Let $\mathrm{X}$ be the projective space $\mathrm{P}^{n}_{K}$ and let $\mathscr{O}(1)$ be the tautological line bundle on $\mathrm{X}$, together with its Weil metric. Let us describe the associated measure, taking the opportunity to add details concerning Berkovich spaces.

As we remarked above, the Weil metric is induced by the tautological line bundle on the projective scheme $\mathfrak{X}=\mathbf{P}^{n}_{K^{\circ}}$ and is smooth. The special fiber of $\mathfrak{X}$ is the projective space $\mathbf{P}^{n}_{\tilde{K}}$ over the residue field of $K^{\circ}$ ; it is in particular irreducible. Moreover, the degree of the tautological line bundle is equal to $1$. The measure $c_{1}(\overline{\mathscr{O}(1)})^{n}$ is therefore equal to the Dirac mass at the unique point of $\mathrm{X}$ which reduces to the generic point of the special fiber. It remains to describe this point more precisely.

The scheme $\mathbf{P}^{n}_{K^{\circ}}$ is the union of $(n+1)$ affine open subsets $\mathfrak{U}_{0},\dots,\mathfrak{U}_{n}$ defined by the non-vanishing of the homogeneous coordinates $x_{0},\dots,x_{n}$. Their generic fibers in the sense of analytic geometry are $n+1$ affinoid subsets $\mathrm{U}_{0},\dots,\mathrm{U}_{n}$, which cover $\mathrm{P}^{n}_{K}$. In fact, $\mathrm{U}_{i}$ corresponds to the set of points $[x_{0}:\dots:x_{n}]$ of $\mathrm{P}^{n}_{K}$ such that $\left|{x_{i}}\right|=\max(\left|{x_{0}}\right|,\dots,\left|{x_{n}}\right|)$.

To fix ideas, let us consider $i=0$. Then, $\mathfrak{U}_{0}=\operatorname{Spec}(K^{\circ}[T_{1},\dots,T_{n}])$ is the affine space ove $K^{\circ}$ with coordinates $T_{j}=x_{j}/x_{0}$. The natural $K$-adic topology on the algebra $K^{\circ}[T_{1},\dots,T_{n}]$, and on its tensor product with $K$, $K[T_{1},\dots,T_{n}]$, is given by the Gauß norm

$$\left\|{f}\right\|=\max_{\mathbf{a}\in{\mathbf{N}}^{n}}\left|{f_{\mathbf{a}}}\right|,\qquad f=\sum f_{\mathbf{a}}T_{1}^{a_{1}}\dots T_{n}^{a_{n}}.$$

The completion of $K[T_{1},\dots,T_{n}]$ for this norm is the Tate algebra $K\langle T_{1},\dots,T_{n}\rangle$ consisting of all power series $f=\sum f_{\mathbf{a}}T_{1}^{a_{1}}\dots T_{n}^{a_{n}}$ with coefficients in $K$ such that $\left|{f_{\mathbf{a}}}\right|\rightarrow 0$ when $\left|{\mathbf{a}}\right|=a_{1}+\dots+a_{n}\rightarrow\infty$ ; it is endowed with the natural extension of the Gauß norm, and is complete. By definition, the generic fiber $\mathrm{U}_{0}$ of $\mathfrak{U}_{0}$ in the sense of analytic geometry is the Berkovich spectrum of the Tate algebra, that is the set of all multiplicative semi-norms on it which are continuous with respect to the topology defined by the Gauß norm. Since the theorem of Gauß asserts that this norm is multiplicative, it defines a point $\gamma\in\mathrm{U}_{0}$, which we like to call the Gauß point.

The reduction map $\mathrm{U}_{0}\rightarrow\mathfrak{U}_{0}\otimes{\tilde{K}}$ is defined as follows. Let $x\in\mathrm{U}_{0}$, let $\mathfrak{p}_{x}\subset K\langle T_{1},\dots,T_{d}\rangle$ be the kernel of the semi-norm $x$, which is also the kernel of the canonical morphism $\theta_{x}\colon K\langle T_{1},\dots,T_{d}\rangle\rightarrow\mathscr{H}(x)$. The images $T_{j}(x)$ of the indeterminates $T_{j}$ are elements of absolute value $\leq 1$ of the complete ultrametric field $\mathscr{H}(x)$ ; they belong to its valuation ring $\mathscr{H}(x)^{\circ}$. Letting $\widetilde{\mathscr{H}(x)}$ to be the residue field, there exists a unique morphism $\theta_{\overline{x}}\colon{\tilde{K}}[T_{1},\dots,T_{d}]\rightarrow\widetilde{\mathscr{H}(x)}$ such that $\theta_{\overline{x}}(T_{i})$ is the image in $\widetilde{\mathscr{H}(x)}$ of $T_{i}(x)$. The kernel of this morphism is a prime ideal of the ring ${\tilde{K}}[T_{1},\dots,T_{d}]$ and defines a point $\overline{x}$ in the scheme $\mathfrak{U}_{0}\otimes{\tilde{K}}$.

Let us now compute the reduction of the Gauß point $\gamma$. By definition, the field $\mathscr{H}(\gamma)$ is the completion of the Tate algebra $K\langle T_{1},\dots,T_{d}\rangle$ for the Gauß norm. I claim that morphism $\theta_{\overline{\gamma}}$ is injective, in other words, that the images of $T_{1}(\gamma),\dots,T_{d}(\gamma)$ in the residue field $\widetilde{\mathscr{H}(\gamma)}$ are algebraically independent. Let $P\in K^{\circ}[T_{1},\dots,T_{d}]$ be any polynomial whose reduction $\overline{P}$ belongs to the kernel of $\theta_{\overline{\gamma}}$ ; this means $\left|{P}\right|_{\gamma}<1$ ; in other words, the Gauß norm of $P$ is $<1$ and each coefficient of $P$ has absolute value $<1$. Consequently, $\overline{P}=0$ and $\theta_{\overline{\gamma}}$ is injective, as claimed. This shows that $\overline{\gamma}$ is the generic point of the scheme $\mathfrak{U}_{0}\otimes{\tilde{K}}$.

We thus have proved the following proposition.