Canonical heights on hyper-K\"ahler varieties and the Kawaguchi-Silverman conjecture

TL;DR

This paper proves the Kawaguchi-Silverman conjecture for certain higher-dimensional varieties, including hyper-K"ahler varieties, by constructing canonical height functions and analyzing dynamical and arithmetic degrees.

Contribution

It establishes the conjecture for all endomorphisms of hyper-K"ahler varieties and develops a canonical height framework in this setting.

Findings

Proved the conjecture for non-uniruled threefolds with degree > 1.

Established the conjecture for all endomorphisms of hyper-K"ahler varieties.

Constructed canonical height functions for automorphisms of hyper-K"ahler varieties.

Abstract

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: $\lambda_1(f) = \alpha_f(P)$. 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-K\"ahler varieties in any dimension. In the latter case, we construct a canonical height function associated to any automorphism $f\colon X \to X$ of a hyper-K\"ahler variety defined over $\overline{\mathbb{Q}}$.