This is a basic example

$$\pi=\pi_{can}=\log|\cdot|:X=({\bf G}_{m}^{an})^{n}\to B_{0}=B={{\bf R}}^{n}$$

described in details in Appendix A. For any algebraic (or analytic) subvariety $Z\subset({\bf G}_{m}^{an})^{n}$ of dimension $m\leq n$ its image $\pi(Z)$ is a non-compact piecewise-linear closed subset of ${\bf R}^{n}$ of real dimension $m$. Smooth points for $\pi_{|Z}$ are dense in $\pi(Z)$.

In particular, if $Z$ is a curve then $\pi(Z)$ is a graph in $B$ with straight edges having rational directions. One can try to make a dictionary which translates the properties of the algebraic variety $Z\subset{\bf G}_{m}^{n}$ to the properties of the PL set $\pi(Z^{an})$ which is the closure of $\pi(Z(\overline{K}))$ in ${\bf R}^{n}$. This circle of ideas is a subject of the so-called “tropical geometry” (see e.g. [Mi]).

Let $\rho:{{\bf Z}}^{n}\to(K^{\times})^{n}$ be a group homomorphism such that the image of the composition $val\circ\rho:{{\bf Z}}^{n}\to{{\bf R}}^{n}$ is a rank $n$ lattice in ${{\bf R}}^{n}$. Group $(K^{\times})^{n}$ acts by translations on the analytic space $({\bf G}_{m}^{an})^{n}$. Restriction of this action to ${{\bf Z}}^{n}$ (via $\rho$) is discrete and cocompact. The quotient is a $K$-analytic space $X$ called Tate torus. There is an obvious map $\pi:X\to B:={{\bf R}}^{n}/(val\circ\rho)({{\bf Z}}^{n})$. All points of $B$ are smooth. The space $X$ depends on $n^{2}$ parameters taking values in $K^{\times}$(cf. with the flat tori example in Section 3.2.1).

For any smooth projective variety $X$ of dimension $n$, and and a snc model ${\cal X}$ of it (see Appendix A) we have a canonical projection to the corresponding Clemens polytope

$$p_{{\cal X}}:X^{an}\rightarrow S_{{\cal X}}\,\,.$$

All interior points of $n$-dimensional simplices of $S_{{\cal X}}$ are $p_{{\cal X}}$-smooth, although there might be other smooth points too. More generally, one can compose projection $p_{{\cal X}}$ with a continuous surjection $\pi^{\prime}:S_{{\cal X}}\twoheadrightarrow B$ where $B$ is a finite CW complex and map $\pi^{\prime}$ is a cell map for some cell subdivision of $S_{{\cal X}}$. We assume that fibers of the composition $\pi:=\pi^{\prime}\circ p_{{\cal X}}:X^{an}\to B$ are connected. This seems to be the most general case of maps from projective varieties over complete local fields to CW complexes relevant for our purposes.

Let $X/K$ be a connected smooth projective curve of genus $g>1$. After passing to a finite extension $K^{\prime}$ of $K$ we may assume that $X$ has a canonical model ${\cal X}$ with stable reduction. The graph $\Gamma^{\prime}$ corresponding to the special fiber ${\cal X}_{0}$ is a retraction of $(X\otimes_{K}K^{\prime})^{an}$. The quotient graph $\Gamma=\Gamma^{\prime}/Gal(K^{\prime}/K)$ is a retraction of the analytic curve $X^{an}$ (see [Be1]). We define $B:=\Gamma$. Then $B^{sm}$ is a complement to a finite set. As in Section 3.2.2, a ${\bf Z}$-affine structure on a graph is the same as a length element (i.e. a metric). Therefore $\Gamma$ is a metrized graph. Notice also that the maximal number of edges of the graph corresponding to a genus $g$ curve is $3g-3$, which is the dimension of the moduli space of genus $g$ curves.

Notice that if in Section 4.2.1 subvariety $Z$ is a curve then its projection is a noncompact metrized graph with unbounded edges corresponding to punctures ${\overline{Z}}\setminus Z$.

Here we will describe a particular case of the construction from Section 4.2.3 (a contraction of a Clemens polytope).

Let field $K$ be ${{\bf C}}((t))$ and $X\subset{\bf P}^{3}_{K}$ be a formal family of complex K3 surfaces given by the equation

$$x_{0}x_{1}x_{2}x_{3}+tP_{4}(x_{0},x_{1},x_{2},x_{3})=0\,\,,$$

where $P_{4}$ is a generic homogeneous polynomial of degree four, and $t$ is a formal parameter.

The special fiber at $t=0$ of this family is singular, it is given by the equation $x_{0}x_{1}x_{2}x_{3}=0$. Let us denote by $\widetilde{{\bf P}^{3}}$ the blow-up of the total space of the trivial ${\bf P}^{3}$-bundle over $Spec({\cal O}_{K})$ at $24$ points $p_{\alpha},1\leq\alpha\leq 24$ of the special fiber, where each $p_{\alpha}$ is a solution of the equation

$$P_{4}(x_{0},x_{1},x_{2},x_{3})=0,\,\,x_{i}=x_{j}=0,\,\,1\leq i<j\leq 4\,\,.$$

The closure ${\cal X}$ of $X$ in $\widetilde{{\bf P}^{3}}$ is a model with simple normal crossings. The associated Clemens polytope $S_{\cal X}$ has $28$ vertices. Four of them correspond to coordinate hyperplanes $x_{i}=0$ in ${\bf P}^{3}$, and $24$ other correspond to divisors sitting at the pre-images of the points $p_{\alpha}$. Therefore $S_{\cal X}$ is the union of the boundary $\partial\Delta^{3}$ of the standard $3$-simplex $\Delta^{3}$ with $24$ copies of the standard $2$-simplex $\Delta^{2}$. Those $24$ triangles $\Delta_{\alpha}^{2},1\leq\alpha\leq 24$ are decomposed into six groups of four triangles in each. All triangles from the same group have a common edge, which is identified with an edge of $\partial\Delta^{3}$ (tetrahedron with $24$ “wings”). As we mentioned in the previous example, there is a continuous map $p:X^{an}\to S_{\cal X}$. We are going to construct $B$ as a retraction of $S_{\cal X}$.

In order to do this we observe that for an edge $e\subset\Delta^{2}$ and a point $a\in e$ one has the canonical retraction $p_{a,e}:\Delta^{2}\to e$. Namely, let us identify the edge $e$ with the interval $[-1,1]$ of the real line, so that $a$ is identified with the point $a=(a_{0},0)$, and $\Delta^{2}$ is bounded by $e$ and the segments $0\leq y\leq 1-|x|$. Then we define $p_{a,e}$ by the formulas (see Figure 2)

$$\begin{array}[]{llcl}(x,y)&\mapsto&(x+y,0)\,,&x+y\leq a_{0}\,\,;\\ (x,y)&\mapsto&(x-y,0)\,,&x-y\geq a_{0}\,\,;\\ (x,y)&\mapsto&(a_{0},0)\,,&\mbox{ otherwise.}\end{array}$$

Now we choose a point $q_{ij},0\leq i<j\leq 3$ in the interior of each edge $e_{ij}$ of $\partial\Delta^{3}$ (here $i,j$ are identified with the vertices of $\partial\Delta^{3}$). There are four “wings” $\Delta_{\alpha}^{2}$ having $e_{ij}$ as a common edge. Then we retract each $\Delta_{\alpha}^{2}$ to $e_{ij}$ by the map $p_{q_{ij},e_{ij}}$. This gives us a retraction $\pi^{\prime}=\pi^{\prime}_{(q_{ij})}:S_{\cal X}\to\partial\Delta^{3}$. Let $\pi:=p_{\cal X}\circ\pi^{\prime}_{(q_{ij})}:X^{an}\to B$ be the composition of the projection $p_{\cal X}:X^{an}\to S_{\cal X}$ with the above retraction. One can show that all points of $B:=\partial\Delta^{3}$ are $\pi$-smooth except of the chosen six points $q_{ij},0\leq i<j\leq 3$. According to Theorem 1 we obtain a ${\bf Z}$-affine structure on $S^{2}\setminus\cup_{1\leq i<j\leq 3}\{q_{ij}\}$. One can show that the local monodromy around each point $q_{ij}$ is conjugate to the matrix

$$\left(\begin{array}[]{cc}1&4\\ 0&1\end{array}\right)\,\,.$$

We skip the computations here.