The proof is inspired from the above-mentioned sources, which in turns is an adaptation of the complex case [41] (see also [52]).

We may replace $L$ by a positive power of itself and assume that it is very ample, induced by a closed embedding of $\mathrm{X}$ in $\mathrm{P}^{n}$, and that the natural map $\Gamma(\mathrm{{\mathbf{P}}}^{n},\mathscr{O}(d))\rightarrow\Gamma(\mathrm{X},\mathscr{O}(d))$ is surjective. Then, there are homogeneous polynomials $(F_{0},\dots,F_{n})$, of degree $d$, with coefficients in $K$, and without common zeroes on $\mathrm{X}$, such that $f([x_{0}:\dots:x_{n}])=[F_{0}(x):\dots:F_{n}(x)]$ for any $x=[x_{0}:\dots:x_{n}]\in\mathrm{P}^{n}$. One considers the polynomial map $F\colon\mathrm{A}^{n+1}\rightarrow\mathrm{A}^{n+1}$ ; it lifts a rational map on $\mathrm{P}^{n}$ which extends the morphism $f$.

For $(x_{0},\dots,x_{n})\in\mathrm{A}^{n+1}$, define $\left\|{x}\right\|=\max(\left|{x_{0}}\right|,\dots,\left|{x_{n}}\right|)$. The Weil metric on $\mathscr{O}(1)$ is given by

where $P$ is an homogeneous polynomial, $s_{P}$ the corresponding global section of $\mathscr{O}(\deg(P))$, and $x$ is a point of $\mathrm{A}^{n+1}$ such that $P(x)\neq 0$. The restriction to $\mathrm{X}$ of this metric is a semi-positive metric $\left\|{\cdot}\right\|_{0}$ on $L$. The construction of the canonical metric on $L$ introduces a sequence of semi-positive metrics $\left\|{\cdot}\right\|_{n}$ on $L$ ; these metrics are given by the following explicit formula

where $F^{(k)}:\mathrm{A}^{n+1}\rightarrow\mathrm{A}^{n+1}$ is the $k$th iterate of $F$.

The convergence of this sequence is therefore equivalent to the convergence of the sequence $(d^{-k}\log\left\|{F^{(k)}}\right\|)_{k}$ towards a continuous fonction on the preimage of $\mathrm{X}$ under the projection map $\mathrm{A}^{n+1}\setminus\{0\}\rightarrow\mathrm{P}^{n}$. The limit is usually called the homogeneous Green function.

For $0\leq i\leq n$, let $\mathrm{V}_{i}$ be the open set of points $x=[x_{0}:\dots:x_{n}]\in\mathrm{P}^{n}$ such that $\left|{x_{i}}\right|>\frac{1}{2}\left\|{x}\right\|$. They form an open covering of $\mathrm{P}^{n}$ ; their intersections with $\mathrm{X}$ form an open covering of $\mathrm{X}$.

Fix $x\in\mathrm{E}_{f}$ and let $\mathrm{U}$ be an open neighbourhood of $x$ such that for any positive integer $k$, there exists $i\in\{0,\dots,n\}$ such that $f^{k}(\mathrm{U})\subset\mathrm{V}_{i}$. For any $i$, let $N_{i}$ be the set of integers $k$ such that $f^{k}(\mathrm{U})\subset\mathrm{V}_{i}$. Let us consider any index $i$ such that $N_{i}$ is infinite ; to fix ideas, let us assume that $i=0$. The canonical norm of a section $s_{P}$ at a point $y\in\mathrm{U}$ is given by

Observe that $[F_{0}^{(k)}(y):\dots:F^{(k)}_{m}(y)]$ are the homogeneous coordinates of the point $f^{k}(y)$. Since $y\in\mathrm{U}$ and $f^{k}(\mathrm{U})\subset\mathrm{V}_{0}$, one has $\left|{F^{(k)}_{i}(y)}\right|\leq 2\left|{F^{(k)}_{0}(y)}\right|$, so that the last term is bounded by $d^{-k}\log 2$ and uniformly converges to $0$ on $\mathrm{U}$. Finally, uniformly on $\mathrm{U}$,

This shows that $\log\left\|{s_{P}}\right\|^{-1}$ is strongly harmonic on $\mathrm{U}$, as claimed. ∎

It suffices to apply Prop. 2.3.3. ∎