Theorem 1’ can be treated as a pair-of-pants decomposition for $V$. We can use this presentation to reconstruct $V$ from $\Pi$. This allows to interpret a maximal balanced polyhedral complex $\Pi$ as the complex encoding the gluing pattern of pairs-of-pants in order to get $V$. Here is the way to reconstruct $V$ from $\Pi$.

For each vertex $v_{j}$ of $\Pi$ take a copy $\bar{Q}_{j}$ of $\bar{\mathcal{P}}_{n}$. This copy can be identified with the localized hyperplane $\bar{Q}^{n}\subset(\mathbb{C}^{*})^{n+1}$. Recall that by Proposition 1.24 $\partial_{1}(\bar{Q}_{j})$ consists of $n+2$ components. Each such component corresponds to a 1-cell of $\Pi$ adjacent to $v_{j}$.

Let $e_{jk}$ be a 1-cell of $\Pi$ connecting the vertices $v_{j}$ and $v_{k}$. For each such 1-cell we identify the closures $F_{j}$ and $F_{k}$ of the corresponding components of $\partial_{1}(\bar{Q}_{j})$ and $\partial_{1}(\bar{Q}_{k})$ in the following way.

Without the loss of generality we may assume that in both copies $\bar{Q}_{j},\bar{Q}_{k}$ of $\bar{Q}^{n}$ the edge $e_{jk}$ corresponds to the facet $x_{n+1}=-R_{n}$ of $\Delta(R_{n})$ (see 3.4). (Note that such correspondence is given by matrices $M_{j},M_{k}$ from Proposition 1.11.) We attach $F_{j}$ to $F_{k}$ by the map

$$(z_{1},\dots,z_{n},z_{n+1})\mapsto(z_{1},\dots,z_{n},\bar{z}_{n+1}),\log|z_{n+1}|=-R_{n},$$

where $\bar{z}_{n+1}$ is the complex conjugate to $z_{n+1}$.

The result $U$ of this gluing is a manifold with boundary. The boundary comes from the unbounded cells of $\Pi$. Denote $W^{\circ}=U\smallsetminus\partial U$. The boundary is formed by the closures $F$ of the components of $\partial_{1}(Q_{j})$ that correspond to unbounded 1-cells in $\Pi$. By Proposition 1.24 each such $F$ is a circle fibration over a union of lower-dimensional pairs-of-pants $\mathcal{P}_{n-1}$. Let $W$ be the result of collapsing all fibers of these fibrations on $\partial U$. Note that $W$ is canonically a smooth manifold since this procedure locally coincides with collapsing the boundary on $\bar{\mathcal{P}}_{n}$ which results in ${\mathbb{C}}{\mathbb{P}}^{n}$.