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.