so called sAND conjecture, which involves the set of "small" arithmetic degrees. Kawaguchi and Silverman showed that a small arithmetic degree is the modulus of an eigenvalue of $f^*\colo…

某些有理自映射的周期点与算术度

Periodic points and arithmetic degrees of certain rational self-maps

本文研究代数数域上的上同调双曲双有理自映射，给出其周期点高度的有界性结果，还将二维有理自映射的Kawaguchi--Silverman猜想归约为动力Mordell--Lang猜想，验证了若干新情形并证明对应Zariski稠密轨道存在。

A rational map with infinitely many points of distinct arithmetic degrees

Let $f \colon X \dashrightarrow X$ be a dominant rational self-map of a smooth projective variety defined over $\overline{\mathbb Q}$. For each point $P\in X(\overline{\mathbb Q})$ whose forward $f$-orbit is well-defined, Silverman introduced the arithmetic degree $α_f(P)$, which measures the growth rate of the heights of the points $f^n(P)$. Kawaguchi and Silverman conjectured that $α_f(P)$ is well-defined and that, as $P$ varies, the set of values obtained by $α_f(P)$ is finite. Based on constructions of Bedford--Kim and McMullen, we give a counterexample to this conjecture when $X=\mathbb P…

椭圆曲线上射影丛的自同态动力学

Dynamics of Endomorphisms for Projective Bundles on Elliptic Curves

本文通过丛的转移函数研究椭圆曲线上射影丛满射自同态的动力学，证明了其Kawaguchi--Silverman猜想，并将结果推广至具有限生成Mori锥的光滑射影簇上的射影分裂丛。

关于动力学Mordell-Lang猜想

Around the dynamical Mordell-Lang conjecture

本文报告了动力学Mordell-Lang猜想的进展，将其已知结果推广至复数域及一致与部分轨道版本，并利用Berkovich空间语言改进了p-adic方法，同时提出了进一步问题。

射影化环面向量丛的动力学

Dynamics of Projectivized Toric Vector Bundles

该研究聚焦环面簇上射影丛的满自同态动力学，通过结构定理、动力次数控制分析及显式转移函数法，完成分类、验证猜想并证明相关丛无非自同构交换满自同态，推进代数几何动力学研究。