Canonical heights for abelian group actions of maximal dynamical rank
Fei Hu, Guolei Zhong
TL;DR
The paper addresses arithmetic dynamics of high-rank abelian group actions on smooth projective varieties over $\overline{\mathbf{Q}}$ by constructing a canonical height for a group $G\cong\mathbf{Z}^{n-1}$ of positive entropy. The authors first obtain $n$ common nef eigen-divisors $D_i$ with associated eigencharacters $\chi_i$ and assemble them into a nef, big divisor $D=\sum_i D_i$, whose augmented base locus $\mathbf{B}_+(D)$ is $G$-invariant. They then define a canonical height $\widehat{h}_G$ as a sum of nef canonical heights $\widehat{h}_{D_i,g_i}$, proving it satisfies the Northcott property on $X\setminus\mathbf{B}_+(D)$ and that $\widehat{h}_G(x)=0$ iff $x$ is $G$-periodic. This framework yields a verification of the Kawaguchi--Silverman conjecture for each $g\in G$ (with arithmetic degrees matching first dynamical degrees for non-periodic orbits) and provides a height counting description for non-periodic points, thereby extending known results from surfaces to higher dimensions. The results hinge on a higher-dimensional canonical height strategy built from commuting cone-preserving linear maps and the augmented base locus analysis, marrying dynamics with arithmetic geometry in maximal dynamical rank scenarios.
Abstract
Let $X$ be a smooth projective variety of dimension $n\geq 2$ and $G\cong\mathbf{Z}^{n-1}$ a free abelian group of automorphisms of $X$ over $\overline{\mathbf{Q}}$. Suppose that $G$ is of positive entropy. We construct a canonical height function $\widehat{h}_G$ associated with $G$, corresponding to a nef and big $\mathbf{R}$-divisor, satisfying the Northcott property. By characterizing its null locus, we prove the Kawaguchi--Silverman conjecture for each element of $G$. As another application, we determine the height counting function for non-periodic points.