Yohsuke Matsuzawa, Shou Yoshikawa
We prove Kawaguchi-Silverman conjecture for all surjective endomorphisms on every smooth rationally connected variety admitting an int-amplified endomorphism.
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Recent advances on Kawaguchi-Silverman conjecture
本文是一篇综述，介绍了关于 Kawaguchi-Silverman 猜想的近期研究进展。
Kawaguchi-Silverman conjecture on automorphisms of projective threefolds
本文将射影三维流形自同构的Kawaguchi-Silverman猜想归结为弱Calabi-Yau与有理连通两种情形，并证明非正则性满足q(X)≥dim X−1时猜想成立，同时探讨了Picard数为2的情形。
Kawaguchi-Silverman conjecture for int-amplified endomorphism
本文在 $\mathbb{Q}$-因子的 klt 射影簇上，对内蕴放大自同态及任意满射自同态证明了 Kawaguchi-Silverman 猜想，将第一动力次数与 Zariski 稠密轨道点的算术次数相等这一结论从极化情形推广到更一般的内蕴放大情形。
The Kawaguchi--Silverman conjecture predicts that if $f\colon X \dashrightarrow X$ is a dominant rational-self map of a projective variety over $\overline{\mathbb{Q}}$, and $P$ is a $\overline{\mathbb{Q}}$-point of $X$ with Zariski-dense orbit, then the dynamical and arithmetic degrees of $f$ coincide: $λ_1(f) = α_f(P)$. We prove this conjecture in several higher-dimensional settings, including all endomorphisms of non-uniruled smooth projective threefolds with degree larger than $1$, and all endomorphisms of hyper-Kähler varieties in any dimension. In the latter case, we construct a canonical…
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.
Let $f\colon X\rightarrow X$ be a surjective endomorphism of a normal projective variety defined over a number field. The dynamics of $f$ may be studied through the dynamics of the linear action $f^*\colon Pic(X)_\mathbb{R}\rightarrow Pic(X)_\mathbb{R}$, which are governed by the spectral theory of $f^*$. Let $λ_1(f)$ be the spectral radius of $f^*$. We study $\mathbb{Q}$-divisors $D$ with $f^*D=λ_1(f) D$ and $κ(D)=0$ where $κ(D)$ is the Iitaka dimension of the divisor $D$. We analyze the base locus of such divisors and interpret the set of small eigenvalues in terms of the canonical heights o…
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。
We collect some results on endomorphisms on projective varieties related with the Kawaguchi-Silverman conjecture. We discuss certain condition 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.
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。
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。
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。
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.
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…
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。
Dynamical Iitaka theory on Fano contractions
本文利用分歧除子的动力 Iitaka 纤维化，建立 Mori 纤维空间上满射自同态的结构定理，并证明射影丛上的 Kawaguchi-Silverman 猜想。
For a dominant rational self-map on a smooth projective variety defined over a number field, Kawaguchi and Silverman conjectured that the (first) dynamical degree is equal to the arithmetic degree at a rational point whose forward orbit is well-defined and Zariski dense. We prove this conjecture for surjective endomorphisms on smooth projective surfaces. For surjective endomorphisms on any smooth projective varieties, we show the existence of rational points whose arithmetic degrees are equal to the dynamical degree. Moreover, we prove that there exists a Zariski dense set of rational points h…
Given a surjective endomorphism $f: X \to X$ on a projective variety over a number field, one can define the arithmetic degree $α_f(x)$ of $f$ at a point $x$ in $X$. The Kawaguchi - Silverman Conjecture (KSC) predicts that any forward $f$-orbit of a point $x$ in $X$ at which the arithmetic degree $α_f(x)$ is strictly smaller than the first dynamical degree $δ_f$ of $f$ is not Zariski dense. We extend the KSC to sAND (= small Arithmetic Non-Density) Conjecture that the locus $Z_f(d)$ of all points of small arithmetic degree is not Zariski dense, and verify this sAND Conjecture for endomorphisms…
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。
Structures theorems and applications of non-isomorphic surjective endomorphisms of smooth projective threefolds
本文研究了光滑射影三维簇的非同构满射自同态，证明了在$δ$-本原条件下双有理极小模型纲领迭代后变为$f$-等变，并据此证明了Kawaguchi-Silverman猜想及将Zariski稠密轨道猜想归结为终端三维簇情形。
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。
Marked points of families of hyperbolic automorphisms of smooth complex projective varieties
本文研究正熵自同构族中标记点轨道的稳定性，分别构造几何标准高度与连续势的正闭 $(1,1)$-流，并在曲线情形证明二者一致，还推出 Kawaguchi–Silverman 猜想的特例。
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