Let $V$ be a finite set, $S\subset 2^{V}$ belongs to the set of $(n+1)$-element subsets of $V$. Then we have a $n$-dimensional simplicial complex $B=\cup_{Y\in S}\Delta^{Y}\subset\Delta^{V}$.

Let us choose a ${\bf Z}$-affine structure on $n$-dimensional faces of $B$ which is compatible with the standard affine structure, and consider all $(n-1)$-dimensional faces which enjoy the following property: they belong to exactly two $n$-dimensional faces. For any two such $n$-dimensional faces $\sigma$ and $\tau$ we choose a ${\bf Z}$-affine structure on $\sigma\cup\tau$ which is compatible with the already chosen ${\bf Z}$-affine structures on $\sigma$ and $\tau$ (such a choice is equivalent to a choice of ${\bf Z}$-affine structure in a neighborhood of the $(n-1)$-dimensional face $\sigma\cap\tau$). In this way we obtain a ${\bf Z}$-affine structure on the union $U$ of the interior points of all $n$-dimensional simplices and also the interior points of $(n-1)$-dimensional faces belonging to exactly two top-dimensional cells.

Proof. Let us proceed inductively by codimension of faces. The induction step reduces to the obvious remark that the extension of the standard ${\bf Z}$-affine structure on ${{\bf R}}^{n}\setminus L$ to a neighborhood of point $p\in L$ in ${\bf R}^{n}$ is unique in the case when $L\subset{\bf R}^{n}$ is an affine subspace, $\dim L\leq n-2\,\,$. $\blacksquare$

It is easy to see that $B^{sm}:=U_{\max}$ with ${\bf Z}$-affine structure on it, compactified by $B$ satisfies both Finiteness and Independence properties.

We introduce PL compactifications both as a “toy model”, and also (as we hope, see Conjecture 6 in Section 6.3) as a sufficiently representative class for applications. In this case we can try to formulate additional desired properties. One of goals is to find a good substitution for the algebro-geometric notion of a canonical singularity (which is, morally, a singularity of a non-collapsing limit of a family of Calabi-Yau manifolds with fixed Kähler class).

For a large class of maximally degenerating Calabi-Yau manifolds there is a proposal by several authors (see [GS] and [HZh]) for a PL compactification $B$ conjecturally related to the Gromov-Hausdorff limit. Space $B$ is topologically a sphere $S^{n}$, it carries two dual cell decompositions. Each of these decompositions is identified with the boundary $\partial P_{1}$ or $\partial P_{2}$ of a convex $(n+1)$-dimensional polytope. Moreover, on each $n$-dimensional face of each polytope we have a ${\bf Z}$-affine structure compatible with the natural affine structure. The assumption is that for any two open $n$-cells $U,U^{\prime}$ from the first and the second cell decompositions, two induced ${\bf Z}$-affine structures on $U\cap U^{\prime}$ coincide. This gives a ${\bf Z}$-affine structure on $B\setminus\left(Sk_{n-1}\cap Sk_{n-1}^{\prime}\right)$ where $Sk_{n-1}$ and $Sk_{n-1}^{\prime}$ are $(n-1)$-skeletons of two CW-structures.