Here we consider the well-known fibrations of the elliptic curve and the K3-surface.

Let $\mathbb{C}E$ be an elliptic curve, i.e. a Riemann surface of genus 1. Since $\mathbb{C}E$ is topologically a torus, there is a trivial $S^{1}$-fibration $\lambda_{E}:\mathbb{C}E\to S^{1}$.

Suppose that the elliptic curve $\mathbb{C}E\subset\mathbb{C}T_{\Delta}$ is presented as a curve in a toric surface $\mathbb{C}T_{\Delta}$, where $\Delta$ is the Newton polygon of a polynomial defining $\mathbb{C}E$. By the genus formula (see [8]), $\operatorname{Int}\Delta$ contains a unique lattice point. By Proposition 1.10 a dual $\Delta$-complex is homotopy equivalent to a circle. It is easy to see that the fibration from Theorem 1’ coincides up to homotopy with the trivial $S^{1}$-fibration $\mathbb{C}E\approx S^{1}\times S^{1}\to S^{1}$.

Another famous fibration $\lambda_{K}:\mathbb{C}K\to S^{2}$ has the K3-surface $\mathbb{C}K$ as its total space. All its fibers, except for 24 of them are Lagrangian tori.

Suppose that the polygon $\Delta$ has exactly one interior lattice point. Then, by Khovanskii’s formula [8], the zero locus $\mathbb{C}K$ of a generic polynomial with the Newton polygon $\Delta$ is a K3-surface. A dual $\Delta$-complex is homotopy equivalent to a sphere $S^{2}$ by Proposition 1.10.

Again, the fibration $\lambda$ can be deformed to a fibration like $\lambda_{K}$ by so-called shelling of $\bar{\Pi}$ ¹¹ 1 A higher-dimensional version of such deformation will be the subject for a future paper..

In higher dimensions, if $\Delta$ is a non-singular polyhedron with a unique interior lattice point, then the corresponding hypersurface $V\subset\mathbb{C}T_{\Delta}$ is a smooth Calabi-Yau manifold. Singular torus fibrations $V\to S^{n}$ were constructed by Zharkov [17]. Ruan [13] noted that such fibrations can be made Lagrangian.

Theorem 1’ constructs in this case a stratified torus fibration over a polyhedral complex homotopy equivalent to $S^{n}$.