Let $X=X/K$ be a smooth proper algebraic variety over a non-archimedean field $K$, $\dim\,X=n$ and $\Omega\in\Gamma(X,\Omega_{X}^{n})$ be a non-zero top degree form on $X$. We will associate canonically with the pair $(X,\Omega)$ a piecewise-linear compact space $Sk(X,\Omega)\subset X^{an}$ such that $\dim_{{\bf R}}Sk(X,\Omega)\leq n$.

Let us assume for simplicity that $K={{\bf C}}((t))$ and $X$ is defined over ${{\bf C}}_{t}^{mer}\subset K$. Analytic space $X^{an}$ contains a dense subset $X_{Div}$ of divisorial points corresponding to irreducible components of special fibers of all snc models ${\cal X}$ of $X$ (see Appendix A):

$$X_{Div}=\cup_{\cal X}i_{\cal X}(V_{S_{\cal X}})\,\,\,,$$

where $V_{S_{\cal X}}$ is the set of vertices of the Clemens polytope $S_{\cal X}$.

Top degree form $\Omega$ gives rise to a map $\psi_{\Omega}:X_{Div}\to{\bf Q}$. Namely, if $p:{\cal X}\to Spec({{\bf C}}_{t}^{mer})$ is a snc model and $D\subset{\cal X}_{0}$ is an irreducible divisor of the special fiber then we define

$$\psi_{\Omega}(D)={{ord_{D}(\Omega\wedge dt/t)}\over{{ord_{D}(p^{\ast}(t))}}}\,\,.$$

Here $\Omega\wedge dt/t$ is a meromorphic top degree form on ${\cal X}/{{\bf C}}$.

It is easy to show that $\psi_{\Omega}(D)$ depends only on the point $i_{\cal X}(D)\in X^{an}$. Function $\psi_{\Omega}$ is (globally) bounded from below.

Let ${\cal X}$ be a snc model. We will explain how to describe $Sk(X,\Omega)$ in terms of ${\cal X}$ and $\Omega$. In fact it is a nonempty simplicial subcomplex of $i_{\cal X}(S_{\cal X})$.

Let us call ${\cal X}$-essential a divisor $D_{i}\subset{\cal X}_{0}$ such that

$$\psi_{\Omega}(D_{i})=\min_{D_{j}\in{\cal X}_{0}}\psi_{\Omega}(D_{j})\,\,.$$

A nonempty collection $D_{i_{1}},...,D_{i_{l}}$ of divisors in ${\cal X}_{0}$ is called ${\cal X}$-essential if all $D_{i_{k}}$ are ${\cal X}$-essential, the intersection $D_{i_{1}}\cap D_{i_{2}}\cap...\cap D_{i_{l}}$ is non-empty and does not belong to the closure of the divisor of zeros of $\Omega$ is ${\cal X}\setminus{\cal X}_{0}$.

Sketch of the proof. Notice that for any snc model ${\cal X}$ the subset $S_{\cal X}({\bf Q})\subset S_{\cal X}$ consisting of points with rational barycentric coordinates is mapped by $i_{\cal X}$ into $X_{Div}$. Namely, we can modify ${\cal X}$ by blowing up at nonempty intersections of irreducible components of the special fiber and then continue this process indefinitely. Divisorial points obtained in this way exhaust all points of $i_{\cal X}(S_{\cal X}({\bf Q}))$.

We will prove that the set of essential points in $X^{an}$ coincides with $i_{\cal X}(S_{\cal X}({\bf Q})\cap Sk({\cal X},\Omega))$. First of all, a direct computation shows that $\psi_{\Omega}$ being restricted to $i_{\cal X}(S_{\cal X}({\bf Q}))$ achieves its absolute minimum on $i_{\cal X}(S_{\cal X}({\bf Q})\cap Sk({\cal X},\Omega))$. Secondly, another straightforward computation shows that the latter set does not change under blow-ups of first and second type (see Section A.5 in Appendix A). This concludes the proof. $\blacksquare$

For Calabi-Yau manifold $X$ we will denote $Sk(X,\Omega)$ simply by $Sk(X)$, as there exists only one (up to a scalar) non-zero top-degree form $\Omega$ on $X$ and $Sk(X,\lambda\Omega)=Sk(X,\Omega)\,\,\,\forall\lambda\in K^{\times}$.

One can prove that the PL space $Sk(X,\Omega)$ is in fact a birational invariant. Moreover the group $Aut^{brt}(X)$ of birational automorphisms of $X$ acts on the skeleton by ${{\bf Z}}PL$ transformations. In order to obtain non-trivial examples of such actions we need Calabi-Yau manifolds with large groups of birational automorphisms. An example of a ${{\bf Z}}PL$-action is considered in the next subsection.