Here we prove that $V_{t}$ is non-singular and that $\lambda_{t}$ satisfies to all hypotheses of Theorem 3 for a large $t>0$.

Note that $||t^{-v(j)}z^{j}||=t^{jx-v(j)}$, where $jx\in\mathbb{R}$ stands for the scalar product, at any $z$ such that $\operatorname{Log}_{t}=x$. Let $F\subset\Pi$ be an open $(n+2-k)$-cell.

This lemma implies that ${V}^{\circ}_{t}$ is non-singular for large $t>0$. Indeed, it is covered by a finite number of open sets and in each set it is a small perturbation of a hyperplane. Furthermore, its compactification $V_{t}\subset\mathbb{C}T_{\Delta}$ is smooth and transverse to the coordinate hyperplanes as the same reasoning with the terms of smaller order applies to the affine charts of $\mathbb{C}T_{\Delta}$.

Our next step is to isotop $V_{t}$ over $N_{j}$ as in 3.4. Recall that $N_{j}$ was defined in 5.5 as a small neighborhood of $\bar{U_{j}}\subset\Pi$ in $\mathbb{R}^{n+1}$. Denote

$$Q^{n}_{j}=M_{j}^{-1}(H_{t}(Q^{n}))\cap\operatorname{Log}_{t}^{-1}(N_{j}).$$

By the last conclusion of Proposition 3.6 these manifolds coincide over $N_{j}\cap N_{k}$ for $t>>0$. We set

$$Q_{\Pi}=\bigcup\limits_{j}Q^{n}_{j}.$$

Note that for $t>>0$ ${V}^{\circ}_{t}$ is isotopic to $Q_{\Pi}$ by the same isotopy as in the proof of Proposition 3.6 since all other monomials of $f_{t}$ have smaller order in $t$. This proves Theorem 4. As in Proposition 3.6 the closure $\bar{Q_{\Pi}}\subset\mathbb{C}T_{\Delta}$ is a smooth manifold. Similarly, $V_{t}$ is isotopic to $\bar{Q_{\Pi}}$ in $\mathbb{C}T_{\Delta}$.

In the proof of Theorem 3 we may assume that ${V}^{\circ}={V}^{\circ}_{t}$ since its closure $V_{t}\subset\mathbb{C}T_{\Delta}$ is smooth and transverse to the coordinate hyperplanes. Similarly, in the proof of Theorems 1, 1’ and 2 we may assume that $V=V_{t}$. We define $\lambda^{\circ}:{V}^{\circ}\to\Pi$ as a composition of the isotopy $V\approx Q_{\Pi}$ (note that since this map is realized by an ambient isotopy it is a symplectomorphism by Moser’s trick), the map $\operatorname{Log}_{t}:(\mathbb{C}^{*})^{n+1}\to\mathbb{R}^{n+1}$ and the projection $\pi_{\mathcal{F}_{\Pi}}:\mathcal{N}\to\Pi$. To define $\lambda:V\to\bar{\Pi}$ we compactify the previous construction by using $\bar{Q_{\Pi}}$ and the reparametrized moment map to $\Delta$ as in 1.3.

This proves Theorem 2, since everything in our construction is equivariant with respect to complex conjugation as long as $a_{j}$ in the patchworking polynomial are real. The fibration $\lambda$ is totally real since it is totally real for a hyperplane.

Also, by Proposition 3.6 this proves the second and the third conclusions in Theorems 1, 1’ and 3. The homotopy type of $\bar{\Pi}$ and $\Pi$ is the wedge of $p_{g}$ copies of $S^{n}$, where $p_{g}=h^{n,0}$ by Proposition 1.10.

To finish the proof of Theorems 1. 1’ and 3 we need to prove injectivity of the induced homomorphism in cohomology and to exhibit the Lagrangian spheres lifting the cycles from $\Pi$.