Related papers: Effective Eigendivisors and the Kawaguchi-Silverma…

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…

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…

We prove the Kawaguchi-Silverman conjecture (KSC), about the equality of arithmetic degree and dynamical degree, for every surjective endomorphism of any (possibly singular) projective surface. In high dimensions, we show that KSC holds for…

Let $X$ be a $\mathbb{Q}$-factorial klt projective variety admitting an int-amplified endomorphism $f$, i.e., the modulus of any eigenvalue of $f^*|_{\text{NS}(X)}$ is greater than $1$. We prove Kawaguchi-Silverman conjecture for $f$ and…

We study the dynamics of surjective endomorphisms of projective bundles on elliptic curves and relate their dynamical properties to the geometry of the bundle. As an application we prove the Kawaguchi--Silverman conjecture for projective…

Under the framework of dynamics on normal projective varieties by Kawamata, Nakayama and Zhang \cite{Kawamata85,Nakayama10,NZ09,NZ10,Zhang16}, Hu and the author \cite{HL21}, we may reduce Kawaguchi-Silverman conjecture for automorphisms $f$…

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…

We study the main open parts of the Kawaguchi--Silverman Conjecture, asserting that for a birational self-map $f$ of a smooth projective variety $X$ defined over $\overline{\mathbb Q}$, the arithmetic degree $\alpha_f(x)$ exists and…

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…

We consider an arbitrary int-amplified surjective endomorphism $f$ of a normal projective variety $X$ over $\mathbb{C}$ and its $f^{-1}$-stable prime divisors. We extend the early result for the case of polarized endomorphisms to the case…

Let $\pi : X\to \Lambda$ be a flat family of smooth complex projective varieties parameterized by a smooth quasi-projective variety $\Lambda$, and let $f: X\to X$ be a family of automorphisms with positive topological entropy. Suppose…

Let $f:X\to X$ be a non-isomorphic (i.e., $\text{deg } f>1$) surjective endomorphism of a smooth projective threefold $X$. We prove that any birational minimal model program becomes $f$-equivariant after iteration, provided that $f$ is…

Given a surjective endomorphism $f: X \to X$ on a projective variety over a number field, one can define the arithmetic degree $\alpha_f(x)$ of $f$ at a point $x$ in $X$. The Kawaguchi - Silverman Conjecture (KSC) predicts that any forward…

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…

Let f : X --> X be a dominant rational map of a projective variety defined over a global field, let d_f be the dynamical degree of f, and let h_X be a Weil height on X relative to an ample divisor. We prove that h_X(f^n(P)) << (d_f + e)^n…

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…

Let $f: X\to X$ be a surjective endomorphism of a projective variety of dimension $d$. The aim of this paper is to study the action of $f$ on the numerical group of divisors. For exmaple, I proved that $f$ is cohomologically hyperbolic if…

Let $\mathbb{K}$ be an algebraically closed field, $X$ a smooth projective variety over $\mathbb{K}$ and $f:X\rightarrow X$ a dominant regular morphism. Let $N^i(X)$ be the group of algebraic cycles modulo numerical equivalence. Let $\chi…

Let f : X --> X be an endomorphism of a normal projective variety defined over a global field K, and let D_0,D_1,D_2,... be divisor classes that form a Jordan block with eigenvalue b for the action of f^* on Pic(X) tensored with C. We…

Let $X$ be a projective variety over an algebraically closed field $k$ of arbitrary characteristic $p \ge 0$. A surjective endomorphism $f$ of $X$ is $q$-polarized if $f^\ast H \sim qH$ for some ample Cartier divisor $H$ and integer $q >…