For a model ${\cal X}$ we can interpret elements of $C_{X^{an}}({\bf Z})$ as paths in ${\cal X}$, i.e. equivalence classes of maps

$$\phi:Spec({{\cal O}}_{L})\to{\cal X},$$

where ${{\cal O}}_{L}$ is the ring of integers in a field $L$ with discrete valuation in ${\bf Z}$, such that the image of $\phi$ does not lie in ${\cal X}$. We define the map

$$p_{\cal X}^{\bf Z}\,:\,C_{X^{an}}({\bf Z})\to C_{\cal X}({\bf Z})$$

as

$$p_{\cal X}^{\bf Z}([\phi]):=\sum_{i}a_{i}\langle D_{i}\rangle,$$

where $a_{i}\in{\bf Z}_{\geq 0}$ is the multiplicity of the intersection of the path $\phi$ with the divisor $D_{i},\,\,i\in I_{\cal X}$.

The following proposition can be derived from [Be1].

We denote by $p_{\cal X}:X^{an}\to S_{\cal X}$ the map induced by $p_{\cal X}^{\bf R}$.

Let $f:{\cal X}^{\prime}\to{\cal X}$ be a dominating map of models. Let us denote by $m_{i,i^{\prime}}\in{{\bf Z}}_{\geq 0}$ the multiplicity of a divisor $D_{i^{\prime}},i^{\prime}\in I_{{\cal X}^{\prime}}$ in the proper pull-back of $D_{i},i\in I_{\cal X}$. We define $p_{{\cal X}^{\prime},{\cal X}}^{{\bf Z}}:C_{X^{an}}({\bf Z})\to C_{\cal X}({\bf Z})$ by the formulas $\sum_{i^{\prime}}a_{i^{\prime}}\langle D_{i^{\prime}}\rangle\mapsto\sum_{i}m_{i,i^{\prime}}a_{i^{\prime}}\langle D_{i}\rangle$. Let $p_{{\cal X}^{\prime},{\cal X}}:S_{X^{an}}({\bf R})\to S_{\cal X}({\bf R})$ be the corresponding by map of Clemens polytopes.

Then we have the following result, which is easy to prove.

With any meromorphic at $t=0$ family of smooth complex projective varieties $X_{t},\,\,\,0<|t|<\epsilon$ one can associate a variety $X$ over the field ${\bf C}((t))$. It is easy to see that for any snc model $\cal X$ one can canonically complete the family $X_{t}$ by adding $S_{\cal X}$ as the fiber over $t=0$. The total space is not a complex manifold by just a Hausdorff locally compact space which maps properly to the dick $\{t\in{\bf C}\,|\,|t|<\epsilon\}$. Passing to the projective limit we see that one can compactify the family $X_{t}$ at $t=0$ by $X^{an}$.