Let $S$ be a closed Riemann surface of genus $g>1$. It is well-known that $S$ admits a decomposition into pairs-of-pants. Namely, there exist $3g-3$ disjoint embedded circles $C_{j}\subset S$ such that $S\smallsetminus\bigcup\limits_{j=1}^{3g-3}C_{j}$ is a disjoint union of $2g-2$ copies of the pair-of-pants $P$. The pair-of-pants surface $P$ is homeomorphic to the Riemann sphere ${\mathbb{C}}{\mathbb{P}}^{1}$ punctured in three points.

To such a decomposition we associate a graph $\Gamma$. The vertices of $\Gamma$ correspond to the pairs-of-pants while the edges correspond to the circles $C_{j}$. Each edge joins the vertices corresponding to the adjacent pairs-of-pants.

There exists a fibration $\pi:S\to\Gamma$ such that the circles $C_{j}$ are inverse images of the midpoints of the edges of $\Gamma$. Such fibration is canonically associated to our decomposition into pairs-of-pants. To construct it we fiber each individual pair-of-pants over a tripod graph as pictured on the left-hand-side of Figure 5.

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 $S^{n}$.

This is a fundamental example for the main theorems. Let $H=\{z_{1}+\dots+z_{n+1}+1=0\}\subset{\mathbb{C}}{\mathbb{P}}^{n+1}$ be a hyperplane. Its toric part $H^{\circ}=H\cap(\mathbb{C}^{*})^{n+1}$ is an open pair-of-pants.

Let $\operatorname{Log}$ be the moment map for $(\mathbb{C}^{*})^{n+1}$ (see (3)).

The complement $\mathbb{R}^{n+1}\smallsetminus\Sigma_{n}$ consists of $n+2$ components. Each component is the region where one of the functions $0,x_{1},\dots,x_{n+1}$ is maximal. In the component corresponding to $x_{j}$ we consider the foliation into straight lines parallel to the gradient of $x_{j}$ (the $j$th basis vector). In the component corresponding to $0$ we consider the foliation into straight lines parallel to $(1,\dots,1)$. These foliations glue to a singular foliation $\mathcal{F}^{\prime}$ which has singularities at $\Sigma_{n}$.

It is easy to smooth out $\mathcal{F}^{\prime}$ (in a symmetric way with respect to the homogeneous coordinates permutations) at the open $n$-cells of $\Sigma_{n}$ (see Figure 6). However, the singularities at the smaller-dimensional cells are essential. The leaves passing through an open $(n-k)$-cell are homeomorphic to the cone over $k+2$ points.

We denote the resulting foliation with $\mathcal{F}$. The foliation $\mathcal{F}$ is a singular fibration and defines the projection $\pi_{\mathcal{F}}:\mathbb{R}^{n+1}\to\Sigma_{n}$.

The following statement is a key lemma in the proof of the main theorems of this paper.

The proof of this lemma occupies the rest of this subsection.

To figure out the fibers of $\lambda_{H}$ we need to understand the critical points of $\operatorname{Log}|_{H^{\circ}}$. Following [6] and [11] for a hypersurface $V^{\circ}\subset(\mathbb{C}^{*})^{n+1}$ we define the logarithmic Gauss map

by taking the composition of a branch of a holomorphic logarithm of each coordinate with the conventional Gauss map. This produces the following formula

where $f$ is the polynomial defining $V^{\circ}$.

Note that the Newton polyhedron of $z_{j}\frac{\partial f}{\partial z_{j}}$ coincides with the Newton polyhedron $\Delta$ of $f$. Therefore, by Kouchnirenko’s formula [9], $\deg\gamma=(n+1)!\operatorname{Vol}\Delta$. In particular, if $V^{\circ}=H^{\circ}$ then $\deg\gamma=1$.

Note that $\pi_{\mathcal{F}}$ is a stratified $[-1,1]$-fibration. Thus, the transversality of $\mathcal{D}$ and $\mathcal{F}$ implies that $\lambda_{H}$ is a stratified fibration for $\Sigma_{n}$. We need to show that the restriction of $\pi_{F}$ to open $n$-cells of $\Sigma$ is a torus fibration.

Consider a point $x=(-t,\dots,-t,0)$ for a large $t>0$. Note that $\mathcal{D}$ is almost horizontal near $x$. Thus the fiber of $\lambda_{H}$ over $x$ is diffeomorphic to the fiber $F$ of a composition of $\operatorname{Log}|_{H^{\circ}}$ and the linear projection onto the first $n$ coordinates. Note that the map $F\to T^{n}$ obtained by taking the arguments of the first $n$ coordinates is a diffeomorphism. Recall that $H^{\circ}$ is given by the equation $z_{1}+\dots+z_{n+1}+1=0$. The absolute values of the coordinates $z_{1},\dots,z_{n}$ are fixed. For any value of their argument we take $z_{n+1}=1-z_{1}-\dots-z_{n}$ to get the unique point from $F$ corresponding to this choice of the arguments. Since $|z_{1}|,\dots,|z_{n}|$ are small $z_{n+1}\neq 0$.

We verify the conclusions of Theorem 3 item-by-item. The first and the last conclusions are vacuous in this case, since $\Sigma_{n}$ (and, therefore, $\bar{\Sigma}_{n}$ as well) is contractible. The second one holds since $H^{\circ}$ is itself an open pair-of-pants.

To make the third conclusion true we have to modify $\lambda_{H}$ a little. The fiber $F$ is not Lagrangian, but it is close to a Lagrangian torus $\Lambda=\{|z_{j}|=\operatorname{const},\ j=1,\dots,n,\ z_{n+1}=-1\}$. We can deform $H^{\circ}$ a little in a neighborhood of $\operatorname{Log}^{-1}(x)$ to make it intersect the fiber of $\pi_{\mathcal{F}}\circ\operatorname{Log}$ along $\Lambda$. Therefore, $F$ is Lagrangian for a nearby symplectic structure. By Moser’s trick (see e.g. [2]) there exists a self-diffeomorphism $h$ of $H^{\circ}$ constant outside of a neighborhood of $\operatorname{Log}^{-1}(x)$ and taking one symplectic structure to another. We redefine $\lambda$ as $\lambda\circ h$. This ensures a Lagrangian fiber over one of the $\begin{pmatrix}n+2\\ 2\end{pmatrix}$ open $n$-cells of $\Sigma_{n}$. We do the same for all other $n$-cells.

The toric part $H^{\circ}\subset(\mathbb{C}^{*})^{n+1}$ of a hyperplane from 3.3 is a nice embedding of $\mathcal{P}_{n}$ to $(\mathbb{C}^{*})^{n+1}$. However for our purposes it is convenient to modify it in a neighborhood of infinity to get a different submanifold $Q^{n}$ which is better suited for gluing.

Note that the symmetric group $S_{n+2}$ acts on $(\mathbb{C}^{*})^{n+1}$ by interchanging the functions $z_{1},\dots,z_{n+1},\frac{1}{z_{1}\dots z_{n+1}}$. This action is inherited from the action of $S_{n+2}$ on ${\mathbb{C}}{\mathbb{P}}^{n+1}$ interchanging the homogeneous coordinates since $(\mathbb{C}^{*})^{n+1}\subset{\mathbb{C}}{\mathbb{P}}^{n+1}$ is invariant. The hyperplane $H^{\circ}$ is invariant with respect to this action.

Denote $\bar{Q}^{n}=Q^{n}\cap\operatorname{Log}^{-1}(\Delta(R_{n}+1))$. This is “the kernel” of $Q^{n}$ and is diffeomorphic to a closed pair-of-pants $\bar{\mathcal{P}_{n}}$ (as a manifold with boundary and corners).