Canonical heights for abelian group actions of maximal dynamical rank

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.