Informally speaking, a proper polyhedral complex in $\mathbb{R}^{n+1}$ is a cellular space where each cell is a convex polyhedron with a rational slope and where some cells are allowed to go to infinity.

As usual, the dimension of $\Pi$ is the maximal dimension of its cells.

Let $\Pi\subset\mathbb{R}^{n+1}$ be a weighted polyhedral $n$-complex. Note that its complement $\mathbb{R}^{n+1}\smallsetminus\Pi$ consists of a finite union of connected components. Let $F\subset\Pi$ be an $n$-cell.

Recall that by Definition 1 the $n$-cell $F$ has a rational slope in $\mathbb{R}^{n+1}$. Therefore, it defines an integer covector

up to its sign. Here are the characteristic properties of $c_{F}$.

• •

  The kernel of $c_{F}$ is parallel to $F$.

• •

  $\frac{1}{w(F)}c_{F}$ is a primitive (i.e. non-divisible) integer covector $\mathbb{Z}^{n+1}\to\mathbb{Z}$.

Furthermore, even the sign of $c_{F}$ becomes well-defined once we co-orient $F\subset\mathbb{R}^{n+1}$.

Polyhedral complexes that appear in this paper have the following additional property.

The following example is a generalization of the previous one. As the following propositions show, it is the fundamental example of balanced polyhedra.

Recall that a polyhedron in $\mathbb{R}^{n+1}$ is called lattice if all its vertices belong to $\mathbb{Z}^{n+1}$. A subdivision of a polyhedron into smaller polyhedra is called lattice if all the polyhedra of the subdivision are lattice.

Note that $\Pi_{v}$ is naturally weighted. Indeed, an $n$-cell $F\subset\Pi_{v}$ comes as a corner between the graphs of two integer linear functions. The difference between these functions is an integer covector $c_{F}$. We define $w(F)\in{\mathbb{N}}$ as the maximum integer divisor of $c_{F}$.

The following proposition shows that Example 2 is fundamental.

This corollary follows from Propositions 1.4 and 1.1.

The next corollary illustrates the strength of the balancing condition that we require just for the $n$-cells. We do not use this corollary elsewhere in the paper.

Let $B$ be a vertex of $\Pi$ and let $E_{1},\dots,E_{k}$ be the edges adjacent to $B$. Let $v_{j}\in\mathbb{Z}^{n+1}$, $j=1,\dots,k$, be the primitive integer vectors in the direction of $E_{j}$. Suppose that each $E_{j}$ is adjacent to exactly $n+2$ connected components of $\mathbb{R}^{n+1}\smallsetminus\Pi$ (note that this is a general position situation).

Here $ASL_{n+1}(\mathbb{Z})$ stands for the group of affine-linear transformations of $\mathbb{R}^{n+1}$ whose rotation part belongs to $SL_{n+1}(\mathbb{Z})$.

Here is a way to canonically cut a maximal complex $\Pi\subset\mathbb{R}^{n+1}$ into standard-looking subsets $U_{j}$. We define the cutting locus $\Xi$ as the following simplicial complex that is partially dual to $\Pi$. The vertices of $\Xi$ are the baricenters of all bounded $k$-cells, $k>0$ from $\Pi$. The simplices of $\Xi$ have the baricenters of positive-dimensional cells $F_{k}\subset\Pi$ in the embedded towers $F_{1}\subset\dots\subset F_{l}$ as its vertices. Note that $\Xi\subset\Pi$ is a finite simplicial $(n-1)$-complex.

Recall that a polyhedral complex $\Pi$ is called generic at a point $x\in\Pi$ from an open $k$-cell if there exists a neighborhood isomorphic to $\mathbb{R}^{k}\times\Sigma_{n-k}$.

Thus, Proposition 1.10 implies that a maximal dual $\Delta$-complex is a generic polyhedron. In topology such polyhedra often appear as the so-called special spines of smooth manifolds. In the next section we see that $\Pi$ can be compactified so as to become a spine of the polyhedron $\Delta$ after puncturing it in the interior lattice points.

Consider the complex algebraic torus $(\mathbb{C}^{*})^{n+1}$, where $\mathbb{C}^{*}=\mathbb{C}\smallsetminus 0$. It is a commutative Lie group under multiplication. The 2-form

is an invariant symplectic form on $(\mathbb{C}^{*})^{n+1}$. There is an action of the real torus $T^{n+1}=S^{1}\times\dots\times S^{1}$ on $(\mathbb{C}^{*})^{n+1}$ by coordinatewise multiplication (we treat $S^{1}\subset\mathbb{C}^{*}$ as the unit circle). The action of $T^{n+1}$ is Hamiltonian and thus we have a well-defined moment map (we refer to [1] for the general definition or to a textbook, e.g. [2]) $\operatorname{Log}:(\mathbb{C}^{*})^{n+1}\to\mathbb{R}^{n+1}$

Let $\Delta\subset\mathbb{R}^{n+1}$ be a convex polyhedron with integer (from $\mathbb{Z}^{n+1}$) vertices. Recall (see e.g. [4]) that there is a complex toric variety $\mathbb{C}T_{\Delta}\supset(\mathbb{C}^{*})^{n+1}$. One way to construct it is to consider the Veronese embedding $(\mathbb{C}^{*})^{n+1}\to{\mathbb{C}}{\mathbb{P}}^{\#(\Delta\cap\mathbb{Z}^{n+1})-1}$ defined by the linear system of monomials associated to $\Delta\cap\mathbb{Z}^{n+1}$. Here we associate to a point $(p_{1},\dots,p_{n+1})$ a monomial $z^{p_{1}}\dots z_{n+1}^{p_{n+1}}$. We define $\mathbb{C}T_{\Delta}$ as the closure of the image of the Veronese embedding. Note that the standard, Fubini-Study, symplectic form on the ambient space ${\mathbb{C}}{\mathbb{P}}^{\#(\Delta\cap\mathbb{Z}^{n+1})-1}$ defines a symplectic form on $\mathbb{C}T_{\Delta}$ (as long as the variety $\mathbb{C}T_{\Delta}$ is non-singular). In particular, it gives a symplectic form $\omega_{\Delta}$ on $(\mathbb{C}^{*})^{n+1}$ that invariant with respect to the action of $T_{\Delta}$. This gives us a moment map with respect to $\omega_{\Delta}$

The image of this embedding is the interior $\operatorname{Int}\Delta$. The map $\mu\Delta$ can be compactified to the moment map $\bar{\mu}_{\Delta}:\mathbb{C}T_{\Delta}\to\Delta$.

The maps $\operatorname{Log}:(\mathbb{C}^{*})^{n+1}\to\mathbb{R}^{n+1}$ and $\mu_{\Delta}:(\mathbb{C}^{*})^{n+1}\to\operatorname{Int}\Delta$ both have the orbits of $T^{n+1}$ as their fibers. Thus, they define a natural reparametrization

We prove this proposition simultaneously with the following proposition describing the behavior of $\Pi$ near infinity. Recall that a supporting vector $\stackrel{{\scriptstyle\to}}{{v}}$ at a face $\Delta^{\prime}\subset\Delta$ is a vector such that $p_{\stackrel{{\scriptstyle\to}}{{v}}}|_{\Delta}$ reaches its maximum precisely over $\Delta^{\prime}$, where $p_{\stackrel{{\scriptstyle\to}}{{v}}}$ is the orthogonal projection in the direction of $\stackrel{{\scriptstyle\to}}{{v}}$.

If $\Pi$ is a maximal dual $\Delta$-complex then it is generic everywhere except at the points of its boundary $\partial\Pi$. The following proposition describes the local topology of $\bar{\Pi}$ near the boundary. It is a corollary of Proposition 1.12.

We call a point with such a neighborhood a $(k,l)$-point of $\bar{\Pi}$.

Let $V$ and $F$ be smooth manifolds, $\Delta\subset\mathbb{R}^{n+1}$ be a lattice polyhedron of full dimension and $\Pi$ be a maximal dual $\Delta$-complex.

The following proposition is a direct corollary of Definition 7.

Let $f:(\mathbb{C}^{*})^{n+1}\to\mathbb{C}$ be a Laurent polynomial

where $z\in(\mathbb{C}^{*})^{n+1}$ and $j\in\mathbb{Z}^{n+1}$ is a multi-index.

We recall that the Newton polyhedron $\Delta$ of $f$ is the convex hull in $\mathbb{R}^{n+1}$ of the set of all indices $j\in\mathbb{Z}^{n+1}$ such that $a_{j}\neq 0$. Since by assumption $f$ is a polynomial this set is finite and $\Delta$ is a bounded convex lattice polyhedron. We also call $\Delta$ the Newton polyhedron of the hypersurface ${V}^{\circ}=\{z\in(\mathbb{C}^{*})^{n+1}\ |\ f(z)=0\}$. According to [4] we call the image $\operatorname{Log}({V}^{\circ})\subset\mathbb{R}^{n+1}$ the amoeba of ${V}^{\circ}$.

For the rest of the paper we assume that $\Delta$ has a non-empty interior in $\mathbb{R}^{n+1}$. Otherwise after a suitable (multiplicative) change of coordinates the polynomial $f$ can be transformed to a polynomial in smaller number of variables.

Let $\mathbb{C}T_{\Delta}$ be the complex toric variety (see e.g. [4]) associated to $\Delta$. We define $V$ as the closure of the hypersurface ${V}^{\circ}=\{z\in(\mathbb{C}^{*})^{n+1}\ |\ f(z)=0\}$ in $\mathbb{C}T_{\Delta}$. Taking the Newton polyhedron for $\Delta$ is a canonical choice. Of course, we can take such compactification for any convex lattice $(n+1)$-polyhedron $\Delta$, even if it was not the Newton polyhedron of ${V}^{\circ}$. However the choice of the Newton polyhedron of ${V}^{\circ}$ as $\Delta$ produces the best results as the next proposition shows. Recall that in the toric construction there is a $k$-dimensional complex toric subvariety $\mathbb{C}T_{\Delta^{\prime}}$ associated to any $k$-dimensional face $\Delta^{\prime}\subset\Delta$.

The following example treats projective hypersurfaces.

Immediately we have the following proposition.

Note that $\mathcal{P}_{1}$ is diffeomorphic to the Riemann sphere punctured 3 times, while $\bar{\mathcal{P}}_{1}$ is diffeomorphic to a closed disk with 2 holes. Thus Definition 8 agrees with the classical, one-dimensional, pair-of-pants definition.

The following proposition describes a natural stratification of the boundary $\partial\bar{\mathcal{P}}$.