Canonical heights for abelian group actions of maximal dynamical rank

Let $X$ be a smooth projective variety of dimension $n\ge 2$ and $G\cong {\mathbf{Z}}^{n-1}$ a free abelian group of automorphisms of $X$ over $\bar{\mathbf{Q}}$. 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.