Let $\Pi$ be a maximal dual $\Delta$-complex and $v:\Delta\cap\mathbb{Z}^{n+1}\to\mathbb{R}$ be the function such that $\Pi=\Pi_{v}$ as in Proposition 1.4. It gives us a patchworking polynomial $f_{t}=\sum\limits_{j\in\Delta\cap\mathbb{Z}^{n+1}}t^{-v(j)}z^{j}$. As before we denote with ${V}^{\circ}_{t}\subset(\mathbb{C}^{*})^{n+1}$ the zero set of this polynomial.

We construct $\lambda_{t}:{V}^{\circ}_{t}\to\Pi$ for a sufficiently large $t$ by gluing the fibrations $\lambda_{H}$ from 3.3.

To do it we construct a singular foliation $\mathcal{F}_{\Pi}$ in a neighborhood $\mathcal{N}\supset\Pi$. By Proposition 1.11 $\Pi$ can be locally identified with $\Sigma_{n}$ by elements of $ASL_{n+1}(\mathbb{Z})$. Recall that an element $M\in ASL_{n+1}(\mathbb{Z})$ is a rotation defined by a unimodular integer $(n+1)\times(n+1)$-matrix $(m_{j,k})$ followed by a translation by $m=(m_{1},\dots,m_{n+1})$ in $\mathbb{R}^{n+1}$. This transformation of $\mathbb{R}^{n+1}$ lifts to $(\mathbb{C}^{*})^{n+1}$ as

$$H_{M}:z_{j}\mapsto m_{j}z_{1}^{m_{j,1}}\dots z_{n+1}^{m_{j,n+1}}.$$

We patch the foliations $\mathcal{F}$ constructed in 3.3 for the primitive $n$-complex $\Sigma_{n}$. Let $v_{j}\in\Pi$ be a vertex. By Proposition 1.11 there exists a neighborhood $U_{j}\ni v_{j}$ in $\Pi$ and $M_{j}\in ASL_{n+1}(\mathbb{Z})$ such that $M_{j}(U_{j})$ is a neighborhood of $0$ in $\Sigma_{n}$. Let $N_{j}$ be a small neighborhood of the closure of $M_{j}(U_{j})$.

Consider the pull-back under $M_{j}$ of the foliation $\mathcal{F}$ constructed in 3.3 restricted to $N_{j}$. Note that $M_{j}^{-1}(N_{j})$ cover $\Pi$. The pull-back foliations at the overlaps $M_{j}^{-1}(N_{j})\cap M_{k}^{-1}(N_{k})$ do agree in general. Nevertheless, they have the same type of singularities at the same points and their non-singular leaves are transverse to $\Pi$. A partition of unity gives a foliation $\mathcal{F}_{\Pi}$ in a neighborhood $\mathcal{N}$ of $\Pi$. Note that we can ensure that $\mathcal{N}$ contains an $\epsilon$-neighborhood of $\Pi$ for some $\epsilon>0$. Following 3.3 we denote $\pi_{\mathcal{F}_{\Pi}}:\mathcal{N}\to\Pi$ the projection along the leaves of $\mathcal{F}_{\Pi}$.

By Corollary 5.4 for a sufficiently large $t>0$ we have $\operatorname{Log}_{t}(V_{t})\subset\mathcal{N},$ and we define

$$\lambda_{t}=\pi_{\mathcal{F}_{\Pi}}\circ\operatorname{Log}_{t}:V_{t}\to\Pi.$$