In that situation, the reduction graph $R(\mathfrak{X})$ is a metrized graph defined as follows. It has for vertices the irreducible components of the special fiber, with as many edges of length $\log\left|{\pi}\right|^{-1}$ between two vertices as the number of intersection points of the corresponding components. In an neighbourghood of a double point, $\mathfrak{X}$ looks like (i.e., has an étale map to) the scheme with equation $xy=\pi$ in the affine plane $\mathbf{A}^{2}_{K^{\circ}}$.

If one replaces the field $K$ by a finite extension $K^{\prime}$, the base change $\mathfrak{X}\otimes_{K^{\circ}}(K^{\prime})^{\circ}$ may no more be regular. Indeed, $\mathfrak{X}\otimes_{K^{\circ}}(K^{\prime})^{\circ}$ is étale locally isomorphic to $xy=(\pi^{\prime})^{e}$, where $\pi^{\prime}$ is a uniformizing element of $K^{\prime}$, and $e$ is the ramification index. When $e>1$, the origin is a singular point of that scheme and one needs to blow it up repeatedly in order to obtain a regular scheme, which is a semi-stable model of $\mathrm{X}_{K^{\prime}}$ over $(K^{\prime})^{\circ}$. The two initial components are replaced by a chain of $e+1$ components, the $e-1$ intermediate ones being projective lines. In other words, $e-1$ vertices have been added, regularly spaced along each edge. One concludes that the reduction graph has not changed, as a topological space. Its metric has not changed neither, since the $e$ edges that partition an original edge (of length $\log\left|{\pi}\right|^{-1}$) have length $\log\left|{\pi^{\prime}}\right|^{-1}=\frac{1}{e}\log\left|{\pi}\right|^{-1}$.

We say that a function on $R(\mathfrak{X})$ is piecewise linear if, up to passing to a finite extension (which replaces each edge by $e$ edges of length equal to $1/e$th of the initial one), it is linear on each edge.

Let us analyse the situation from the Berkovich viewpoint. As we have seen, the generic points of the special fiber are the reductions of canonical points of $\mathrm{X}$ : the vertices of the graph $R(\mathfrak{X})$ naturally live in $\mathrm{X}$. The same holds for the edges, but is a bit more subtle. As we have seen, blowing-up intersection points of components in the special fiber gives rise to new components, hence to new points of $\mathrm{X}$. Would we enlarge the ground field and blow-up indefinitely, the constellation of points in $\mathrm{X}$ that we draw converges to a graph which is isomorphic to $R(\mathfrak{X})$.

According to Berkovich [12], a far more precise result holds. Let us consider a neighborhood $\mathfrak{U}$ of a singular point of the special fiber, pretending it is isomorphic to the locus defined by the equation $xy-\pi$ in $\mathbf{A}^{2}$ ; so $\mathfrak{U}=\operatorname{Spec}(K^{\circ}[x,y]/(xy-\pi))$. Its generic fibre is the affinoid space $\mathrm{U}$ defined by the inequality $\left|{xy}\right|=\left|{\pi}\right|$ in the unit polydisk $\mathrm{B}^{2}=\mathscr{M}(K\langle x,y\rangle$. The affinoid algebra of $\mathrm{U}$ is the quotient

whose elements $f$ are (non-uniquely) represented by a series

with $a_{m,n}\rightarrow 0$ when $m+n\rightarrow\infty$. However, observing that $x$ is invertible in this algebra, with inverse $\pi^{-1}y$, so that $y=\pi x^{-1}$, we can replace each product $x^{m}y^{n}$ by $\pi^{n}x^{m-n}$, leading to an expression of the form

where $\left|{a_{n}}\right|\rightarrow 0$ when $n\rightarrow+\infty$ and $\left|{a_{n}}\right|\pi^{-n}\rightarrow 0$ when $n\rightarrow-\infty$. Such an expression is now unique, and is called the Laurent expansion of $f$.

It leads to a natural family $(\gamma_{r})_{r\in[0,\log\left|{\pi}\right|^{-1}]}$ of multiplicative seminorms on the algebra $\mathscr{O}(\mathrm{U})$, parametrized by the unit interval in ${\mathbf{R}}$. Namely, for each real number $r\in[0,\log\left|{\pi}\right|^{-1}]$, we can set

Obviously, $\gamma_{r}$ is a norm on $\mathscr{O}(\mathrm{U})$ which extends the absolute value of $K$ ; its multiplicativity is proved analogously that of the Gauß norm. It is easy to check that the map $[0,\log\left|{\pi}\right|^{-1}]\rightarrow\mathrm{U}$ defined by $r\mapsto\gamma_{r}$ is continuous (this amounts to the fact that the maps $r\mapsto\gamma_{r}(f)$ are continuous), hence defines an parametrized path in the topological space $\mathrm{U}$.

Let $S(\mathfrak{U})$ be its image (with the induced distance) ; Berkovich calls it the skeleton of the formal scheme obtained by completing $\mathfrak{U}$ along its special fibre. A point $u$ in $\mathrm{U}$ has two coordinates $(x(u),y(u))$ in the completed residue field $\mathscr{H}(u)$ which are elements of absolute value $\leq 1$ satisfying $x(u)y(u)=\pi$. In particular,

The map $\rho\colon u\mapsto\gamma_{r(u)}$ is a continuous from $\mathrm{U}$ to $S(\mathfrak{U})$.

Let us compute the image of $\gamma_{r}$ by this map. By definition of $\gamma_{r}$, one has

hence $r(\gamma_{r})=r$ and $\rho(\gamma_{r})=\gamma_{r}$. In other words, the map $\rho$ is a retraction of $\mathrm{U}$ onto the skeleton $S(\mathfrak{U})$.

The special fiber of $\mathfrak{U}$ is defined by the equation $xy=0$ in $\mathbf{A}^{2}_{\tilde{K}}$, hence has two components. One can check that the point $\gamma_{0}$ reduces to the generic point of the component with equation $y=0$, while $\gamma_{\log\left|{\pi}\right|^{-1}}$ reduces to the generic point of the component with equation $x=0$.

These constructions have to be done around each singular point of the special fiber of $\mathfrak{X}$, locally for the étale topology of $\mathfrak{X}$. Berkovich proves that they can be glued, so that the graph $R(\mathfrak{X})$ is again canonically interpreted as an actual metrized graph drawn on the analytic space $\mathrm{X}$ ; we write $\iota\colon R(\mathfrak{X})\hookrightarrow\mathrm{X}$ for the canonical embedding. The map $\iota$ admits a continuous retraction $\rho\colon\mathrm{X}\rightarrow R(\mathfrak{X})$.

Although we will not use this fact, we must mention that the retraction $\rho$ is a deformation retraction. (For any $t\in[0,1]$ and any $x\in\mathrm{X}$, $\rho_{t}(x)$ is the semi-norm $x^{1-t}\rho(x)^{t}$.)

A construction of S. Zhang [57], building on prior results of Chinburg–Rumely [20], furnishes continuous metrics on divisors from continuous functions on the reduction graph $R(\mathfrak{X})$. It works as follows. First of all, if $P\in\mathrm{X}(K)$ is a rational point, there is a unique morphism $\varepsilon_{P}\colon\operatorname{Spec}K^{\circ}\rightarrow\mathfrak{X}$ which extends the point $P$ viewed as a morphism from $\operatorname{Spec}K$ to $X$. The image of this section is a divisor $D_{P}$ on $\mathfrak{X}$ and the line bundle $\mathscr{O}(D_{P})$ on $\mathfrak{X}$ defines a smooth metric on $\mathscr{O}(P)$ ; we write $\overline{\mathscr{O}(P)}_{\mathfrak{X}}$ for the corresponding metrized line bundle. We also define $\mu_{P}$ as the Dirac measure at the vertex of the graph corresponding to the (unique) irreducible component of the special fiber by which $D_{P}$ passes through. The construction and the notation is extended by additivity for divisors which are sums of rational points. More generally, if $P$ is only a closed point of $X$, we do this construction after the finite extension $K(P)/K$, so that $P$ becomes a sum of rational points, using for model the minimal resolution of $\mathfrak{X}\otimes K(P)^{\circ}$ described earlier.

If $f$ is any continuous function on $R(\mathfrak{X})$ and $D$ a divisor on $\mathrm{X}$, the metrized line bundle $\mathscr{O}(D+f)_{\mathfrak{X}}$ is deduced from $\overline{\mathscr{O}(D)}_{\mathfrak{X}}$ by multiplying the metric by $e^{-f}$. When $f$ is piecewise linear, this metrized line bundle is smooth. To prove that, we may extend the scalars and assume that $D$ is a sum of rational points $\sum n_{j}P_{j}$ and that $f$ is linear on each edge corresponding to an intersection point of components of the special fiber. Letting $(V_{i})$ being the family of these components, and writing $v_{i}$ for the vertex of $R(\mathfrak{X})$ corresponding to $V_{i}$, the divisor

defines the metrized line bundle $\mathscr{O}(D+f)_{\mathfrak{X}}$.

In this context, Zhang has defined a curvature operator, which associates to a metrized line bundle a distribution on the graph $R(\mathfrak{X})$, defined in such a way that

• —

  for any divisor $D$ on $\mathrm{X}$, ${\operatorname{curv}}(\overline{\mathscr{O}(D)}_{\mathfrak{X}})=\mu_{D}$ ;

• —

  for any continuous function $f$, ${\operatorname{curv}}(\overline{O}(f)_{\mathfrak{X}})=-\Delta f$, where $\Delta$ is the Laplacian operator of the graph $R(\mathfrak{X})$,

and depending linearly on the metrized line bundle. The following lemma compares this construction with the general one on Berkovich spaces.

In other words, the measure $c_{1}(\overline{L})$ is supported by the graph $R(\mathfrak{X})$ where it coincides essentially with Zhang’s curvature.