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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 the zero locus of 
$\widehat {h}_G$, we prove the Kawaguchi–Silverman conjecture for each element of G. As for other applications, we determine the height counting function for non-periodic points and show that X satisfies potential density.
