The Lagrangian spheres will come from components of certain real hypersurfaces whose complexification is isotopic to $V$.

Let $j$ be a lattice point of $\Delta$. We define

$$f_{t}^{(j)}=\sum\limits_{k\neq j}|a_{k}|t^{v(k)}z^{k}-|a_{j}|t^{v(j)}z^{j}.$$

Denote with $V_{t}^{(j)}\subset(\mathbb{C}^{*})^{n+1}$ the zero set of $f_{t}^{(j)}$ and with $\mathbb{R}V_{t}^{(j)}\subset(\mathbb{R}^{*})^{n+1}$ its real part. The Viro patchworking theorem [15] (see also [4] for a special case of combinatorial patchworking and [5] for an elementary description in the case of curves) implies that $\mathbb{R}V_{t}^{(j)}\cap\mathbb{R}_{+}^{n+1}$ is diffeomorphic to a sphere $S^{n}$. This sphere $S^{n}_{j}\subset V_{t}^{(j)}$ is Lagrangian as a component of the real part and it maps under $\operatorname{Log}_{t}$ to $\mathcal{N}\supset\Pi$ for $t>>0$. Furthermore, it realizes in $H_{n}(\Pi)$ the class corresponding to $j$ according to Proposition 1.10.

By 5.6 $V_{t}^{(j)}$ is smooth. Thus, it is isotopic to $V_{t}$ and we have a diffeomorphism $h:V_{t}^{(j)}\to V_{t}$. Moreover, we can choose an isotopy among the hypersurfaces defined by such polynomials that the norm of all monomials is constant in the course of deformation. All such hypersurfaces are smooth and their image under $\operatorname{Log}_{t}$ is contained in $\mathcal{N}\supset\Pi$ by 5.6. Therefore, the image $h(S^{n}_{j})$ projects to the same class in $H_{n}(\Pi)$.

By Moser’s trick, $h$ is isotopic to a symplectomorphism. This gives a Lagrangian sphere in $V_{t}$ which projects to the class in $H_{n}(\Pi)$ corresponding to $j$. Thus the last conclusion of Theorems 1 and 1’ is proved.

Existence of such spheres also implies the first conclusion of Theorems 1 and 1’. The map $\lambda^{*}$ is injective since we can distinguish the images in $H^{n}(V;\mathbb{Z})$ by their evaluations on these Lagrangian spheres.

The proof of Theorem 3 is the same since these spheres belong to the toric part $\mathbb{R}{V}^{\circ}_{t}$ of $\mathbb{R}V$.