Let $V\subset{\mathbb{C}}{\mathbb{P}}^{n+1}$ be a smooth hypersurface of degree $d$. We choose homogeneous coordinates $[Z_{0}:\dots:Z_{n+1}]$ so that $V$ is transverse to coordinate hyperplanes $Z_{j}=0$ and all their intersections. The complement of the coordinate hyperplanes in ${\mathbb{C}}{\mathbb{P}}^{n+1}$ is $(\mathbb{C}^{*})^{n+1}$. Denote ${V}^{\circ}=V\cap(\mathbb{C}^{*})^{n+1}$. Then the hypersurface ${V}^{\circ}\in(\mathbb{C}^{*})^{n+1}$ is given by equation $f(z)=0$, where

stands for affine coordinates in $(\mathbb{C}^{*})^{n+1}$ and $f$ is a polynomial with the Newton polyhedron $\Delta_{d}$ from Example 5. Recall that we denote the real $n$-dimensional torus with $T^{n}=S^{1}\times\dots\times S^{1}$.

This theorem admits a straightforward generalization to toric varieties other than ${\mathbb{C}}{\mathbb{P}}^{n+1}$. Let $\Delta$ be a bounded convex lattice polyhedron such that all singularities of the toric variety $\mathbb{C}T_{\Delta}$ are isolated. Note that the isolated singular points of $\mathbb{C}T_{\Delta}$ necessarily correspond to some vertices of $\Delta$. Consider the space $(\mathbb{C}^{*})^{\#(\Delta\cap\mathbb{Z}^{n+1})}$ of all polynomials of the type $f(z)=\sum\limits_{j\in\Delta}a_{j}z^{j}$ such that $a_{j}\neq 0$. Then for a generic choice of a polynomial $f$ from this space the closure $V$ in $\mathbb{C}T_{\Delta}$ of the zero set of $f$ is a smooth hypersurface transverse to all toric subvarieties $\mathbb{C}T_{\Delta^{\prime}}$ corresponding to the faces $\Delta^{\prime}\subset\Delta$. All such $V$ are diffeomorphic and, if we equip them with the symplectic form from $\mathbb{C}T_{\Delta}$, are symplectomorphic varieties.

By Remark 1.9 not all convex lattice polyhedra have maximal dual complexes. However, in the case of $\Delta_{d}$ (the polyhedra corresponding to the projective space), such subdivisions exists for any $d$. Maximal subdivisions also exist for products of different $\Delta_{d}$ (this corresponds to hypersurfaces in the product of projective spaces). It is conjectured that for any lattice polyhedron $\Delta$ there exists a sufficiently large integer $N$ that $N\Delta$ (the result of scaling of $\Delta$ by $N$) has a maximal subdivision.

The next theorem describes the behavior of the fibration $\lambda$ with respect to a complex structure on $V$. Recall that, unlike the smooth and symplectic structures, the complex structure on $V$ depends on the polynomial $f$ and not just on $\Delta$.

Recall that a map $\lambda:V\to\bar{\Pi}$ is called a totally real fibration if for any $z\in V$ the tangent space to the fiber through $z$ is totally real i.e. contains no positive-dimensional complex subspaces (as long as the fiber is smooth near $z$). We say that a hypersurface $V\subset\mathbb{C}T_{\Delta}$ is defined over $\mathbb{R}$ if it can be obtained as the closure of the zero set of a polynomial $f:(\mathbb{C}^{*})^{n+1}\to\mathbb{C}$ whose coefficients are real.

Theorems 1’ and 2 can be extended further to polyhedra $\Delta$ corresponding to toric varieties with non-isolated singularities. However, in order to do that, one has to modify the definition of stratified fibrations to include singular total spaces $V$. We do not do that. In the next theorem we no longer have any restrictions on the convex lattice polyhedron $\Delta$, but its statement concerns only the toric, non-singular, part ${V}^{\circ}\subset(\mathbb{C}^{*})^{n+1}$ of the hypersurface $V$.