Research ContributionArnold Mathematical Journal

Received: 17 October 2015 / Revised: 12 November 2015 / Accepted: 23 June 2016

# Volume Polynomials and Duality Algebras of Multi-Fans

Anton Ayzenberg Faculty of Mathematics,
National Research University
Higher School of Economics, Moscow, Russia
ayzenberga@gmail.com; aizenbergaa@yandex.ru
+7-916-647-52-24
Mikiya Masuda Department of Mathematics, Osaka City University, Osaka, Japan

### Abstract

We introduce a theory of volume polynomials and corresponding duality algebras of multi-fans. Any complete simplicial multi-fan $\Delta$ determines a volume polynomial $V_{\Delta}$ whose values are the volumes of multi-polytopes based on $\Delta$. This homogeneous polynomial is further used to construct a Poincare duality algebra $\mathcal{A}^{*}(\Delta)$. We study the structure and properties of $V_{\Delta}$ and $\mathcal{A}^{*}(\Delta)$ and give applications and connections to other subjects, such as Macaulay duality, Novik–Swartz theory of face rings of simplicial manifolds, generalizations of Minkowski’s theorem on convex polytopes, cohomology of torus manifolds, computations of volumes, and linear relations on the powers of linear forms. In particular, we prove that the analogue of the $g$-theorem does not hold for multi-polytopes.

###### Keywords
Multi-fan, Multi-polytope, Volume polynomial,Poincare duality algebra, Macaulay duality, Stanley–Reisnerring, Minkowski theorem,
Minkowski relations, Torusmanifold
###### Mathematics Subject Classification
52A39, 52B11, 05E45, 52C35, 05E40, 13H10, 52B05, 52B40, 52B70, 57N65, 55N91, 28A75, 51M25, 13A02

## 1 Introduction

There is a fundamental correspondence in algebraic geometry ([Fulton1993]):

 $\displaystyle\left\{{\begin{array}[]{c}\hbox{Toric varieties}\\ \hbox{of complex dimension }n\end{array}}\right\}\leftrightsquigarrow\left\{% \hbox{Rational fans in }\mathbb{R}^{n}\right\}.$

One can read the information about toric variety from its fan. Complete toric varieties correspond to complete fans, non-singular varieties correspond to non-singular fans, and projective toric varieties correspond to normal fans of convex polytopes. Combinatorics of a fan and geometry of a toric variety are closely connected. In particular, the rays of a fan correspond to the divisors on toric variety and higher dimensional cones correspond to the intersections of divisors.

In the work [Hattori and Masuda2003] expanded this setting to topological category and generalized the above-mentioned correspondence in the following way:

 $\displaystyle\left\{{\begin{array}[]{c}\hbox{Torus manifolds}\\ \hbox{of real dimension }2n\end{array}}\right\}\rightsquigarrow\left\{\hbox{ Nonsingular multi-fans in }\mathbb{R}^{n}\right\},$ (1.1)

which will be explained in a minute.

Let $X$ be a smooth closed oriented $2n$-manifold with an effective action of an $n$-dimensional compact torus $T$ and at least one fixed point. A closed, connected, codimension two submanifold of $X$ will be called characteristic if it is a connected component of the fixed point set of a certain circle subgroup $S$ of $T$, and if it contains at least one $T$-fixed point. The manifold $X$ together with a preferred orientation of each characteristic submanifold is called a torus manifold. Characteristic submanifolds are the analogues of divisors on a toric variety.

Note, that there is no one-to-one correspondence in (1.1): there may be different (in any sense) torus manifolds producing the same multi-fan. Nevertheless, multi-fans provide a convenient tool to study such manifolds.

A multi-fan is the central object of this paper. We recall the precise definition later. Informally, a multi-fan is a collection of cones in $V\cong\mathbb{R}^{n}$ with apex at the origin, coming with multiplicities and satisfying certain restrictions. Sometimes it is convenient to assume that there is a fixed lattice $N\subset V$, and the rays of $\Delta$ are rational with respect to $N$. The cones of a multi-fan may overlap nontrivially, which makes a multi-fan more general and flexible object than an ordinary fan, and provides many nontrivial examples.

A multi-polytope is defined as follows. Let $\Delta$ be a simplicial multi-fan in $V\cong\mathbb{R}^{n}$. For each ray $l_{i}\in\Delta$, we specify an affine hyperplane $H_{i}\subset V^{*}$ orthogonal to the linear span of $l_{i}$. A tuple $P=(\Delta,H_{1},\ldots,H_{m})$ is called a simple multi-polytope based on $\Delta$. The relation of the multi-polytope to the multi-fan on which it is based, is exactly the same as the relation of a polytope to its normal fan.

For any multi-polytope $P\subset V^{*}$ there is a function $\operatorname{DH}_{P}\colon V^{*}{\setminus}\bigcup H_{i}\to\mathbb{Z}$ (the notation stands for Duistermaat–Heckman, see [Hattori and Masuda2003]). Informally, for a generic point $x\in V^{*}$ the value $\operatorname{DH}_{P}(x)$ indicates how many times the “boundary” of $P$ wraps around $x$. The precise definition is given in Sect. 3. For an ordinary simple convex polytope this function takes value 1 inside the polytope, and 0 outside.

A multi-fan $\Delta$ is called complete if it satisfies certain mild conditions (see [Hattori and Masuda2003] or Definition 2.5 below). For multi-polytopes based on complete simplicial multi-fans, the function $\operatorname{DH}_{P}$ is compactly supported. We can define the volume of a multi-polytope $P$ as an integral

 $\displaystyle\operatorname{Vol}(P):=\int_{V^{*}}\operatorname{DH}_{P}d\mu$

(the measure $\mu$ is chosen such that the volume of a fundamental domain of the dual lattice $N^{*}$ is 1).

For a given simplicial multi-fan $\Delta$ consider the space $\operatorname{Poly}(\Delta)$ of all multi-polytopes based on $\Delta$. Following [Timorin1999] we call it the space of analogous multi-polytopes. To specify an affine hyperplane orthogonal to a line $\langle l_{i}\rangle\subset V$ one needs a single real number $c_{i}$, the normalized distance from $H_{i}$ to the origin taken with sign. This number is called the support parameter. Thus the space $\operatorname{Poly}(\Delta)$ is isomorphic to $\mathbb{R}^{m}$, where $m$ is the number of rays of $\Delta$. Support parameters $(c_{1},\ldots,c_{m})$ provide the canonical coordinates on $\operatorname{Poly}(\Delta)$.

If $\Delta$ is complete, the volume gives a function on the space of analogous polytopes: $\operatorname{Poly}(\Delta)\to\mathbb{R}$, $P\mapsto\operatorname{Vol}(P)$. Similarly to the case of actual convex polytopes, studied by Pukhlikov–Khovanskii ([Khovanskii and Pukhlikov1992a]), this function is a homogeneous polynomial in the support parameters.

###### Theorem 1.1.

([Hattori and Masuda2003]) Let $\Delta$ be a complete simplicial multi-fan in $\mathbb{R}^{n}$ with $m$ rays. There exists a homogeneous polynomial $V_{\Delta}\in\mathbb{R}[c_{1},\ldots,c_{m}]$ of degree $n$ such that $V_{\Delta}(c_{1},\ldots,c_{m})=\operatorname{Vol}(P)$ for a multi-polytope $P\in\operatorname{Poly}(\Delta)$ with support parameters $(c_{1},\ldots,c_{m})$.

Following the approach introduced in [Khovanskii and Pukhlikov1992b] and developed in [Timorin1999], we proceed as follows. Consider the ring $\mathcal{D}$ of differential operators with constant coefficients, acting on $\mathbb{R}[c_{1},\ldots,c_{m}]$. We have $\mathcal{D}=\mathbb{R}[\partial_{1},\ldots,\partial_{m}]$, where $\partial_{i}=\frac{\partial}{\partial c_{i}}$. It is convenient to double the degree, so we assume that $\deg\partial_{i}=2$. Given any nonzero homogeneous polynomial $\Psi\in\mathbb{R}[c_{1},\ldots,c_{m}]$ of degree $n$, consider the subspace $\operatorname{Ann}(\Psi)\subset\mathcal{D}$, $\operatorname{Ann}(\Psi)=\{D\in\mathcal{D}\mid D\Psi=0\}$. It is easily seen, that $\operatorname{Ann}(\Psi)$ is a graded ideal, and the quotient algebra $\mathcal{D}/\operatorname{Ann}(\Psi)$ is finite-dimensional and vanishes in degrees ${>}2n$. Moreover, $\mathcal{D}/\operatorname{Ann}(\Psi)$ is a commutative Poincare duality algebra of formal dimension $2n$ (see e.g. [Timorin1999], Prop. 2.5.1).

Now consider a complete simplicial multi-fan $\Delta$ and apply this construction to the volume polynomial $V_{\Delta}$. In result we obtain a Poincare duality algebra $\mathcal{A}^{*}(\Delta):=\mathcal{D}{/}{\!}\operatorname{Ann}(V_{\Delta})$ associated with a multi-fan $\Delta$. The main goal of this work is to study the volume polynomials and investigate the structure of the corresponding algebras and to show their relation to other topics in combinatorics, convex geometry, commutative algebra, and topology.

The work has the following structure. In Sects. 2 and 3 we review the basic notions of the theory of multi-fans and in Sect. 4 we review the notion of the index map which is the key ingredient in the construction of the volume polynomial. In the work [Hattori and Masuda2003], introducing multi-fans, the existence of a lattice $N\cong\mathbb{Z}^{n}\subset V$ was assumed, so that multi-fans are non-singular (or at least rational) with respect to this lattice. In our paper we consider general multi-fans, probably non-rational. Instead of a lattice we assume that the ambient space $V$ has a fixed inner product. This allows, in particular, to define and compute volumes of multi-polytopes in $V^{*}=V$ of dimensions smaller than $n$ (dealing with lattices, only unimodular volumes make sense). The exposition of the multi-fan theory is built to comply with this continuous setting. Nevertheless, all statements in the introductory sections follow from their lattice analogues discussed in [Hattori and Masuda2003].

In Sect. 5 we prove the basic enumerative properties of the volume polynomial. While the values of $V_{\Delta}$ are the volumes of multi-polytopes, the values of its partial derivatives are the volumes of proper faces of these multi-polytopes up to certain constants. These relations will be used further in Sect. 9.

In Sect. 6 we prove a general formula (actually, a family of formulas) for the volume polynomial, and indicate a geometrical procedure which allows to find non-trivial linear identities on the powers of linear forms. For actual convex polytopes our formula coincides with the Lawrence’s formula ([Lawrence1991]), which is well known in computational geometry.

In Sect. 7 we review the general correspondence between homogeneous polynomials and Poincare duality algebras, known as the Macaulay duality. Using this correspondence we obtain an algebra $\mathcal{A}^{*}(\Delta)$ as a Poincare duality algebra corresponding to the volume polynomial $V_{\Delta}$. One way to obtain this algebra is via differential operators as discussed above. Another way involves the index map of a multi-fan.

The structure of multi-fan algebras in some particular cases is described in Sect. 8. Every (complete simplicial) multi-fan has an underlying simplicial cycle. If this cycle is a homology sphere $K$, then $\mathcal{A}^{*}(\Delta)$ is the quotient of the Stanley–Reisner algebra of $K$ by a linear system of parameters, and the dimensions of its graded components are the $h$-numbers of $K$. This is similar to ordinary fans. If the underlying simplicial cycle is a homology manifold, the algebra $\mathcal{A}^{*}(\Delta)$ is the quotient of the Stanley–Reisner algebra by the linear system of parameters and by the certain ideal introduced and studied by [Novik and Swartz2009a]; [Novik and Swartz2009b]. In this case the dimensions of the graded components of $\mathcal{A}^{*}(\Delta)$ are the $h^{\prime\prime}$-numbers of $K$. A short exposition of the Novik–Swartz theory is provided.

Section 9 aims to generalize a classical Minkowski theorem on convex polytopes to multi-polytopes. The direct Minkowski theorem has a straightforward generalization which can be used to obtain linear relations in the algebra $\mathcal{A}^{*}(\Delta)$. On the other hand, the inverse Minkowski theorem, properly formulated, is controlled by the power map $\mathcal{A}^{2}(\Delta)\to\mathcal{A}^{2n-2}(\Delta)$, $a\mapsto a^{n-1}$.

In Sect. 10 we answer the question which polynomials are volume polynomials of multi-fans, and which Poincare duality algebras are algebras of multi-fans. We prove that every Poincare duality algebra generated in degree 2 is isomorphic to $\mathcal{A}^{*}(\Delta)$ for some complete simplicial multi-fan $\Delta$. This proves that the analogue of the $g$-theorem fails for multi-fans.

The basic operations on multi-fans, such as flips and connected sums, and their effects to multi-fan algebras are described in Sect. 11. In particular, we prove that, under flips, the dimensions of graded components of $\mathcal{A}^{*}(\Delta)$ change similarly to $h$-numbers of simplicial complexes.

Finally, in Sect. 12 we discuss the relation of $\mathcal{A}^{*}(\Delta)$ to the cohomology of torus manifolds. It is known that, for complete smooth toric variety $X$, the cohomology ring $H^{*}(X;\mathbb{R})$ coincides with the algebra $\mathcal{A}^{*}(\Delta_{X})$ of the corresponding fan. Situation with general torus manifolds and their multi-fans is more complicated. Nevertheless, in a certain sense, the multi-fan algebra $\mathcal{A}^{*}(\Delta_{X})$ gives a “lower bound” for the cohomology of a torus manifold $X$.

## 2 Definitions: Multi-Fans

### 2.1 Multi-Fans as Parametrized Collections of Cones

Let us recall the definition and basic properties of multi-fans. This exposition follows the lines of [Hattori and Masuda2003].

Consider an oriented vector space $V\cong\mathbb{R}^{n}$ with a lattice $N\subset V$, $N\cong\mathbb{Z}^{n}$. A subset of the form $\kappa=\{r_{1}v_{1}+\cdots+r_{k}v_{k}\mid r_{i}\geqslant 0\}$ for given $v_{1},\ldots,v_{k}\in V$ is called a cone in $V$. Dimension of $\kappa$ is the dimension of the linear hull of $\kappa$. A cone is called strongly convex if it contains no line through the origin. In the following all cones are assumed strongly convex.

Using classical construction of supporting hyperplane one can define the faces of $\kappa$, which are also the cones of smaller dimensions. If the generating set $v_{1},\ldots,v_{k}$ may be chosen linearly independent (resp. rational, the part of basis of the lattice $N$), $\kappa$ is called simplicial (resp. rational, unimodular). Let $\operatorname{Cone}(V)$ denote the set of all cones in $V$. This set obtains a partial order: $\kappa_{1}\prec\kappa_{2}$ whenever $\kappa_{1}$ is a face of $\kappa_{2}$.

Let $\Sigma$ be a finite partially ordered set with the minimal element $*$. Suppose there is a map $C\colon\Sigma\to\operatorname{Cone}(V)$ such that

1. 1.

$C(*)=\{0\}$;

2. 2.

If $I<J$ for $I,J\in\Sigma$, then $C(I)\prec C(J)$;

3. 3.

For any $J\in\Sigma$ the map $C$ restricted on $\{I\in\Sigma\mid I\leqslant J\}$ is an isomorphism of ordered sets onto $\{\kappa\in\operatorname{Cone}(V)\mid\kappa\preceq C(J)\}$.

The image $C(\Sigma)$ is a finite set of cones in $V$. We may think of a pair $(\Sigma,C)$ as a set of cones in $V$ labeled by the ordered set $\Sigma$.

The poset $\Sigma$ obtains a rank function: $\operatorname{rk}(I):=\dim C(I)$. The set of elements in $\Sigma$ having maximal rank $n$ is denoted $\Sigma^{\langle n\rangle}$.

Consider an arbitrary function $\sigma\colon\Sigma^{\langle n\rangle}\to\{-1,+1\}$ called a sign function.

###### Definition 2.1.

(Old definition) The triple $\Delta:=(\Sigma,C,\sigma)$ is called a multi-fan in $V$. The number $n=\dim V$ is called the dimension of $\Delta$.

Multi-fan $\Delta$ is called simplicial (resp. rational, non-singular) if the values of $C$ are simplicial (resp. rational, unimodular) cones. In the following we will always assume that $\Delta$ is simplicial. Then every cone of $\Delta$ is simplicial and property (3) of the map $C$ implies that $\Sigma$ is a simplicial poset. Recall that a poset $\Sigma$ is called simplicial if any lower order ideal $\Sigma_{\leqslant J}:=\{I\in\Sigma\mid I\leqslant J\}$ is isomorphic to the poset of faces of a simplex (i.e. a boolean lattice).

### 2.2 Multi-Fans as Pairs of Weight and Characteristic Functions

Note that Definition 2.1 of a multi-fan slightly differs from the definition of multi-fan given in [Hattori and Masuda2003]. To establish the correspondence consider the following construction. Let $[m]=\{1,\ldots,m\}$ denote the set of vertices of $\Sigma$.

The signs of maximal simplices in $\Sigma$ determine two functions on ${[m]\choose n}$, the set of all $n$-subsets of $[m]$:

 $\displaystyle w^{\pm}\colon{[m]\choose n}\to\mathbb{Z}_{\geqslant 0},$

where $w^{+}(\{i_{1},\ldots,i_{n}\})$ (resp. $w^{-}(\{i_{1},\ldots,i_{n}\})$) equals the number of simplices $I\in\Sigma^{\langle n\rangle}$ on the vertices $\{i_{1},\ldots,i_{n}\}$ having sign $+1$ (resp. $-1$). Although both functions $w^{+},w^{-}$ are important by topological reasons (see [Hattori and Masuda2003]), only their difference $w:=w^{+}-w^{-}$ is relevant to our work. So far $w$ is a function which assigns an integral number to each $n$-subset of $[m]$. Let us consider a pure simplicial complex $K$ on the set $[m]$ whose maximal simplices $K^{\langle n\rangle}$ are the subsets $I\subset[m]$ satisfying $w(I)\neq 0$. To reach greater generality we allow $w$ to take real values, thus

 $\displaystyle w\colon{[m]\choose n}\to\mathbb{R}.$

Each vertex $i\in[m]$ corresponds to a ray (i.e. 1-dimensional cone) of $\Delta$. We choose a generator in each ray. This gives a so called characteristic function $\lambda\colon[m]\to V$, such that the ray $C(i)$ is generated by $\lambda(i)$ for every $i\in[m]$. It satisfies the following property:

 $\displaystyle\hbox{if }\{i_{1},\ldots,i_{k}\}\in K,\hbox{ then }\lambda(i_{1})% ,\ldots,\lambda(i_{k})\in V\hbox{ are linearly independent.}$

This condition is called $*$-condition.

Note that in [Hattori and Masuda2003] all multi-fans were assumed rational. In this case the generator $\lambda(i)$ can be chosen canonically as a unique primitive integral vector contained in $C(i)$. Since we want to include non-rational simplicial multi-fans in our consideration, we should specify the generators somehow in order for the subsequent calculations to make sense.

Finally we get to the following definition

###### Definition 2.2.

(New definition) A triple $(K,w,\lambda)$ is called a simplicial multi-fan in $V$. Here $w\colon{[m]\choose n}\to\mathbb{R}$ is a weight function, $K$ is a simplicial complex which is the support of $w$, and $\lambda\colon[m]\to V$ is a characteristic function. Characteristic function satisfies $*$-condition with respect to $K$: if $I=\{i_{1},\ldots,i_{k}\}\in K$, then the vectors $\lambda(i_{1}),\ldots,\lambda(i_{k})$ are linearly independent in $V$.

Here $K$ may have ghost vertices, i.e. $i\in[m]$ such that $\{i\}\notin K$. The value of characteristic function in such vertices may be arbitrary (even zero). In the following we will not pay too much attention to ghost vertices since their presence does not affect the calculations.

Strictly speaking, the new definition is not equivalent to the old one, since we cannot restore the poset $\Sigma$ and the sign function $\sigma\colon\Sigma^{\langle n\rangle}\to\{\pm 1\}$ when $w$ takes non-integral values. Even in the integral case we cannot restore $\Sigma$ uniquely. On the other hand, as was shown above, every multi-fan in the sense of old definition determines a multi-fan in the sense of new definition. We will work with the new definition most of the time.

###### Remark 2.3.

When passing from the old definition to the new one, we may lose an important information. For example consider the multi-fan in $\mathbb{R}^{2}=\langle e_{1},e_{2}\rangle$ whose maximal cones are two copies of the non-negative cone (i.e. the cone generated by basis vectors $e_{1},e_{2}$), and two rays are generated by $e_{1}$ and $e_{2}$. One of the maximal cones is taken with the sign $+1$ and the other with the sign $-1$. We remark that such multi-fan corresponds to the torus manifold $S^{4}$ ([Hattori and Masuda2003]). We have $w^{+}(\{1,2\})=w^{-}(\{1,2\})=1$, therefore $w(\{1,2\})=0$. Thus $K$ is empty (equivalently, $w\colon{[m]\choose n}\to\mathbb{R}$ vanishes).

One way to avoid such situations is to assume in the beginning that $\Sigma$ itself is a simplicial complex rather than a general simplicial poset. In this case $K$ coincides with $\Sigma$ and the weight function $w$ on $K$ coincides with the sign function $\sigma$. In particular, $w$ takes the value $\pm 1$ on each maximal simplex of $K$ (see Example 2.9 below).

### 2.3 Underlying Simplicial Chain

Let $\triangle_{[m]}$ denote an abstract simplex on the vertex set $[m]$, and let $\triangle_{[m]}^{(n-1)}$ be its $(n-1)$-skeleton. Every subset $I\subset[m]$, $|I|=n$ may be considered as a maximal simplex of $\triangle_{[m]}^{(n-1)}$. If $I\in K^{\langle n\rangle}$, then we can orient $I$ as follows: we say that the order of vertices $(i_{1},\ldots,i_{n})$ of $I$ is positive if and only if the basis $(\lambda(i_{1}),\ldots,\lambda(i_{n}))$ determines the positive orientation of $V$.

###### Definition 2.4.

The element

 $\displaystyle w_{ch}=\sum_{I\in K^{\langle n\rangle}}w(I)I\in C_{n-1}(K;% \mathbb{R})\subseteq C_{n-1}\left(\triangle_{[m]}^{(n-1)};\mathbb{R}\right)$

is called the underlying chain of a multi-fan $\Delta$. Here $C_{n-1}(K;\mathbb{R})$ denotes the group of simplicial chains of $K$.

### 2.4 Complete Multi-Fans

Let us briefly recall the notion of projected multi-fan. We give the construction in terms of new definition of multi-fan although the similar construction may be given in terms of simplicial posets and sign functions.

Let $\Delta=(K,w,\lambda)$ be a simplicial multi-fan in the space $V$, and let $I=\{i_{1},\ldots,i_{k}\}\in K$ be a simplex. Let $V_{I}$ denote the quotient vector space $V/\langle\lambda(i_{1}),\ldots,{}\lambda(i_{k})\rangle$. Consider the multi-fan $\Delta_{I}=(\operatorname{lk}_{K}I,w_{I},\lambda_{I})$ in $V_{I}$ defined as follows:

• -

$\operatorname{lk}_{K}I:=\{J\subset[m]{\setminus}I\mid I\cup J\in K\}$ is the link of the simplex $I$ in $K$.

• -

$w_{I}(J):=w(I\cup J)$ for every $J\in\operatorname{lk}_{K}I$, $|J|=n-|I|$.

• -

$\lambda_{I}(j)$ is the image of $\lambda(j)\in V$ under the natural projection $V\to V_{I}=V/\langle\lambda(i_{1}),\ldots,\lambda(i_{k})\rangle$. It is easily seen that $\lambda_{I}$ satisfies $*$-condition.

If we choose some orientation of a simplex $I\in K$, the space $V_{I}$ obtains an orientation induced from $V$. To be precise, let us say that the basis $([v_{1}],\ldots,[v_{n-k}])$ determines a positive orientation of $V_{I}$ if the basis $(v_{1},\ldots,v_{n-k},\lambda(i_{1}),\ldots,\lambda(i_{k}))$ is a positive basis of $V$ for a chosen positive order $(i_{1},\ldots,i_{k})$ of vertices of $I$.

We call $\Delta_{I}$ the projected multi-fan of $\Delta$. The construction satisfies the hereditary relation $(\Delta_{I_{1}})_{I_{2}}=\Delta_{I_{1}\sqcup I_{2}}$ whenever it makes sense, and there holds $\Delta_{\varnothing}=\Delta$.

Let us call a vector $v\in V$ generic with respect to $\Delta$ if it is not contained in the vector subspaces spanned by the cones of $\Delta$ of dimensions $<n$. For any such $v$ define the number $d_{v}=\sum w(I)\in\mathbb{R}$, where the sum is taken over all subsets $I=\{i_{1},\ldots,i_{n}\}\subset[m]$ such that the cone generated by $\lambda(i_{1}),\ldots,\lambda(i_{n})$ contains $v$.

###### Definition 2.5.

The multi-fan $\Delta$ is called pre-complete if $d_{v}$ does not depend on a generic vector $v\in V$. In this case $d_{v}$ is called the degree of $\Delta$. The multi-fan $\Delta$ is called complete if the projected multi-fan $\Delta_{I}$ is pre-complete for any simplex $I\in K$.

###### Remark 2.6.

Note that this definition allows $w$ to be constantly zero. We call a multi-fan zero if its weight function is constantly zero. A zero multi-fan is pre-complete and therefore complete.

###### Proposition 2.7.

A multi-fan $\Delta$ is complete if and only if its underlying simplicial chain $w_{ch}\in C_{n-1}(\triangle_{[m]}^{(n-1)};\mathbb{R})$ is a cycle$,$ that is $dw_{ch}=0$ for the standard simplicial differential $d\colon C_{n-1}(\triangle_{[m]}^{(n-1)};\mathbb{R})\to C_{n-2}(\triangle_{[m]}% ^{(n-1)};\mathbb{R})$ $($if $n=1,$ we assume that $d\colon C_{0}(\triangle_{[m]}^{(n-1)};\mathbb{R})\to\mathbb{R}$ is the augmentation map$).$

###### Proof.

In the case when $w$ takes only integral values, the statement is proved in [Hattori and Masuda2003], Sec. 6. If $w$ takes only rational values, scaling the values of $w$ by a common denominator reduces the task to the integral case. It remains to prove the statement for real-valued $w$. Both conditions “$\Delta$ is complete” and “$dw_{ch}=0$” determine rational vector subspaces in the space of all possible weight functions (it is not difficult to define the pre-completeness condition in terms of the “wall-crossing relations”, which are linear relations on $w(I)$ with integral coefficients). Thus the rational case implies the real case. $\square$

For convenience we summarize the discussion by the following definition.

###### Definition 2.8.

(Complete simplicial multi-fan) A complete simplicial multi-fan is a pair $(w_{ch},\lambda)$, where $w_{ch}=\sum_{I\subset[m],|I|=n}w(I)I\in Z_{n-1}(\triangle_{[m]}^{(n-1)})$ is a simplicial cycle on $m$ vertices, and $\lambda\colon[m]\to V$ is any function satisfying the condition: $\{\lambda(i)\}_{i\in I}$ is a basis of $V$ if $|I|=n$ and $w(I)\neq 0$.

For a complete multi-fan $\Delta$ the corresponding homology class $[w_{ch}]\in\widetilde{H}_{n-1}(K;\mathbb{R}){}\subset\widetilde{H}_{n-1}(% \triangle_{[m]}^{(n-1)};\mathbb{R})$ will be denoted $[\Delta]$ and called the underlying homology class of $\Delta$. Since $C_{n}(\triangle_{[m]}^{(n-1)};\mathbb{R})=0$, the groups $Z_{n-1}(\triangle_{[m]}^{(n-1)})$ and $\widetilde{H}_{n-1}(\triangle_{[m]}^{(n-1)})$ may be identified. Thus $w_{ch}$ and $[\Delta]$ are just two different notations for the same object.

###### Example 2.9.

A pseudomanifold is a pure simplicial complex such that each simplex of codimension one is contained in exactly two maximal simplices. It is called orientable if all its maximal simplices can be simultaneously oriented so that orientations of neighboring simplices are compatible. One obvious way to obtain a complete multi-fan is to start with any oriented pseudomanifold $K$ of dimension $n-1$ on the set of vertices $[m]$, and take any characteristic function $\lambda\colon[m]\to V$. Since $K$ is oriented, every maximal simplex $I$ of $K$ becomes oriented, but this orientation may be different from the one determined by characteristic function (see Sect. 2.3). Let $w(I)$ be $+1$ or $-1$ depending on whether these two orientations agree or not. Let us extend the weight function by zeroes to non-simplices of $K$. The corresponding simplicial chain $w_{ch}=\sum_{I}w(I)I\in C_{n-1}(\triangle_{[m]}^{(n-1)};\mathbb{R})$ is closed, since it is exactly the fundamental chain of $K$ in $\triangle_{[m]}^{(n-1)}$. Therefore, $(w_{ch},\lambda)$ is a complete simplicial fan.

###### Example 2.10.

The previous example may be restricted to the case when $K$ is a homology sphere or homology manifold. We will study these two cases in more detail in Sect. 8.

We say that $\Delta$ is based on an orientable simplicial pseudomanifold $K$ if the corresponding simplicial cycle is given by $K$.

There is one interesting feature of (complete) multi-fans revealed by Definitions 2.2 and 2.8. The multi-fans with the given set of vertices $[m]$ and the given characteristic function $\lambda$ form a vector space: we may add them by adding their weights and multiply by real numbers by scaling their weights. Let $\operatorname{MultiFans}_{\lambda}$ denote the vector space of complete multi-fans with the given characteristic function $\lambda$. This space may be identified with certain vector subspace of $Z_{n-1}(\triangle_{[m]}^{(n-1)};\mathbb{R})$. We will discuss this subspace in detail in Sect. 10.3. The set of multi-fans with integral weights forms a lattice inside $\operatorname{MultiFans}_{\lambda}$ which is a certain sublattice of $Z_{n-1}(\triangle_{[m]}^{(n-1)};\mathbb{Z})$.

## 3 Definitions: Multi-Polytopes

### 3.1 Multi-Polytopes

Let $\Delta$ be a simplicial multi-fan with characteristic function $\lambda\colon[m]\to V$. Let $HP(V^{*})$ denote the set of all affine hyperplanes in the dual vector space $V^{*}$.

For each $i\in[m]$ choose an affine hyperplane $\mathcal{H}(i)\subset V^{*}$ in the dual space which is orthogonal to the linear hull of the $i$-th cone. In other words, $\mathcal{H}(i)$ is defined by equation $\mathcal{H}(i)=\{u\in V^{*}\mid\langle u,\lambda(i)\rangle=c_{i}\}$ for some constant $c_{i}\in\mathbb{R}$ called the support parameter of $\mathcal{H}(i)$.

###### Definition 3.1.

A multi-polytope $P$ is a pair $(\Delta,\mathcal{H})$, where $\Delta$ is a multi-fan, and $\mathcal{H}\colon[m]\to HP(V^{*})$ is a function such that $\mathcal{H}(i)$ is orthogonal to $\lambda(i)$ for any $i\in[m]$. We say that $P$ is based on the multi-fan $\Delta$.

Although the definition may be stated in general, we restrict to simplicial multi-fans $\Delta$, in which case $P$ is called a simple multi-polytope.

Let us denote the set of all multi-polytopes based on $\Delta$ by $\operatorname{Poly}(\Delta)$. Every such multi-polytope is completely determined by its support parameters $c_{1},\ldots,c_{m}$. Thus $\operatorname{Poly}(\Delta)$ has natural coordinates $(c_{1},\ldots,c_{m})$ and may be identified with $\mathbb{R}^{m}$. This space is called the space of analogous multi-polytopes based on $\Delta$.

To simplify notation, we denote $\mathcal{H}(i)$ by $H_{i}$ and set

 $\displaystyle H_{I}:=\bigcap\nolimits_{i\in I}H_{i}\quad\hbox{for }I\in K.$

$H_{I}$ is a codimension $|I|$ affine subspace in $V^{*}$, since the normals of the hyperplanes $H_{i}$, $i\in I$ are linearly independent by $*$-condition. In particular, when $I$ is a maximal simplex, $I\in K^{\langle n\rangle}$, $H_{I}$ is a point in $V^{*}$ which is called the vertex of $P$.

###### Definition 3.2.

Let $\Delta$ be a simplicial multi-fan in $V$ with the underlying simplicial complex $K$ and let $P$ be a simple multi-polytope based on $\Delta$. Let $I\in K$. Consider a simple multi-polytope $F_{I}=(\Delta_{I},\mathcal{H}_{I})$ in the space $H_{I}\subset V^{*}$. Note that the projected multi-fan $\Delta_{I}$ is defined in the space $V_{I}$ (see Sect. 2.4), so the multi-polytope based on $\Delta_{I}$ should formally lie in $V_{I}^{*}$. Nevertheless, we may identify $H_{I}$ with $V_{I}^{*}$. The supporting hyperplanes of $F_{I}$ are defined as follows: $\mathcal{H}_{I}(j)=H_{I}\cap H_{j}$ for any vertex $j$ of $\operatorname{lk}_{K}I$. The multi-polytope $F_{I}$ is called the face of $P$ dual to $I$.

### 3.2 Duistermaat–Heckman Function of a Multi-Polytope

Suppose $I\in K^{\langle n\rangle}$. Then the set $\{\lambda(i)\mid i\in I\}$ is a basis of $V$. Denote its dual basis of $V^{*}$ by $\{u_{i}^{I}\mid i\in I\}$, i.e. $\langle u_{i}^{I},\lambda(j)\rangle=\delta_{ij}$ where $\delta_{ij}$ denotes the Kronecker delta. Take a generic vector $v\in V$. Then $\langle u_{i}^{I},v\rangle\neq 0$ for all $I\in K^{\langle n\rangle}$ and $i\in I$. Set

 $\displaystyle(-1)^{I}:=(-1)^{\sharp\{i\in I\mid\langle u_{i}^{I},v\rangle>0\}}% \quad\hbox{and}\quad(u_{i}^{I})^{+}:=\begin{cases}u_{i}^{I}&\hbox{if }\langle u% _{i}^{I},v\rangle>0,\\ -u_{i}^{I}&\hbox{if }\langle u_{i}^{I},v\rangle<0.\end{cases}$

We denote by $C^{*}(I)^{+}$ the cone in $V^{*}$ spanned by $(u_{i}^{I})^{+}$’s ($i\in I$) with apex at a vertex $H_{I}$ of a multi-polytope $P$, and by $\phi_{I}$ the function on $V^{*}$ which takes value 1 inside $C^{*}(I)^{+}$ and 0 outside (the indicator function of the cone $C^{*}(I)^{+}$).

###### Definition 3.3.

A function $\operatorname{DH}_{P}$ on $V^{*}{\setminus}\bigcup_{i=1}^{m}H_{i}$ defined by

 $\displaystyle\sum_{I\in K^{\langle n\rangle}}(-1)^{I}w(I)\phi_{I}$

is called a Duistermaat–Heckman function associated with $P$.

The summands in the definition depend on the choice of a generic vector $v\in V$. Nevertheless, the function itself is independent of $v$ when $\Delta$ is complete (we refer to [Hattori and Masuda2003] when $w$ is integral-valued and note that the same argument works for real weights).

The function $\operatorname{DH}_{P}$ for a simple multi-polytope $P$ based on a complete multi-fan has the following geometrical interpretation. Let $S$ be the realization of first barycentric subdivision of $K$ and let $G_{I}\subset S$ be the dual face of $I\in K$, $I\neq\varnothing$, i.e. a realization of the set $\{\{I<I_{1}<\cdots<I_{k}\}\in K^{\prime}\}$. If $I\in K^{\langle n\rangle}$, then $G_{I}$ is a point. For a given multi-polytope $P$ based on $\Delta$ there exists a continuous map $\psi\colon S\to V^{*}$ such that $\psi(G_{I})\subset H_{I}$ for any $I\in K$, $I\neq\varnothing$ (in particular, when $I\in K^{\langle n\rangle}$, this map sends the point $G_{I}\in S$ to the vertex $H_{I}$ of a multi-polytope $P$). This map is unique up to homotopy preserving the stratifications.

Let us take any point $u\in V^{*}{\setminus}\bigcup_{i=1}^{m}H_{i}$. Then $u$ is not contained in the image of $\psi$ by the construction of $\psi$. Thus we may consider the induced map in homology:

 $\displaystyle\psi_{*}\colon\widetilde{H}_{n-1}(S;\mathbb{R})\to\widetilde{H}_{% n-1}(V^{*}{\setminus}\{u\};\mathbb{R}).$

The underlying simplicial cycle $[\Delta]$ may be considered as an element of the group $\widetilde{H}_{n-1}(S;\mathbb{R})$. Since $V^{*}$ is oriented, we have the fundamental class $[V^{*}{\setminus}\{u\}]\in\widetilde{H}_{n-1}(V^{*}{\setminus}\{u\};\mathbb{R})$. Thus

 $\displaystyle\psi_{*}([\Delta])=\operatorname{WN}_{P}(u)\cdot[V^{*}{\setminus}% \{u\}],$

for some number $\operatorname{WN}_{P}(u)\in\mathbb{R}$. This number has a natural meaning of winding number of cycle $[\Delta]$ around $u$. It happens that this number is exactly the value of $\operatorname{DH}_{P}$ at the point $u\in V^{*}$ (see details in [Hattori and Masuda2003], Sec. 6).

It is easily seen from the above consideration that $\operatorname{DH}_{P}$ has a compact support when $\Delta$ is complete. Thus in the case of complete multi-fan we may define the volume of a multi-polytope $P$ as

 $\displaystyle\operatorname{Vol}(P)=\int_{V^{*}}\operatorname{DH}_{P}(u)d\mu$ (3.1)

with respect to some Euclidean measure on $V^{*}$ (in a presence of a lattice $N\subset V$ the measure is normalized so that the fundamental domain of $N^{*}\subset V^{*}$ has volume 1).

Finally, we may consider the volume as a function on the space $\operatorname{Poly}(\Delta)\cong\mathbb{R}^{m}$ of analogous multi-polytopes. We have a function $V_{\Delta}\colon\mathbb{R}^{m}\to\mathbb{R}$ whose value at $(c_{1},\ldots,c_{m})$ equals $\operatorname{Vol}(P)$ for the multi-polytope $P$ with the support parameters $c_{1},\ldots,c_{m}$. The goal of the next section is to study this function using equivariant localization ideas and prove Theorem 1.1.

###### Remark 3.4.

Needless to say that in case of actual simple convex polytopes the notions introduced above coincide with the classical ones. If $P$ is a simple convex polytope and $\Delta$ is its normal fan, then $\operatorname{DH}_{P}$ takes the value 1 inside $P$ and 0 outside. The volume of $P$ is just the usual volume. Note that even if $\Delta$ is an actual fan, not all multi-polytopes based on $\Delta$ are actual convex polytopes. Nevertheless, the notion of volume and Duistermaat–Heckman function have transparent geometrical meanings for all of them.

###### Example 3.5.

Consider the two-dimensional multi-fan $\Delta$ with $m=5$ and $V=\mathbb{R}^{2}$ depicted on Fig. 1, left. Its characteristic function is the following: $\lambda(1)=(1,0)$, $\lambda(2)=(-2,1)$, $\lambda(3)=(1,-2)$, $\lambda(4)=(0,1)$, $\lambda(5)=(-1,-1)$. The weight function takes the value 1 on the subsets $\{1,2\}$, $\{2,3\}$, $\{3,4\}$, $\{4,5\}$, $\{1,5\}$ and the value 0 on all other subsets. Geometrically this indicates the fact that in the multi-fan we have the cones generated by $\{\lambda(1),\lambda(2)\}$, $\{\lambda(2),\lambda(3)\}$, etc. with multiplicity one, and do not have the cones generated by $\{\lambda(1),\lambda(3)\}$, $\{\lambda(1),\lambda(4)\}$, and so on. It can be seen that every generic point of $V=\mathbb{R}^{2}$ is covered by exactly two cones, therefore $\Delta$ is pre-complete of degree 2. Moreover, a simple check shows that all its projected multi-fans are complete. Hence $\Delta$ is complete. The underlying chain of $\Delta$ has the form $(1,2)+(2,3)+(3,4)+(4,5)+(5,1)\in C_{1}(\triangle_{[5]}^{(1)};\mathbb{R})$ which is obviously a simplicial cycle. The underlying complex $K$ of $\Delta$ is a circle made of 5 segments, and $[\Delta]$ is its fundamental class.

An example of a multi-polytope $P$ based on $\Delta$ is shown at Fig. 1, right. Each hyperplane $H_{i}$ is orthogonal to the linear span of the corresponding ray $\lambda(i)$ of $\Delta$, $i\in[5]$. The Duistermaat–Heckman function of $P$ is shown on Fig. 2. The function is constant on the chambers: it takes value 2 in the middle pentagon since the multi-polytope “winds” around the points of this region twice, and takes value 1 on triangles adjacent to the central pentagon. The value of $\operatorname{DH}_{P}$ in all other chambers is 0. The volume of a multi-polytope is therefore not just the volume of the five-point star: the points in the central region contribute to the volume twice.

## 4 Volume Polynomial from the Index Map

### 4.1 Index Map

Let $\Delta=(w_{ch},\lambda)$ be a simplicial multi-fan in $V\cong\mathbb{R}^{n}$ with $m$ rays. The characteristic function $\lambda\colon[m]\to V$ may be considered as a linear map $\lambda\colon\mathbb{R}^{m}\to V$ which sends the basis vector $e_{i}\in\mathbb{R}^{m}$, $i\in[m]$ to $\lambda(i)$. Let $\{x_{i}\}_{i\in[m]}$ be the basis of $(\mathbb{R}^{m})^{*}$ dual to $\{e_{i}\}_{i\in[m]}$, so that $(\mathbb{R}^{m})^{*}=\langle x_{1},\ldots,x_{m}\rangle$. Let us also consider the adjoint map $\lambda^{\top}\colon V^{*}\to(\mathbb{R}^{m})^{*}$. By definition it sends the vector $u\in V^{*}$ to

 $\displaystyle\sum_{i=1}^{m}\langle u,\lambda(i)\rangle x_{i}.$

For any maximal simplex $I=\{i_{1},\ldots,i_{n}\}\in K^{\langle n\rangle}$ the vectors $\{\lambda(i)\}_{i\in I}$ form a basis of $V$ according to $*$-condition, defined in Sect. 2.2. Let $\{u_{i}^{I}\}_{i\in I}$ be the dual basis of $V^{*}$. Let $\iota_{I}\colon(\mathbb{R}^{m})^{*}\to V^{*}$ be the linear map defined by

 $\displaystyle\iota_{I}(x_{i})=\begin{cases}u^{I}_{i}&\quad\hbox{if}\quad i\in I% ,\\ 0,&\quad\hbox{if}\quad i\notin K.\end{cases}$ (4.1)

Consider $\mathbb{R}[x_{1},\ldots,x_{m}]$, the algebra of polynomials on $(\mathbb{R}^{m})^{*}$. Also let $\mathbb{R}[V^{*}]$ denote the algebra of polynomials on $V^{*}$. Both polynomial algebras are graded, where we set the degrees of the generating spaces $(\mathbb{R}^{m})^{*}$ and $V^{*}$ to 2. The linear map $\iota_{I}$ induces the graded algebra homomorphism

 $\displaystyle\iota_{I}\colon\mathbb{R}[x_{1},\ldots,x_{m}]\to\mathbb{R}[V^{*}],$

denoted by the same letter. In the following, if $A$ is a graded algebra, we denote by $A_{j}$ its homogeneous part of degree $j$.

Let $S^{-1}\mathbb{R}[V^{*}]$ denote the ring of rational functions over $\mathbb{R}[V^{*}]$ graded in a natural way. Given a weight function $w\colon K^{\langle n\rangle}\to\mathbb{R}$ we can define the linear map $\pi_{!}^{\Delta}\colon\mathbb{R}[x_{1},\ldots,x_{m}]\to S^{-1}\mathbb{R}[V^{*}]$ as the following weighted sum:

 $\displaystyle\pi_{!}^{\Delta}(x)=\sum_{I\in K^{\langle n\rangle}}\dfrac{w(I)% \iota_{I}(x)}{|\det\lambda_{I}|\prod_{i\in I}\iota_{I}(x_{i})}$ (4.2)

for $x\in\mathbb{R}[x_{1},\ldots,x_{m}]$. We assume that an inner product is fixed on $V$, so that $|\det\lambda_{I}|=|\det(\lambda(i)_{i\in I})|$ is well-defined even if there is no lattice in $V$. The inner product on $V$ induces a Euclidean measure on $V^{*}$ and $|\det\lambda_{I}|$ is the volume of the parallelepiped spanned by $\{\lambda(i)\}_{i\in I}$. The translation invariant measure on $V^{*}$ is assumed the same as in (3.1). The map $\pi_{!}^{\Delta}$ is well-defined since $\lambda_{I}$ are isomorphisms. It can be seen that $\pi_{!}^{\Delta}$ is homogeneous of degree $-2n$. It is called the index map of a multi-fan $\Delta=(K,w,\lambda)$.

###### Theorem 4.1.

The following properties of $\Delta$ are equivalent$:$

1. 1.

The image of $\pi_{!}^{\Delta}$ lies in $\mathbb{R}[V^{*}]\subset S^{-1}\mathbb{R}[V^{*}];$

2. 2.

The underlying chain $w_{ch}=\sum_{I\in K^{\langle n\rangle}}w(I)I$ is closed $;$

3. 3.

The multi-fan $\Delta=(w_{ch},\lambda)$ is complete.

###### Proof.

Equivalence of (2) and (3) was already shown in Proposition 2.7. The implication (2) $\Rightarrow$ (1), in case when $\lambda$ takes values in the lattice and $w$ is integer-valued, is proved in [Hattori and Masuda2003], Lm. 8.4. It should be noted that in this case $|\det\lambda_{I}|$ appearing in the denominator is nothing but the order of the finite group $G_{I}=N/N_{I}$, where $N\subset V$ is the lattice and $N_{I}$ is a sublattice generated by $\{\lambda(i)\}_{i\in I}$. The situation when $\lambda$ and $w$ are rational is reduced to the integral case by multiplying all values of $\lambda$ and $w$ by a common denominator (both conditions (1) and (2) are invariant under rescaling). The real case follows by continuity. Indeed, the subset of simplicial cycles with rational coefficients, $Z_{n-1}(K;\mathbb{Q})$, is dense in $Z_{n-1}(K;\mathbb{R})$; the right hand side of (4.2) is continuous with respect to $\lambda$ and $w$; and the subset $\mathbb{R}[V^{*}]$ is closed in $S^{-1}\mathbb{R}[V^{*}]$. Therefore, arbitrary complete multi-fan $(w,\lambda)$ can be approximated by a sequence of rational complete multi-fans $\Delta_{\alpha}=(w_{\alpha},\lambda_{\alpha})$ which implies that the values of $\pi_{!}^{\Delta}$ are approximated by the values of $\pi_{!}^{\Delta_{\alpha}}$. Since the values of $\pi_{!}^{\Delta_{\alpha}}$ are polynomials, so are the values of $\pi_{!}^{\Delta}$.

Let us prove that (1) implies (2). Take any simplex $J\in K$ such that $|J|=n-1$ and consider the monomial $x_{J}=\prod_{i\in J}x_{j}$ of degree $2(n-1)$ lying in $\mathbb{R}[x_{1},\ldots,x_{m}]$. The map $\pi_{!}^{\Delta}$ lowers the degree by $2n$ thus we have $\deg\pi_{!}^{\Delta}(x_{J})=-2$. Condition (1) implies that $\pi_{!}^{\Delta}(x_{J})$ is a polynomial, thus $\pi_{!}^{\Delta}(x_{J})=0$. By definition, we have

 $\displaystyle\pi_{!}^{\Delta}(x_{J})=\sum_{I\in K^{\langle n\rangle}}\dfrac{w(% I)\iota_{I}(x_{J})}{|\det\lambda_{I}|\prod_{i\in I}\iota_{I}(x_{i})}$

Note that $\iota_{I}$ is a ring homomorphism and $\iota_{I}(x_{i})=0$ if $i\notin I$ by (4.1). Therefore,

 $\displaystyle\pi_{!}^{\Delta}(x_{J})=\sum_{I\in K^{\langle n\rangle},J\subset I% }\dfrac{w(I)\prod_{i\in J}\iota_{I}(x_{i})}{|\det\lambda_{I}|\prod_{i\in I}% \iota_{I}(x_{i})}=\sum_{j\in[m]{\setminus}J,I:=J\cup\{j\}\in K^{\langle n% \rangle}}\dfrac{w(I)}{|\det\lambda_{I}|\iota_{I}(x_{j})}$

Recall that $\iota_{I}(x_{i})=u_{i}^{I}$, where $\{u_{i}^{I}\}_{i\in I}$ is the basis of $V^{*}$ dual to the basis $\{\lambda(i)\}_{i\in I}$ of $V$. Consider the linear functional $\varrho\in V^{*}$ taking the value $\varrho(v)=\det((\lambda(i))_{i\in J},v)$ for any $v\in V$. It can be seen that $|\det\lambda_{I}|\iota_{I}(x_{j})\in V^{*}$, where $I=J\cup\{j\}$, coincides with $\varrho$ up to sign. More precisely $|\det\lambda_{I}|\iota_{I}(x_{j})={[I\!:\!J]}\varrho$, where ${[I\!:\!J]}$ is the incidence sign of two simplices of $K$ (it appears because we need to permute the vectors $((\lambda(i))_{i\in J},\lambda(j))$ in order to get the positive determinant). Therefore,

 $\displaystyle 0=\pi_{!}^{\Delta}(x_{J})=\dfrac{1}{\varrho}\sum_{I\in K^{% \langle n\rangle},J\subset I}{[I\!:\!J]}w(I)$

It remains to notice that the sum in this expression is exactly the coefficient of $J$ in the simplicial chain $dw_{ch}\in C_{n-2}(K;\mathbb{R})$. This calculation applies to any $J\in K$, $|J|=n-2$, therefore $dw_{ch}=0$. $\square$

The map $\lambda^{\top}\colon V^{*}\to(\mathbb{R}^{m})^{*}$, the adjoint of $\lambda$, induces the ring homomorphism $\mathbb{R}[V^{*}]\to\mathbb{R}[x_{1},\ldots,x_{m}]$. Hence $\mathbb{R}[x_{1},\ldots,x_{m}]$ obtains the structure of $\mathbb{R}[V^{*}]$-module. It can be checked that $\lambda^{\top}$ is the right inverse of each $\iota_{I}\colon(\mathbb{R}^{m})^{*}\to V^{*}$, therefore all ring homomorphisms $\iota_{I}\colon\mathbb{R}[x_{1},\ldots,x_{m}]\to\mathbb{R}[V^{*}]$ are the $\mathbb{R}[V^{*}]$-module homomorphisms. Thus $\pi_{!}^{\Delta}$ is also a homomorphism of $\mathbb{R}[V^{*}]$-modules (even in the case $w_{ch}$ is not closed).

###### Remark 4.2.

Note that conditions (1) and (2) in Theorem 4.1 make sense over an arbitrary field $\Bbbk$. We may start with a $\Bbbk$-valued chain $w_{ch}\in C_{n-1}(K;\Bbbk)$ and a characteristic function valued in $\Bbbk^{n}$. These data allow to define the maps $\iota_{I}$ and $\pi_{!}^{\Delta}$ absolutely similar to the real case.

###### Problem 4.3.

Does equivalence of (1) and (2) in Theorem 4.1 hold for arbitrary fields?

For general fields we cannot reduce the task to the integral case but it is likely that there exists a straightforward algebraical proof.

### 4.2 Stanley–Reisner Rings

Let us recall the definition of the Stanley–Reisner ring.

###### Definition 4.4.

Let $K$ be a simplicial complex on the vertex set $[m]$ and $\Bbbk$ be a ground ring (either $\mathbb{Z}$ or a field). The Stanley–Reisner ring is the quotient of a polynomial ring by the Stanley–Reisner ideal:

 $\displaystyle\Bbbk[K]:=\Bbbk[x_{1},\ldots,x_{m}]/I_{SR},\quad\hbox{where }I_{% SR}=(x_{i_{1}}\cdot\ldots\cdot x_{i_{k}}\mid\{i_{1},\ldots,i_{k}\}\notin K),$

endowed with the grading $\deg x_{i}=2$ and the natural structure of graded $\Bbbk[x_{1},\ldots,x_{m}]$-module.

For now let us concentrate on the case $\Bbbk=\mathbb{R}$. Given a characteristic function $\lambda$ on $K$ we may define a certain ideal in $\mathbb{R}[K]$ generated by linear forms. As before, let $\lambda^{\top}\colon V^{*}\to(\mathbb{R}^{m})^{*}=\langle x_{1},\ldots,x_{m}\rangle$ denote the adjoint map of $\lambda\colon\mathbb{R}^{m}\to V$. Let $\Theta$ denote the ideal of $\mathbb{R}[x_{1},\ldots,x_{m}]$ generated by the image of $\lambda^{\top}$. By abuse of notation we denote the corresponding ideal in $\mathbb{R}[K]$ with the same letter $\Theta$.

Let us state things in the coordinate form. Fix a basis $f_{1},\ldots,f_{n}$ of $V$. Then every characteristic value $\lambda(i)$, $i\in[m]$ is written as a row-vector $(\lambda_{i,1},\ldots,\lambda_{i,n})$, where $\lambda_{i,j}\in\mathbb{R}$. The $*$-condition for $\lambda$ (see Sect. 2.2) states that the square matrix formed by row-vectors $(\lambda_{i,1},\ldots,\lambda_{i,n})_{i\in I}$ is non-degenerate for any $I\in K^{\langle n\rangle}$.

If we consider the dual basis $\bar{f}_{1},\ldots,\bar{f}_{n}$ in the dual space $V^{*}$, then its image under $\lambda^{\top}\colon V^{*}\to(\mathbb{R}^{m})^{*}=\langle x_{1},\ldots,x_{m}\rangle$ has the form

 $\displaystyle\theta_{j}:=\lambda^{\top}(\bar{f}_{j})=\lambda_{1,j}x_{1}+% \lambda_{2,j}x_{2}+\cdots+\lambda_{m,j}x_{m}$

for $j=1,\ldots,n$. Thus $\Theta$ (as an ideal either in $\mathbb{R}[x_{1},\ldots,x_{m}]$ or $\mathbb{R}[K]$) is generated by the elements $\theta_{1},\ldots,\theta_{n}$. In particular, if $\lambda$ is integer-valued, then $\Theta=(\theta_{1},\ldots,\theta_{n})$ may be considered as a well-defined ideal in $\mathbb{Z}[K]$ or $\mathbb{Z}[x_{1},\ldots,x_{m}]$.

It is known that the Krull dimension of $\mathbb{R}[K]$ equals $\dim K+1=n$ (see e.g. [Stanley1996]), and $\theta_{1},\ldots,\theta_{n}$ is a linear system of parameters in $\mathbb{R}[K]$ for any characteristic function $\lambda$ and every choice of a basis in $V$ (e.g. [Buchstaber and Panov2015], Lm. 3.3.2). Thus $\mathbb{R}[K]/\Theta$ has Krull dimension 0, which in our case is equivalent to saying that $\mathbb{R}[K]/\Theta$ is a finite-dimensional vector space. Moreover, it is known (see e.g. [Hattori and Masuda2003], Lm. 8.1 or [Ayzenberg2016], Lm. 3.5) that the classes of monomials $x_{I}=x_{i_{1}}\cdot\ldots\cdot x_{i_{k}}$ taken for each simplex $I=\{i_{1},\ldots,i_{k}\}\in K$ linearly span $\mathbb{R}[K]/\Theta$ (however there exist relations on these classes!).

We introduce the following notation to make the exposition consistent with that of [Hattori and Masuda2003]:

 $\displaystyle H_{T}^{*}(\Delta;\Bbbk):=\Bbbk[K],\quad H^{*}(\Delta;\Bbbk):=% \Bbbk[K]/\Theta,$ (4.3)

and, for short, $H^{*}_{T}(\Delta):=H^{*}_{T}(\Delta;\mathbb{R})$ and $H^{*}(\Delta):=H^{*}(\Delta;\mathbb{R})$.

### 4.3 Evaluation on Fundamental Class

Let $x=x_{i_{1}}^{j_{1}}\cdot\cdots\cdot x_{i_{k}}^{j_{k}}$ be a monomial whose index set $\{i_{1},\ldots,i_{k}\}$ is not a simplex of $K$. Then $\iota_{I}(x)=0$ for any $I\in K^{\langle n\rangle}$, according to (4.1). Therefore $\pi_{!}^{\Delta}(x)=0$. Hence $\pi_{!}^{\Delta}$ vanishes on the Stanley–Reisner ideal $I_{SR}$ and descends to the map

 $\displaystyle\pi_{!}^{\Delta}\colon\mathbb{R}[K]\to\mathbb{R}[V^{*}].$

Since $\pi_{!}^{\Delta}$ is a map of $\mathbb{R}[V^{*}]$-modules, we may apply $\otimes_{\mathbb{R}[V^{*}]}\mathbb{R}$ to $\pi_{!}^{\Delta}$. This gives a linear map

 $\displaystyle\int_{\Delta}\colon H^{2n}(\Delta)\to\mathbb{R}\cong\mathbb{R}[V^% {*}]/\mathbb{R}[V^{*}]^{+}$
###### Definition 4.5.

Let $\Delta=(K,w,\lambda)$ be a complete simplicial multi-fan. The map $\int_{\Delta}\colon H^{2n}(\Delta)\to\mathbb{R}$ is called “the evaluation on the fundamental class of $\Delta$”.

We denote the composite map $\mathbb{R}[x_{1},\ldots,x_{m}]\twoheadrightarrow H^{2n}(\Delta)\to\mathbb{R}$ by $\int_{\Delta,\mathbb{R}[m]}$.

### 4.4 Chern Class of a Multi-Polytope

Let $P$ be a multi-polytope based on a complete simplicial multi-fan $\Delta=(K,w,\lambda)$ of dimension $n$, and let $c_{1},\ldots,c_{m}\in\mathbb{R}$ be the support parameters of $P$. The element

 $\displaystyle c_{1}(P):=c_{1}x_{1}+\cdots+c_{m}x_{m}\in H^{2}(\Delta)$

is called the first Chern class of $P$.

###### Proposition 4.6.
 $\displaystyle\operatorname{Vol}(P)=\frac{1}{n!}\int_{\Delta}c_{1}(P)^{n}.$ (4.4)
###### Proof.

If $\lambda$ and $w$ are integral, the statement is proved in [Hattori and Masuda2003], Lm. 8.6. The rational case follows from the integral case by the following arguments. (1) In the rational case we may choose a refined lattice such that $\lambda$ becomes integral with respect to this lattice (this would change the Euclidean measure on $V^{*}$, but this change affects both sides of (4.4) in the same way). (2) A rational weight $w$ may be turned into an integral weight by rescaling (both sides of (4.4) depend linearly on $w$, thus rescaling of $w$ preserves (4.4)). Real case follows by continuity, since both sides of (4.4) depend continuously on $\lambda$ and $w$. $\square$

It is easily seen that, for a given $\Delta$, the expression on the right hand side of (4.4) is a homogeneous polynomial of degree $n$ in the variables $c_{1},\ldots,c_{m}$:

 $\displaystyle V_{\Delta}(c_{1},\ldots,c_{m})=\frac{1}{n!}\int_{\Delta}(c_{1}x_% {1}+\cdots+c_{m}x_{m})^{n}.$

Thus Proposition 4.6 implies Theorem 1.1.

## 5 Basic Properties of Volume Polynomials

### 5.1 Partial Derivatives of Volume Polynomial

We continue to assume that there is a fixed inner product in $V$ which makes the integral lattice in $V$ unnecessary. The inner product allows to identify $V$ and $V^{*}$ and to introduce a measure on each affine subspace of $V$ or $V^{*}$. Consider the space $\Lambda^{k}V$ of exterior forms on $V$. Given an inner product in $V$ we obtain an inner product on $\Lambda^{k}V$.

Suppose that every simplex $I\in K$ is oriented somehow. For a characteristic function $\lambda\colon[m]\to V$ on $K$ and $I=\{i_{1},\ldots,i_{k}\}\in K$ let $\lambda(I)$ denote the skew form $\lambda(i_{1})\wedge\cdots\wedge\lambda(i_{k})\in\Lambda^{k}V$, where $(i_{1},\ldots,i_{k})$ is the positive order of vertices of $I$ (that is the order which defines the positive orientation of a simplex). Denote the norm of $\lambda(I)$ by $\operatorname{covol}(I)$:

 $\displaystyle\operatorname{covol}(I):=\|\lambda(I)\|=\|\lambda(i_{1})\wedge% \cdots\wedge\lambda(i_{k})\|.$

Recall from Sect. 3 the notion of a face of a multi-polytope. If $P$ is a multi-polytope of dimension $n$ and $I\in K$ then $F_{I}$ is a multi-polytope of dimension $n-|I|$ sitting in the affine subspace $H_{I}\subset V^{*}$. There is a measure on $H_{I}$ determined by the inner product, hence we may define the volume of $F_{I}$. The following lemma shows that we can compute the volumes of faces from the volume polynomial.

###### Lemma 5.1.

(cf. [Timorin1999], Thm. 2.4.3) Let $J\subset[m]$ and let $\partial_{J}$ denote the differential operator $\prod_{i\in J}\partial_{i}$. Consider the homogeneous polynomial $\partial_{J}V_{\Delta}$ of degree $n-|J|$. Then

1. 1.

Let $\theta_{u}$ denote the linear differential operator $\sum_{i=1}^{m}\langle u,\lambda(i)\rangle\partial_{i}$ for $u\in V^{*}$. Then $\theta_{u}V_{\Delta}=0$.

2. 2.

If $J\notin K,$ then $\partial_{J}V_{\Delta}=0;$

3. 3.

If $J\in K,$ then the value of the polynomial $\partial_{J}V_{\Delta}$ at a point $(\tilde{c}_{1},\ldots,\tilde{c}_{m})\in\mathbb{R}^{m}$ is equal to

 $\displaystyle\dfrac{\operatorname{Vol}F_{J}}{\operatorname{covol}(J)}$ (5.1)

when $|J|<n$ and

 $\displaystyle\dfrac{w(J)}{\operatorname{covol}(J)}=\dfrac{w(J)}{|\det\lambda_{% J}|}$ (5.2)

when $|J|=n$. Here $\tilde{c}_{i}$ are the support parameters of a multi-polytope $P$ and $F_{J}$ are its faces.

###### Proof.

(1) We have

 $\displaystyle\theta_{u}V_{\Delta}$ $\displaystyle=$ $\displaystyle\frac{1}{n!}\int\left(\sum_{i=1}^{m}\langle u,\lambda(i)\rangle% \frac{\partial}{\partial c_{i}}\right)(c_{1}x_{1}+\cdots+c_{m}x_{m})^{n}$ $\displaystyle=$ $\displaystyle\frac{1}{(n-1)!}\int\left(\sum_{i=1}^{m}\langle u,\lambda(i)% \rangle x_{i}\right)\cdot(c_{1}x_{1}+\cdots+c_{m}x_{m})^{n-1}=0,$

since $\sum_{i=1}^{m}\langle u,\lambda(i)\rangle x_{i}=0$ in $H^{*}(\Delta)$.

(2) The proof of second statement is completely similar to (1). We have

 $\displaystyle\partial_{J}V_{\Delta}$ $\displaystyle=$ $\displaystyle\frac{1}{n!}\int\left(\prod_{i\in J}\frac{\partial}{\partial c_{i% }}\right)(c_{1}x_{1}+\cdots+c_{m}x_{m})^{n}$ $\displaystyle=$ $\displaystyle\frac{1}{(n-|J|)!}\int\left(\prod_{i\in J}x_{i}\right)\cdot(c_{1}% x_{1}+\cdots+c_{m}x_{m})^{n-|J|}=0,$

since $x_{J}=\prod_{i\in J}x_{i}=0$ in $H^{*}(\Delta)$ for $J\notin K$.

(3) The second claim requires some technical work. At first, let $|J|=n$, i.e. $J\in K^{\langle n\rangle}$. We have

 $\displaystyle\partial_{J}V_{\Delta}=\partial_{J}\frac{1}{n!}\int_{\Delta}(c_{1% }x_{1}+\cdots+c_{m}x_{m})^{n}=\int_{\Delta}x_{J}=\pi_{!}^{\Delta}(x_{J}),$

where $x_{J}=\prod_{i\in J}x_{i}$. By the definition of the index map (4.2) we have

 $\displaystyle\pi_{!}^{\Delta}(x_{J})=\sum_{I\in K^{\langle n\rangle}}\dfrac{w(% I)\iota_{I}(x_{J})}{|\det\lambda_{I}|\prod_{i\in I}\iota_{I}(x_{i})}.$

If $I\neq J$, the corresponding summand vanishes, since $\iota_{I}(x_{j})=0$ for $j\notin I$ by (4.1). The summand corresponding to $I=J$ contributes $\frac{w(J)}{|\det\lambda_{J}|}$ which proves the statement.

Let us prove the case $|J|<n$. Recall that the projected multi-fan $\Delta_{J}=(\operatorname{lk}_{K}J,w_{J},\lambda_{J})$ is the multi-fan in the vector space $V_{J}=V/\langle\lambda(j)\mid j\in J\rangle$. There exists a “restriction” map

 $\displaystyle\varphi_{J}\colon H^{*}(\Delta)\to H^{*}(\Delta_{J}),$

defined as follows:

 $\displaystyle\varphi_{J}(x_{j})=\begin{cases}x_{j},&\hbox{if }j\in% \operatorname{lk}_{K}J;\\ -\sum_{i\in\operatorname{lk}_{K}J}p^{J}_{i,j}x_{i},&\hbox{if }j\in J;\\ 0,&\hbox{otherwise}\end{cases}$

Here the constants $p^{J}_{i,j}$ for $j\in J$ and $i\in\operatorname{lk}_{K}J$ are defined by

 $\displaystyle\operatorname{proj}_{J}\lambda(i)=\sum_{j\in J}p^{J}_{i,j}\lambda% (j),$ (5.3)

where $\operatorname{proj}_{J}\lambda(i)$ is the orthogonal projection of the vector $\lambda(i)$ to the linear subspace spanned by $\lambda(j)$ $(j\in J)$.

The homomorphism $\varphi_{J}$ is now defined on the level of polynomial algebras.

###### Claim 5.2.

$\varphi_{J}$ is a well-defined ring homomorphism from $H^{*}(\Delta)=\mathbb{R}[K]/\Theta$ to $H^{*}(\Delta_{J})=\mathbb{R}[\operatorname{lk}_{K}J]/\Theta_{J}$.

###### Proof.

The proof is a routine check. First let us prove that Stanley–Reisner relations in $\Delta$ are mapped to the Stanley–Reisner ideal of $\Delta_{J}$. Let $I$ be a non-simplex of $K$. The definition of $\varphi_{J}$ implies that $\varphi_{J}(x_{I})=0$ unless $I\subset J\cup\operatorname{Vert}(\operatorname{lk}_{K}J)$. If $I\subset J\cup\operatorname{Vert}(\operatorname{lk}_{K}J)$, we have that $I\cap\operatorname{Vert}(\operatorname{lk}_{K}J)$ is a non-simplex of $\operatorname{lk}_{K}J$ (otherwise we would have $I\in K$ contradicting the assumption). Then the element $\varphi_{J}(x_{I})=\varphi_{J}\big(\prod_{i\in I\cap\operatorname{Vert}(% \operatorname{lk}_{K}J)}x_{i}\big)\cdot\varphi_{J}\big(\prod_{i\in I\cap J}x_{% i}\big)=\prod_{i\in I\cap\operatorname{Vert}(\operatorname{lk}_{K}J)}x_{i}% \cdot\varphi_{J}\big(\prod_{i\in I\cap J}x_{i}\big)$ lies in the Stanley–Reisner ideal of $\operatorname{lk}_{K}J$.

Let us check that linear relations in $H^{*}(\Delta)$ are mapped into linear relations of $H^{*}(\Delta_{J})$. A general linear relation in $H^{*}(\Delta)$ has the form $\sum_{i\in[m]}\langle u,\lambda(i)\rangle x_{i}$ for some $u\in V^{*}$. The map $\varphi_{J}$ sends it to the element

 $\displaystyle\sum_{i\in\operatorname{Vert}(\operatorname{lk}_{K}J)}\left(% \langle u,\lambda(i)\rangle-\sum_{j\in J}p^{J}_{i,j}\langle u,\lambda(j)% \rangle\right)x_{i}$ $\displaystyle\quad=\sum_{i\in\operatorname{Vert}(\operatorname{lk}_{K}J)}\left \langle u,\lambda(i)-\sum_{j\in J}p^{J}_{i,j}\lambda(j)\right\rangle x_{i}= \sum_{i\in\operatorname{Vert}(\operatorname{lk}_{K}J)}\langle u,\lambda_{J}(i) \rangle x_{i}.$

(note that $\lambda(i)-\sum_{j\in J}p^{J}_{i,j}\lambda(j)=\lambda(i)-\operatorname{proj}_{% J}\lambda(i)=\lambda_{J}(i)$ is the projection of $\lambda(i)$ to the plane orthogonal to $\langle\lambda(j)\rangle_{j\in J}$). The last expression is zero in $H^{*}(\Delta_{J})$. $\square$

Next we show that restriction homomorphism is compatible with the first Chern classes of the multi-polytopes.

###### Claim 5.3.

$\varphi_{J}$ sends $c_{1}(P)$ to $c_{1}(F_{J})$.

###### Proof.

Recall that $H_{J}$ denotes the ambient space of the face $F_{J}$ of the multi-polytope $P$. The supporting hyperplanes of $F_{J}$ are given by intersections $H_{J}\cap H_{i}$, where $H_{i}$ is the supporting hyperplane of $P$ for $i\in\operatorname{lk}_{K}J$.

Let us denote by $U_{J}$ the subspace spanned by $\lambda(j)$’s $(j\in J)$ so that $V_{J}=V/U_{J}$. By the definition (see Sect. 2.4), $\lambda_{J}(i)$ is the projection image of $\lambda(i)$ on $V_{J}$ if $i$ is the vertex of $\operatorname{lk}_{K}J$. As in the proof of previous claim we identify the quotient space $V_{J}=V/U_{J}$ with the orthogonal complement $U_{J}^{\bot}$ of $U_{J}$. The projected vector $\lambda_{J}(i)$ can be considered as the element in $V$ and we have

 $\displaystyle\lambda(i)=\lambda_{J}(i)+\operatorname{proj}_{J}\lambda(i)$ (5.4)

with respect to the orthogonal decomposition $V=U_{J}^{\bot}\oplus U_{J}$.

The affine hyperplane $H_{i}$ is given by $\{u\in V^{*}\mid\langle u,\lambda(i)\rangle=c_{i}\}$. The affine plane $H_{J}$ is given by $\{u\in V^{*}\mid\langle u,\lambda(j)\rangle=c_{j}$, for all $j\in J\}$. By using (5.3) and (5.4) we may write the intersection $H_{i}\cap H_{J}$ as

 $\displaystyle\{u\in H_{J}\mid\langle u,\lambda_{J}(i)+\operatorname{proj}_{J}% \lambda(i)\rangle=c_{i}\}$ $\displaystyle\quad=\left\{u\in H_{J}\Bigg|\left\langle u,\lambda_{J}(i)+\sum_{% j\in J}p^{J}_{i,j}\lambda(j)\right\rangle=c_{i}\right\}$ $\displaystyle\quad=\left\{u\in H_{J}\mid\langle u,\lambda_{J}(i)\rangle=c_{i}-% \sum_{j\in J}p^{J}_{i,j}c_{j}\right\}.$

Therefore the $i$-th support parameter of $F_{J}$ is $c_{i}-\sum_{j\in J}p^{J}_{i,j}c_{j}$ for $i\in\operatorname{lk}_{K}J$. Now it remains to note that the coefficient of $x_{i}$ in the projected class $\varphi_{J}(c_{1}(P))$ is exactly $c_{i}-\sum_{j\in J}p^{J}_{i,j}c_{j}$. Thus $\varphi_{J}(c_{1}(P))=c_{1}(F_{J})$. $\square$

Now we prove the following

###### Claim 5.4.
 $\displaystyle\int_{\Delta}y\prod_{j\in J}x_{j}=\frac{1}{\operatorname{covol}(J% )}\int_{\Delta_{J}}\varphi_{J}(y)\quad\text{for any }y\in H^{*}(\Delta).$
###### Proof.

Let us denote by $\operatorname{Vol}S$ the volume of the parallelepiped formed by a set of vectors $S$. Then $\operatorname{covol}(J)=\operatorname{Vol}\{\lambda(i)\}_{i\in J}$ and the index map can be written as

 $\displaystyle\pi_{!}^{\Delta}(x)=\sum_{I\in K^{\langle n\rangle}}\frac{w(I)% \iota_{I}(x)}{\operatorname{Vol}\{\lambda(i)\}_{i\in I}\prod_{i\in I}\iota_{I}% (x_{i})}.$ (5.5)

Let $\tilde{I}\in\operatorname{lk}_{K}J$ and, therefore, $\tilde{I}\sqcup J\in K$. Then

 $\displaystyle\operatorname{Vol}\{\lambda(i)\}_{i\in\tilde{I}\sqcup J}=% \operatorname{Vol}\{\lambda(i)\}_{i\in J}\cdot\operatorname{Vol}\{\lambda_{J}(% i)\}_{i\in\tilde{I}}=\operatorname{covol}(J)\cdot\operatorname{Vol}\{\lambda_{% J}(i)\}_{i\in\tilde{I}}.$

This together with (5.5) implies the lemma. $\square$

Applying Claim 5.4 to $y=c_{1}(P)^{n-|J|}$ and using Claim 5.3, we obtain

 $\displaystyle\partial_{J}V_{\Delta}=\frac{1}{(n-|J|)!}\int_{\Delta}c_{1}(P)^{n% -|J|}\prod_{j\in J}x_{j}=\frac{1}{\operatorname{covol}(J)}\frac{1}{(n-|J|)!}% \int_{\Delta_{J}}c_{1}(F_{J})^{n-|J|}.$

Expression at the right evaluates to $\frac{\operatorname{Vol}(F_{J})}{\operatorname{covol}(J)}$ which finishes the proof of Lemma 5.1.

###### Corollary 5.5.

Let $\Delta=(w_{ch},\lambda)$ be a complete multi-fan. Then $V_{\Delta}=0$ implies $w_{ch}=0$.

###### Proof.

If $V_{\Delta}=0$, then $\partial_{J}V_{\Delta}=0$ for any $J\in K^{\langle n\rangle}$. This implies $w_{ch}=0$. $\square$

###### Remark 5.6.

Of course, according to Proposition 7.2 the polynomial $V_{\Delta}$ is non-zero if and only if the map $\int_{\Delta}$ is non-zero. The fact that $\int_{\Delta}$ is non-zero for every non-zero $w_{ch}$ is proved by applying this map to all monomials $x_{I}$, $I\in K^{\langle n\rangle}$ (recall that these monomials span $H^{2n}(\Delta)$). This procedure is essentially the same as applying differential operators $\partial_{I}$ to $V_{\Delta}$.

###### Corollary 5.7.

Let $\partial_{P}$ denote the linear differential operator $\sum_{i\in[m]}\tilde{c}_{i}\partial_{i}$ where $\tilde{c}_{i}$ are the support parameters of a multi-polytope $P$. Then

 $\displaystyle\displaystyle\frac{1}{n!}\partial_{P}^{n}V_{\Delta}=\operatorname% {Vol}(P),$ $\displaystyle\displaystyle\frac{1}{(n-|I|)!}\partial_{P}^{n-|I|}\partial_{I}V_% {\Delta}=\dfrac{\operatorname{Vol}(F_{I})}{\operatorname{covol}(I)}.$
###### Proof.

Both formulas follow from Lemma 5.1 and a simple observation: if $\Psi\in\mathbb{R}[c_{1},\ldots,c_{m}]_{k}$ is a homogeneous polynomial of degree $k$, then

 $\displaystyle\left(\sum_{i\in[m]}\tilde{c}_{i}\partial_{i}\right)^{k}\Psi=k!% \Psi(\tilde{c}_{1},\ldots,\tilde{c}_{m}).$

(evaluation at a point coincides with the result of differentiation up to $k!$). $\square$

### 5.2 Recovering Multi-Fans from Volume Polynomials

When we associate a volume polynomial to a complete simplicial multi-fan, the numbering of the one-dimensional cones by $[m]$ is incorporated in the data of the multi-fan. We call a multi-fan with the numbering a based multi-fan. Two based multi-fans $\Delta$ and $\Delta^{\prime}$ are said to be equivalent if there is an automorphism of $V$ which induces an isomorphism between $\Delta$ and $\Delta^{\prime}$ preserving the numbering. In the presence of a lattice $N\subset V$ there should be an automorphism of the lattice with this property. Equivalent complete simplicial based multi-fans have the same volume polynomial. We will see that the converse holds for complete simplicial based multi-fans $\Delta$ whose underlying simplicial complexes are oriented strongly connected pseudo-manifolds. Strong connectedness of $K$ means that for any two maximal simplices $I,I^{\prime}\in K^{\langle n\rangle}$ there exists a sequence of maximal simplices $I=I_{0},I_{1},\ldots,I_{k}=I^{\prime}$ such that $|I_{s}\cap I_{s+1}|=n-1$ for $0\leqslant s\leqslant k-1$.

We assume that the volume polynomial $V_{\Delta}$ associated to $\Delta$ is non-zero. Then the class $[\Delta]$ is non-zero. Since $K$ is assumed to be a pseudo-manifold, $w(I)\neq 0$ for any $I\in K^{\langle n\rangle}$. Then Lemma 5.1 shows that $V_{\Delta}$ recovers $K$.

Remember that

 $\displaystyle\sum_{i=1}^{m}\langle u,\lambda(i)\rangle x_{i}=0\quad\text{in }H% ^{*}(\Delta)\hbox{ for any }u\in V^{*}.$ (5.6)

Let $J\in K$, $|J|=n-1$. Since $K$ is assumed to be a pseudo-manifold, there are exactly two elements $i_{1}$ and $i_{2}$ in $[m]$ such that $J\cup\{i_{1}\}$ and $J\cup\{i_{2}\}$ are in $K^{\langle n\rangle}$. Multiplying $x_{J}=\prod_{i\in J}x_{i}$ to the both sides in (5.6), we obtain

 $\displaystyle\sum_{j\in J}\langle u,\lambda(j)\rangle x_{j}x_{J}+\langle u,% \lambda(i_{1})\rangle x_{i_{1}}x_{J}+\langle u,\lambda(i_{2})\rangle x_{i_{2}}% x_{J}=0\quad\text{for all }u\in V^{*}.$

Applying $\int_{\Delta}$ to the above identity, we have

 $\displaystyle\left\langle u,\sum_{j\in J}\left(\int_{\Delta}x_{j}x_{J}\right)% \lambda(j)+\left(\int_{\Delta}x_{i_{1}}x_{J}\right)\lambda(i_{1})+\left(\int_{% \Delta}x_{i_{2}}x_{J}\right)\lambda(i_{2})\right\rangle=0.$

Since this holds for all $u\in V^{*}$, one can conclude

 $\displaystyle\sum_{j\in J}\left(\int_{\Delta}x_{j}x_{J}\right)\lambda(j)+\left (\int_{\Delta}x_{i_{1}}x_{J}\right)\lambda(i_{1})+\left(\int_{\Delta}x_{i_{2}} x_{J}\right)\lambda(i_{2})=0.$ (5.7)

Note that the numbers $\int_{\Delta}x_{i_{1}}x_{J}$ and $\int_{\Delta}x_{i_{2}}x_{J}$ are non-zero. Identity (5.7) shows that once basis vectors $\{\lambda(i)\}_{i\in I}$ for some $I\in K^{\langle n\rangle}$ are determined, then the other vectors $\lambda(k)$’s will be determined by the intersection numbers $\int_{\Delta}x_{\mathcal{I}}$ where $\mathcal{I}$ consists of elements in $[m]$ with $|\mathcal{I}|=n$ (an element in $\mathcal{I}$ may appear more than once). On the other hand, since

 $\displaystyle V_{\Delta}=\frac{1}{n!}\int_{\Delta}(c_{1}x_{1}+\cdots+c_{m}x_{m% })^{n},$

the coefficient of $c^{\mathcal{I}}$ agrees with $\int_{\Delta}x_{\mathcal{I}}$ up to some non-zero constant independent of $\Delta$. These show that $V_{\Delta}$ determines $\Delta$ up to equivalence.

###### Proposition 5.8.

Two complete simplicial toric varieties are isomorphic if and only if their volume polynomials agree up to permutations of variables. Here it is assumed that all $\lambda(i)$’s are the primitive generators of the rays.

###### Proof.

This follows from the above observation and the fact that two toric varieties are isomorphic if and only if their fans are isomorphic ([Berchtold2003]).11We are grateful to Ivan Arzhantsev from whom we learned this fact. $\square$

## 6 A Formula for the Volume Polynomial

We say that the set $S$ of $n+1$ vectors in $V\cong\mathbb{R}^{n}$ is in general position, if any $n$ of them are linearly independent. Any such set determines a multi-fan whose underlying simplicial complex is a boundary of a simplex $K=\partial\triangle_{[n+1]}$. The weights of all maximal simplices are the same up to sign due to closedness condition $dw_{ch}=0$. Thus without loss of generality we may assume that every weight is 1 or $-1$ depending on the orientation. We call such multi-fan an elementary multi-fan and denote it $\Delta^{el}(S)$.

###### Lemma 6.1.

Let $\Delta$ be an elementary multi-fan determined by the vectors $\lambda(1),\ldots,{}\lambda(n+1)\in V$. Let $0\neq(\alpha_{1},\ldots,\alpha_{n+1})\in\mathbb{R}^{n+1}$ be a nonzero linear relation on these vectors$,$ i.e. $\sum_{i=1}^{n+1}\alpha_{i}\lambda(i)=0$. Then

 $\displaystyle V_{\Delta}(c_{1},\ldots,c_{n+1})=\operatorname{const}\cdot(% \alpha_{1}c_{1}+\cdots+\alpha_{n+1}c_{n+1})^{n}.$ (6.1)

for some constant $\operatorname{const}$.

We postpone the proof to Sect. 8.3.

###### Remark 6.2.

It is not difficult to compute the constant: just apply the differential operator $\partial_{J}$ for $J\subset[n+1]$, $|J|=n$ to both sides of (6.1) and use Lemma 5.1. However, we do not need this constant at the moment and ignore it to simplify the exposition.

###### Theorem 6.3.

Let $\Delta=(w_{ch},\lambda)$ be a complete multi-fan. Let $v\in V$ be a generic vector. Then

 $\displaystyle V_{\Delta}(c_{1},\ldots,c_{m})=\frac{1}{n!}\sum_{I=\{i_{1},% \ldots,i_{n}\}\in K}\frac{w(I)}{|\det\lambda_{I}|\prod_{j=1}^{n}\alpha_{I,j}}(% \alpha_{I,1}c_{i_{1}}+\cdots+\alpha_{I,n}c_{i_{n}})^{n},$

where $\alpha_{I,1},\ldots,\alpha_{I,n}$ are the coordinates of $v$ in the basis $(\lambda(i_{1}),\ldots,\lambda(i_{n})),$ and $w(I)$ is the weight.

###### Proof.

We derive a more general family of formulas, and (6.3) will be a particular case. Let $[m^{\prime}]$ be a set containing $[m]$ and let

 $\displaystyle z_{ch}=\sum_{J\in{[m^{\prime}]\choose n+1}}z(J)J\in C_{n}(% \triangle_{[m^{\prime}]};\mathbb{R})$

be a simplicial chain such that $dz_{ch}=w_{ch}$ (it exists since $w_{ch}$, considered as an element in $C_{n}(\triangle_{[m^{\prime}]};\mathbb{R})$, is closed hence exact). Consider any function $\eta\colon[m^{\prime}]\to V$ which extends $\lambda\colon[m]\to V$ and satisfies the condition: for any $J=\{j_{1},\ldots,j_{n+1}\}$ with $z(J)\neq 0$ the vectors $\eta(j_{1}),\ldots,\eta(j_{n+1})$ are in general position. Thus for any such $J$ we can construct an elementary multi-fan $\Delta^{el}(\eta(J))$.

In the group of multi-fans we have a relation $\Delta=\sum_{J\in{[m^{\prime}]\choose n+1}}z(J)\Delta^{el}(\eta(J))$, if $\Delta$ is considered as a multi-fan on $[m^{\prime}]$ (recall that multi-fans with the same characteristic function can be added to each other and multiplied by integers by performing this operations on their weights, and therefore such multi-fans form an abelian group). Volume polynomial is additive, hence we get

 $\displaystyle V_{\Delta}=\sum_{J\in{[m^{\prime}]\choose n+1}}z(J)V_{\Delta^{el% }(\eta(J))}.$ (6.3)

Therefore, any simplicial chain whose boundary is $w_{ch}$ gives a formula for the volume polynomial. Now let us consider the particular case, namely, the cone over $w_{ch}$. Let $[m^{\prime}]=[m]\sqcup\{r\}$ and set $\eta(r)=v$, for a generic vector $v\in V$. So the phrase “$v$ is a generic vector” means that the set $\lambda(I)\sqcup\{v\}$ is in general position for any $I\subset[m]$ such that $|I|=n$ and $w(I)\neq 0$. The function $z$ on the cone is defined in an obvious way: $z(I\sqcup\{r\}):=w(I)$.

Relation (6.3) and Lemma 6.1 imply

 $\displaystyle V_{\Delta}=\sum_{I=\{i_{1},\ldots,i_{n}\}\subset[m]}% \operatorname{const}\cdot w(I)\cdot(\alpha_{I,1}c_{i_{1}}+\cdots+\alpha_{I,n}c% _{i_{n}}+\beta_{I}c_{r})^{n}.$ (6.4)

The tuple $(\alpha_{I,1},\ldots,\alpha_{I,n},\beta_{I})$ is a linear relation on the vectors $\lambda(i_{1}),\ldots,\lambda(i_{n}),v$. Therefore we may assume that $\beta_{I}=-1$ and $(\alpha_{I,1},\ldots,\alpha_{I,n})$ are the coordinates of $v$ in the basis $\lambda(i_{1}),\ldots,\lambda(i_{n})$.

Left hand side of (6.4) does not depend on $c_{r}$ (it is a redundant support parameter), therefore we may put $c_{r}=0$:

 $\displaystyle V_{\Delta}=\sum_{I=\{i_{1},\ldots,i_{n}\}\subset[m]}A_{I}\cdot w% (I)\cdot(\alpha_{I,1}c_{i_{1}}+\cdots+\alpha_{I,n}c_{i_{n}})^{n}.$ (6.5)

To compute the constants $A_{I}$ take any $J=\{j_{1},\ldots,j_{n}\}\in K$ and apply the differential operator $\partial_{J}=\frac{\partial}{\partial c_{j_{1}}}\cdot\cdots\cdot\frac{\partial% }{\partial c_{j_{n}}}$ to the identity (6.5). On the left we have $\partial_{J}V_{\Delta}=\frac{w(J)}{|\det\lambda_{J}|}$, according to Lemma 5.1. On the right side all summands with $I\neq J$ vanish, and the one with $I=J$ contributes $n!\cdot A_{J}\cdot w(J)\prod_{i}\alpha_{J,i}$. Thus $A_{J}=\frac{1}{n!}\frac{1}{|\det\lambda_{J}|\cdot\prod_{i}\alpha_{J,i}}$ and the statement follows. $\square$

###### Remark 6.4.

Note that the formula (6.3) can be applied to compute the volume of a simple convex polytope in the case when the polytope is described as the intersection of half-spaces with the given equations. In this case the formula is known as Lawrence’s formula ([Lawrence1991]). It has found applications in explicit volumes’ calculations.

###### Example 6.5.

Consider the standard fan $\Delta$ of $\mathbb{C}P^{2}$, generated by the vectors $\lambda(1)=(1,0)$, $\lambda(2)=(0,1)$, $\lambda(3)=(-1,-1)$. Take the generic vector $v=(1,2)$. We have

 $\displaystyle v=\lambda(1)+2\lambda(2)=\lambda(2)-\lambda(3)=-2\lambda(3)-% \lambda(1).$

Theorem 6.3 implies

 $\displaystyle V_{\Delta}=\frac{1}{2}\left(\frac{1}{2}(c_{1}+2c_{2})^{2}-(c_{2}% -c_{3})^{2}+\frac{1}{2}(-c_{1}-2c_{3})^{2}\right).$

This expression equals $\frac{1}{2}(c_{1}+c_{2}+c_{3})^{2}$. The same expression is given by Lemma 6.1.

###### Example 6.6.

Consider the normal fan of the standard $n$-cube. The underlying simplicial complex is isomorphic to the boundary of cross-polytope. Let $\{1,\ldots,n,-1,\ldots,{}-n\}$ be its set of vertices, so the maximal simplices have the form $\{\pm 1,\ldots,\pm n\}$. We have $\lambda(\pm i)=\pm e_{i}$. Take the generic vector $v=e_{1}+\cdots+e_{n}$. Then Theorem 6.3 implies

 $\displaystyle V_{\Delta}=\frac{1}{n!}\sum_{(\epsilon_{1},\ldots,\epsilon_{n})% \in\{+,-\}^{n}}\frac{1}{\prod_{i=1}^{n}\epsilon_{i}}(\epsilon_{1}c_{\epsilon_{% 1}1}+\cdots+\epsilon_{n}c_{\epsilon_{n}n})^{n}.$

On the other hand, we have $V_{\Delta}=\prod_{i}(c_{i}+c_{-i})$ by geometrical reasons. Indeed, the polytope dual to $\Delta$ is the brick with sides $\{c_{i}+c_{-i}\}_{i\in[n]}$. By setting $c_{-i}=0$ for each $i$ we get the identity

 $\displaystyle\prod_{i=1}^{n}c_{i}=\frac{1}{n!}\sum_{I\subseteq[n]}(-1)^{n-|I|}% c_{I}^{n},$ (6.6)

where $c_{I}=\sum_{i\in I}c_{i}$. This identity is well known as the discrete polarization identity.

###### Remark 6.7.

The proof of Theorem 6.3 implies the following consideration. Take two simplicial $n$-chains $z_{ch,1},z_{ch,2}\in C_{n}(\triangle_{[m]};\mathbb{R})$ endowed with functions $\eta_{1},\eta_{2}\colon[m]\to\mathbb{R}^{n}$ such that $\eta_{\epsilon}(J)$ is in general position for any simplex $J$ of the chain $z_{ch,\epsilon}$, $\epsilon=1,2$. Assume that $dz_{ch,1}=dz_{ch,2}$ and the functions $\eta_{1},\eta_{2}$ agree on the vertices of the boundary. Then the volume polynomial of the multi-fan $\Delta=(dz_{ch,1},\eta_{1})=(dz_{ch,2},\eta_{2})$ can be expressed by two formulas:

 $\displaystyle\sum_{J=(j_{1},\ldots,j_{n+1})\subset[m^{\prime}]}\operatorname{% const}\cdot z_{1}(J)(\alpha_{J,1}c_{j_{1}}+\cdots+\alpha_{J,n+1}c_{j_{n+1}})^{% n}=V_{\Delta}$ $\displaystyle\quad=\sum_{J=(j_{1},\ldots,j_{n+1})\subset[m^{\prime}]}% \operatorname{const}\cdot z_{2}(J)(\alpha_{J,1}c_{j_{1}}+\cdots+\alpha_{J,n+1}% c_{j_{n+1}})^{n}.$

We may take a difference of the left and right parts and summarize as follows. Let us take any closed simplicial $n$-chain $z_{ch}$, $dz_{ch}=0$, on the vertex set $[m^{\prime}]$, and endow it with a function $\eta\colon[m^{\prime}]\to\mathbb{R}^{n}$ which is in general position on any simplex $J$ of the chain. Then we get an identity

 $\displaystyle\sum_{J=(j_{1},\ldots,j_{n+1})\subset[m^{\prime}]}\operatorname{% const}\cdot z(J)(\alpha_{J,1}c_{j_{1}}+\cdots+\alpha_{J,n+1}c_{j_{n+1}})^{n}=0$

(the constants may be computed by the same method as we used previously). This seems to be a quite general way to construct algebraical identities from geometrical data.

This idea can be illustrated by a simple identity obtained in Example 6.5:

 $\displaystyle\frac{1}{2}(c_{1}+2c_{2}-c_{4})^{2}-(c_{2}-c_{3}-c_{4})^{2}+\frac% {1}{2}(-c_{1}-2c_{3}-c_{4})^{2}=(c_{1}+c_{2}+c_{3})^{2}.$

This identity is induced by the schematic picture shown on Fig. 3. Note that the last step in the proof of Theorem 6.3 was to specialize $c_{4}=0$, but even without this specialization the identity holds true.

## 7 Poincare Duality Algebra of a Multi-Fan

### 7.1 Poincare Duality Algebras

###### Definition 7.1.

Let $\Bbbk$ be a field. Let $\mathcal{A}^{*}=\bigoplus_{j=0}^{n}\mathcal{A}^{2j}$ be a finite-dimensional graded commutative $\Bbbk$-algebra such that

• -

there exists an isomorphism $\int_{\mathcal{A}}\colon\mathcal{A}^{2n}\to\Bbbk$;

• -

the pairing $\mathcal{A}^{2p}\otimes\mathcal{A}^{2n-2p}\to\Bbbk$, $a\otimes b\mapsto\int_{\mathcal{A}}(a\cdot b)$ is non-degenerate.

Then $\mathcal{A}$ is called a Poincare duality algebra of formal dimension $2n$.

Let $\partial_{i}=\frac{\partial}{\partial c_{i}}$, $i\in[m]$ be the differential operator acting on the ring of polynomials $\mathbb{R}[c_{1},\ldots,c_{m}]$ in a standard way. For a subset $I\subset[m]$ let $\partial_{I}$ denote the product $\prod_{i\in I}\partial_{i}$.

Consider the algebra of differential operators with constant coefficients $\mathcal{D}:=\mathbb{R}[\partial_{1},\ldots,\partial_{m}]$. It will be convenient to double the degree, so we assume $\deg\partial_{i}=2$, $i\in[m]$ (while still assuming that $\deg c_{i}=1$). For any non-zero homogeneous polynomial $\Psi\in\mathbb{R}[c_{1},\ldots,c_{m}]$ of degree $n$ we may consider the following ideal in $\mathcal{D}$:

 $\displaystyle\operatorname{Ann}\Psi:=\{D\in\mathcal{D}\mid D\Psi=0\}.$

It is not difficult to check that the quotient $\mathcal{D}/\operatorname{Ann}\Psi$ is a Poincare duality algebra of formal dimension $2n$ (see [Timorin1999], Prop. 2.5.1), where the “integration map” assigns the number $D\Psi\in\mathbb{R}$ to any differential operator of rank $n$ (i.e. of formal degree $2n$ in our setting).

It happens that every Poincare duality algebra generated by degree two can be obtained by this construction as the following proposition shows.

###### Proposition 7.2.

Suppose $\operatorname{char}\Bbbk=0$ and let $\Bbbk[m]=\Bbbk[x_{1},\ldots,x_{m}]$ be a polynomial ring$,$ where $\deg x_{i}=2$. Then the following three sets of objects are naturally equivalent$:$

1. 1.

Poincare duality algebras $\mathcal{A}^{*}$ of formal dimension $2n$ which are the quotients of the polynomial ring $\Bbbk[m];$

2. 2.

Non-zero homogeneous polynomials $\Psi\in\Bbbk[c_{1},\ldots,c_{m}]$ of degree $n$ $($where $\deg c_{i}=1)$ up to multiplication by a non-zero constant$;$

3. 3.

Non-zero linear maps $\int\colon\Bbbk[m]_{2n}\to\Bbbk$ up to multiplication by a non-zero constant.

###### Proof.

We give a very brief sketch of the proof. For details the reader is referred to the monograph ([Meyer and Smith2005]) which, among other things, describes the case $\operatorname{char}\Bbbk\neq 0$ (for general fields instead of a polynomial $\Psi$ one should take an element of divided power algebra). Also we would like to mention that an equivalence of (1) and (2) is a manifestation of the well-known phenomenon called Macaulay duality (or its extended version, Matlis duality).

(1) $\Rightarrow$ (3). Let $\mathcal{A}^{*}\cong\Bbbk[m]/\mathcal{I}$ be a Poincare duality quotient of the ring of polynomials. Then we have a linear isomorphism $\int_{\mathcal{A}}\colon\mathcal{A}^{2n}\to\Bbbk$. The composite

 $\displaystyle\Bbbk[m]_{2n}\twoheadrightarrow\mathcal{A}^{2n}\to\Bbbk$

is the required linear map.

(3) $\Rightarrow$ (1). Given a linear map $\int\colon\Bbbk[m]_{2n}\to\Bbbk$ we may define a pairing $\Bbbk[m]_{2p}\otimes\Bbbk[m]_{2n-2p}\to\Bbbk$ by $a\otimes b\mapsto\int a\cdot b$. This pairing is degenerate and we define its kernel:

 $\displaystyle W^{*}=\bigoplus_{p}W^{2p},\quad W^{2p}=\left\{x\in\Bbbk[m]_{2p} \mid\int x\cdot\Bbbk[m]_{2n-2p}=0\right\}.$

It is easy to check that $W^{*}\subset\Bbbk[m]$ is an ideal and $\Bbbk[m]/W^{*}$ is a Poincare duality algebra.

(3) $\Rightarrow$ (2). We construct a polynomial $\Psi$ in $c_{1},\ldots,c_{m}$ by

 $\displaystyle\Psi:=\frac{1}{n!}\int(c_{1}x_{1}+\cdots+c_{m}x_{m})^{n}.$

This polynomial is non-zero. Indeed, $\Bbbk[m]_{2n}$ is additively generated by the monomials of degree $n$ in the variables $x_{i}$. Each monomial can be expressed as a linear combination of expressions of the form $(c_{1}x_{1}+\cdots+c_{m}x_{m})^{n}$ for some constants $c_{i}$ as follows from the polarization identity (see (6.6) in Example 6.6). Thus expressions of the form $(c_{1}x_{1}+\cdots+c_{m}x_{m})^{n}$ linearly span $\Bbbk[m]_{2n}$ and therefore, since $\int$ is non-zero, the polynomial $\Psi$ is not a constant zero as well.

(2) $\Rightarrow$ (1). Given a homogeneous polynomial $\Psi$ in the variables $c_{1},\ldots,c_{m}$, we may construct a Poincare duality quotient $\Bbbk[\partial_{1},\ldots,\partial_{m}]/\operatorname{Ann}\Psi$, where the action of $\partial_{i}=\frac{\partial}{\partial c_{i}}$ on polynomials is defined formally in the usual way.

The consistency of all these constructions is a routine check. $\square$

The same arguments can be used to prove that there is a one-to-one correspondence between Poincare duality quotients of formal dimension $2n$ of an algebra $\mathcal{B}^{*}$ and the non-zero linear functionals on the linear space $\mathcal{B}^{2n}$. For this correspondence we do not need the assumptions that $\mathcal{B}$ is generated by degree 2 and $\operatorname{char}\Bbbk=0$. This motivates the following definition.

###### Definition 7.3.

Let $\mathcal{B}^{*}=\bigoplus_{j}\mathcal{B}^{2j}$ be a graded commutative $\Bbbk$-algebra and suppose that for some $n>0$ a non-zero linear map $\int\colon\mathcal{B}^{2n}\to\Bbbk$ is given. The corresponding Poincare duality quotient of $\mathcal{B}^{*}$, i.e. the algebra

 $\displaystyle\mathcal{B}^{*}/W^{*},\quad W^{2p}=\left\{b\in\mathcal{B}^{2p}% \mid\int b\cdot\mathcal{B}^{2n-2p}=0\right\}.$

is denoted by $\operatorname{PD}(\mathcal{B}^{*},\int)$ and called Poincare dualization of $\mathcal{B}^{*}$ (w.r.t. $\int$).

###### Lemma 7.4.

Consider two algebras $\mathcal{B}_{1}^{*},\mathcal{B}_{2}^{*}$ with the given non-zero linear maps $\int_{1}\colon\mathcal{B}_{1}^{2n}\to\Bbbk,$ $\int_{2}\colon\mathcal{B}_{2}^{2n}\to\Bbbk$. Let $\varphi\colon\mathcal{B}_{1}^{*}\twoheadrightarrow\mathcal{B}_{2}^{*}$ be an epimorphism of algebras consistent with the integration maps$:$ $\int_{2}\circ\varphi|_{\mathcal{B}_{1}^{2n}}=\int_{1}$. Then $\varphi$ induces an isomorphism $\operatorname{PD}(\mathcal{B}_{1}^{*},\int_{1})\cong\operatorname{PD}(\mathcal{B}_{2}^{*},\int_{2})$.

###### Proof.

From the surjectivity of $\varphi$ it easily follows that the kernel $W^{*}_{1}$ of the intersection pairing in the first algebra maps to the kernel $W_{2}^{*}$ of the second algebra. Thus the homomorphism $\widetilde{\varphi}\colon\operatorname{PD}(\mathcal{B}_{1}^{*},\int_{1})\to% \operatorname{PD}(\mathcal{B}_{2}^{*},\int_{2})$ is well defined. Obviously it is surjective. Let us prove that $\widetilde{\varphi}$ is injective. The map $\widetilde{\varphi}$ is an isomorphism in degree $2n$. Suppose that $\widetilde{\varphi}(a)=0$ for some $0\neq a\in\operatorname{PD}(\mathcal{B}_{1}^{*},\int_{1})_{2p}$. By the definition of Poincare duality algebra, there exists $b\in\operatorname{PD}(\mathcal{B}_{1}^{*},\int_{1})_{2n-2p}$ such that $ab\neq 0$. But then we have $\widetilde{\varphi}(ab)=\widetilde{\varphi}(a)\widetilde{\varphi}(b)=0$ which gives a contradiction. $\square$

In the following let $\mathcal{A}^{*}(\Psi)=\mathcal{D}{/}{\!}\operatorname{Ann}\Psi$ denote the Poincare duality algebra corresponding to the homogeneous polynomial $\Psi$ of degree $n$.

### 7.2 Algebras Associated with Multi-Fans

The linear maps $\int_{\Delta}\colon H^{2n}(\Delta)\to\mathbb{R}$ and $\int_{\Delta,\mathbb{R}[m]}\colon\mathbb{R}[x_{1},\ldots,x_{m}]_{2n}\to\mathbb {R}$ are consistent with the natural projection $\mathbb{R}[x_{1},\ldots,x_{m}]\to H^{*}(\Delta)$. Thus Lemma 7.4 implies an isomorphism

 $\displaystyle\operatorname{PD}\left(H^{*}(\Delta),\int_{\Delta}\right)\cong% \operatorname{PD}\left(\mathbb{R}[m],\int_{\Delta,\mathbb{R}[m]}\right).$

According to the constructions mentioned in the proof of Proposition 7.2, this Poincare duality algebra is also isomorphic to $\mathcal{A}^{*}(V_{\Delta})=\mathcal{D}/\operatorname{Ann}V_{\Delta}$, where $V_{\Delta}$ is the volume polynomial.

###### Definition 7.5.

Let $\Delta$ be a complete simplicial multi-fan of dimension $n$ with $m$ rays. Then the algebra

 $\displaystyle\mathcal{A}^{*}(\Delta):=\mathcal{D}/\operatorname{Ann}V_{\Delta}% \cong\operatorname{PD}\left(H^{*}(\Delta),\int_{\Delta}\right)\cong% \operatorname{PD}\left(\mathbb{R}[m],\int_{\Delta,\mathbb{R}[m]}\right)$

is called a multi-fan algebra of $\Delta$.

###### Remark 7.6.

The constructions above show that there is a ring epimorphism from $H^{*}(\Delta)\cong\mathbb{R}[K]/\Theta$ to $\mathcal{A}^{*}(\Delta)\cong\operatorname{PD}(H^{*}(\Delta),\int_{\Delta})$, sending $x_{i}$ to $\partial_{i}$ for each $i\in[m]$. Therefore $\mathcal{A}^{*}(\Delta)$ can be considered as a quotient of $H^{*}(\Delta)$, and all the relations in $\mathbb{R}[K]/\Theta$ are inherited by $\mathcal{A}^{*}(\Delta)$. We have

 $\displaystyle\partial_{J}V_{\Delta}=0\quad\hbox{for }J\notin K\quad\hbox{(% Stanley--Reisner relations),}$ $\displaystyle\left(\sum_{i\in[m]}\lambda_{i,j}\partial_{i}\right)V_{\Delta}=0% \quad\hbox{for }j=1,\ldots,n\quad(\hbox{Linear relations}).$

This proves points 1 and 2 of Lemma 5.1 in a more conceptual way.

## 8 Structure of Multi-Fan Algebra in Particular Cases

### 8.1 Ordinary Fans

As was mentioned in the introduction, when $\Delta$ is a normal fan of a simple convex polytope $P$, the construction of the algebra $\mathcal{A}^{*}(\Delta)=\mathcal{D}/\operatorname{Ann}V_{\Delta}$ was introduced by [Khovanskii and Pukhlikov1992b] and studied extensively by [Timorin1999]. In this case the underlying simplicial complex of $\Delta$ is a sphere and the weight function takes value $+1$ on all maximal simplices of $K$. Using purely combinatorial and geometrical considerations Timorin proved that $\mathcal{A}^{*}(\Delta)\cong\mathbb{R}[K]/\Theta$. This means, in particular, that dimensions $d_{i}=\dim\mathcal{A}^{2i}(\Delta)$ are equal to $h_{i}$, the $h$-numbers of $K$ (see the definition below). The developed technique is applied to prove that $\mathcal{A}^{*}(\Delta)$ is a Lefschetz algebra, meaning that there exists an element $\omega\in\mathcal{A}^{2}(\Delta)$ such that

 $\displaystyle\times\omega^{n-2k}\colon\mathcal{A}^{2k}\to\mathcal{A}^{2n-2k}$

is an isomorphism for each $k=0,\ldots,[n/2]$. In particular this implies that the distribution of $h$-numbers of convex simplicial spheres is unimodal, i.e.

 $\displaystyle h_{0}\leqslant h_{1}\leqslant\cdots\leqslant h_{[n/2]}=h_{n-[n/2% ]}\geqslant\cdots\geqslant h_{n-1}\geqslant h_{n}.$

According to Timorin’s result, Lefschetz element $\omega$ may be chosen in the form $c_{1}(P)=c_{1}\partial_{1}+\cdots+c_{m}\partial_{m}\in\mathcal{A}^{2}(\Delta)$ where $P$ is any convex simple polytope with the normal fan $\Delta$ and support parameters $c_{1},\ldots,c_{m}$.

For complete non-singular fans the algebra $\mathcal{A}^{*}[\Delta]\cong\mathbb{R}[K]/\Theta$ coincides with the cohomology algebra $H^{*}(X_{\Delta};\mathbb{R})$ of the corresponding toric variety. It was the original observation of [Stanley1980], that in the case when a fan $\Delta$ is polytopal, the corresponding complete toric variety $X_{\Delta}$ is projective, therefore there exists a Lefschetz element in its cohomology ring according to hard Lefschetz theorem.

After Stanley’s work, several approaches were developed to prove the existence of Lefschetz elements in elementary terms, i.e. without referring to hard Lefschetz theorem. These approaches include in particular McMullen’s construction of the polytope algebra ([McMullen1993]), the approach based on continuous piece-wise polynomial functions ([Brion1996]), and the approach based on the volume polynomial and differential operators ([Timorin1999]).

We will see that ordinary fans are not the only examples of multi-fans for which the structure of $\mathcal{A}^{*}(\Delta)$ can be explicitly described. On the other hand, $\mathcal{A}^{*}(\Delta)$ is always a Poincare duality algebra, so it is natural to ask if it is Lefschetz (or at least if the dimension vector $(d_{0},d_{1},\ldots,d_{n})$ is unimodal). We will show that this is not true in general, see Theorem 10.1 below.

### 8.2 Combinatorial Preliminaries

For now we concentrate on multi-fans based on oriented pseudomanifolds as described in Example 2.9. Let $K$ be a pure simplicial complex of dimension $n-1$ on the vertex set $[m]$.

Let $f_{j}$ denote the number of $j$-dimensional simplices of $K$ for $j=-1,0,\ldots,n-1$, in particular we assume that $f_{-1}=1$ (this reflects the fact that the empty simplex formally has dimension $-1$). The $h$-numbers of $K$ are defined by the formula:

 $\displaystyle\sum_{j=0}^{n}h_{j}t^{n-j}=\sum_{j=0}^{n}f_{j-1}(t-1)^{n-j},$ (8.1)

where $t$ is a formal variable. Let $\widetilde{\beta}_{j}(K)$ denote the reduced Betti number $\dim\widetilde{H}_{j}(K)$ of $K$. The $h^{\prime}$- and $h^{\prime\prime}$-numbers of $K$ are defined by the formulas

 $\displaystyle h_{j}^{\prime}=h_{j}+{n\choose j}\left(\sum_{s=1}^{j-1}(-1)^{j-s% -1}\widetilde{\beta}_{s-1}(K)\right)\quad\hbox{for }0\leqslant j\leqslant n;$ $\displaystyle h_{j}^{\prime\prime}=h_{j}^{\prime}-{n\choose j}\widetilde{\beta% }_{j-1}(K)=h_{j}+{n\choose j}\left(\sum_{s=1}^{j}(-1)^{j-s-1}\widetilde{\beta}% _{s-1}(K)\right)$ (8.2)

for $0\leqslant j\leqslant n-1$, and $h^{\prime\prime}_{n}=h^{\prime}_{n}$. The sum over an empty set is assumed zero.

### 8.3 Homology Spheres

###### Definition 8.1.

$K$ is called Cohen–Macaulay (over $\Bbbk$), if $\widetilde{H}_{j}(\operatorname{lk}_{K}I;\Bbbk)=0$ for any $I\in K$ and $j<\dim\operatorname{lk}_{K}I=n-1-|I|$. If, moreover, $\widetilde{H}_{n-1-|I|}(\operatorname{lk}_{K}I;\Bbbk)\cong\Bbbk$ for any $I\in K$, then $K$ is called Gorenstein* or (generalized) homology sphere.

The famous theorems of Reisner and Stanley (the reader is referred to the monograph ([Stanley1996])) tell that whenever $K$ is Cohen–Macaulay (resp. Gorenstein*), its Stanley–Reisner algebra $\Bbbk[K]$ is Cohen–Macaulay (resp. Gorenstein).

Given a characteristic function $\lambda\colon[m]\to V\cong\mathbb{R}^{n}$ we obtain a linear system of parameters $\theta_{1},\ldots,\theta_{n}\in\mathbb{R}[K]$. It generates an ideal which we denoted by $\Theta\subset\mathbb{R}[K]$ in Sect. 4.2. In Cohen–Macaulay case every linear system of parameters is a regular sequence. This implies ([Stanley1996]):

 $\displaystyle\dim(\mathbb{R}[K]/\Theta)_{2j}=h_{j}.$

If $K$ is a homology sphere, then $\mathbb{R}[K]$ is Gorenstein. Thus its quotient by a linear system of parameters $\mathbb{R}[K]/\Theta$ is a Gorenstein algebra of Krull dimension zero. This implies that $\mathbb{R}[K]/\Theta$ is a Poincare duality algebra ([Meyer and Smith2005], Part 1).

Now let $\Delta$ be a complete multi-fan based on a homology sphere $K$. We have the ring epimorphism $\mathbb{R}[K]/\Theta\to\mathcal{A}^{*}(\Delta)$ (see Remark 7.6). Since both algebras have Poincare duality, this map is an isomorphism (see Lemma 7.4). This proves the following

###### Theorem 8.2.

Let $\Delta$ be a complete multi-fan based on a homology sphere $K$. Then $\mathcal{A}^{*}(\Delta)\cong\mathbb{R}[K]/\Theta$. It follows that $\dim\mathcal{A}^{2j}(\Delta)=h_{j},$ the $h$-number of $K$.

Note that Poincare duality implies the well-known Dehn–Sommerville relations for homology spheres: $h_{j}=h_{n-j}$.

We are in position to prove Lemma 6.1 which states that the volume polynomial of an elementary multi-fan $\Delta$ on the vectors $\lambda(i)\in V$ $(i=1,\ldots,n+1)$, is equal, up to multiplicative constant, to $(\sum_{i=1}^{n+1}\alpha_{i}c_{i})^{n}$, where $(\alpha_{1},\ldots,\alpha_{n+1})$ is a linear relation on $\lambda(i)$’s.

###### Proof of Lemma 6.1.

The underlying simplicial complex of $\Delta$ is the boundary of a simplex, which is a sphere. Therefore, by Theorem 8.2 we have $\mathcal{A}^{*}(\Delta)\cong\mathbb{R}[\partial\triangle_{[n+1]}]/\Theta$. Hence the ideal $\operatorname{Ann}V_{\Delta}\subset\mathbb{R}[\partial_{1},\ldots,\partial_{n+% 1}]$ is generated by $\prod_{i=1}^{n+1}\partial_{i}$ (Stanley–Reisner relation) and linear differential operators $\theta_{j}=\sum_{i=1}^{n+1}\lambda_{i,j}\partial_{i}$ for $j=1,\ldots,n$. Here $(\lambda_{i,j})_{j=1}^{n}$ are the coordinates of the vector $\lambda(i)$ for each $i=1,\ldots,n+1$. Since $\sum_{i=1}^{n+1}\alpha_{i}\lambda(i)=0$ we have a linear relation $\sum_{i=1}^{n+1}\alpha_{i}\lambda_{i,j}=0$ for each $j=1,\ldots,n$. Now it is easy to check that the differential operators $\prod_{i=1}^{n+1}\partial_{i}$ and $\theta_{j}=\sum_{i=1}^{n+1}\lambda_{i,j}\partial_{i}$, $j=1,\ldots,n$ annihilate the polynomial $(\alpha_{1}c_{1}+\cdots+\alpha_{n+1}c_{n+1})^{n}$. Thus, according to Proposition 7.2, $V_{\Delta}$ coincides with $(\alpha_{1}c_{1}+\cdots+\alpha_{n+1}c_{n+1})^{n}$ up to constant. $\square$

### 8.4 Homology Manifolds

###### Definition 8.3.

$K$ is called Buchsbaum (over $\Bbbk$), if $\widetilde{H}_{j}(\operatorname{lk}_{K}I;\Bbbk)=0$ for any $I\in K$, $I\neq\varnothing$ and $j<\dim\operatorname{lk}_{K}I=n-1-|I|$. If, moreover, $\widetilde{H}_{n-1-|I|}(\operatorname{lk}_{K}I;\Bbbk)\cong\Bbbk$ for any $I\in K$, $I\neq\varnothing$, then $K$ is called a homology manifold. $K$ is called an orientable homology manifold if $\widetilde{H}_{n-1}(K;\mathbb{Z})\cong\mathbb{Z}$.

The difference from the Cohen–Macaulay case is that there are no restrictions on the topology of $K=\operatorname{lk}_{K}\varnothing$ itself. Similar to Cohen–Macaulay property, the term “Buchsbaum” indicates that the corresponding algebra $\Bbbk[K]$ is Buchsbaum (the result of [Schenzel1981]). In Buchsbaum case linear system of parameters is no longer a regular sequence. Nevertheless, Buchsbaum complexes are extensively studied. First, Schenzel’s theorem ([Schenzel1981]) tells that if $K$ is a Buchsbaum complex, then

 $\displaystyle\dim(\Bbbk[K]/\Theta)_{2j}=h^{\prime}_{j}$

for $j=0,\ldots,n$ and the $h^{\prime}$-numbers determined by (8.2). Second, there is a theory of socles of Buchsbaum complexes introduced by [Novik and Swartz2009a] which we briefly review next.

Let $\mathcal{M}$ be a module over the graded polynomial ring $\Bbbk[m]:=\Bbbk[x_{1},\ldots,x_{m}]$. The socle of $\mathcal{M}$ is the following subspace

 $\displaystyle\operatorname{Soc}\mathcal{M}:=\{a\in\mathcal{M}\mid a\cdot\Bbbk[% m]_{+}=0\}.$

which is obviously a $\Bbbk[m]$-submodule of $\mathcal{M}$.

If $K$ is Buchsbaum, then there exists a submodule $I_{NS}\subset\operatorname{Soc}(\Bbbk[K]/\Theta)$ such that

 $\displaystyle(I_{NS})_{2j}\cong{n\choose j}\widetilde{H}^{j-1}(K;\Bbbk),$

where the right hand side means the direct sum of ${n\choose j}$ copies of $\widetilde{H}^{j-1}(K;\Bbbk)$. Moreover, the result of [Novik and Swartz2009b] tells that whenever $K$ is an orientable connected homology manifold, then $I_{NS}$ coincides with $\operatorname{Soc}(\Bbbk[K]/\Theta)$ and the quotient

 $\displaystyle(\Bbbk[K]/\Theta)/I_{NS}^{<2n}$

is a Gorenstein algebra (thus Poincare duality algebra). Here $I_{NS}^{<2n}$ is the part of $I_{NS}$ taken in all degrees except the top one, $2n$. The definition of $h^{\prime\prime}$-numbers (8.2) implies that

 $\displaystyle\dim((\Bbbk[K]/\Theta)/I_{NS}^{<2n})_{2j}=h^{\prime\prime}_{j}% \quad\hbox{for }0\leqslant j\leqslant n.$

Now let $\Delta$ be a complete multi-fan based on an orientable connected homology manifold $K$. Recall from Definition 7.3 that $W^{*}$ denotes the subspace of $H^{*}(\Delta)=\mathbb{R}[K]/\Theta$ whose graded components are

 $\displaystyle W^{2j}$ $\displaystyle=$ $\displaystyle\left\{a\in(\mathbb{R}[K]/\Theta)_{2j}\mid\int a\cdot(\mathbb{R}[% K]/\Theta)_{2n-2j}=0\right\}$ (8.3) $\displaystyle=$ $\displaystyle\left\{a\in(\mathbb{R}[K]/\Theta)_{2j}\mid\int a\cdot\mathbb{R}[m% ]_{2n-2j}=0\right\}.$

By definition, $\mathcal{A}^{*}(\Delta)=\operatorname{PD}(\mathbb{R}[K]/\Theta)=(\mathbb{R}[K]% /\Theta)/W^{*}$. The socle $\operatorname{Soc}(\mathbb{R}[K]/\Theta)$ lies in $W^{*}$ in all degrees except the top one since it is killed by $\mathbb{R}[m]_{+}$. Therefore we have a well-defined ring epimorphism

 $\displaystyle(\mathbb{R}[K]/\Theta)/I_{NS}^{<2n}\twoheadrightarrow\mathcal{A}^% {*}(\Delta).$

Again, since both algebras have Poincare duality, there holds

###### Theorem 8.4.

Let $\Delta$ be a complete multi-fan based on oriented connected homology manifold $K$. Then $\mathcal{A}^{*}(\Delta)\cong(\mathbb{R}[K]/\Theta)/I_{NS}^{<2n}$. It follows that $\dim\mathcal{A}^{2j}(\Delta)=h_{j}^{\prime\prime},$ the $h^{\prime\prime}$-number of $K$.

In this case Poincare duality implies the well-known generalized Dehn–Sommerville relations for oriented homology manifolds: $h_{j}^{\prime\prime}=h_{n-j}^{\prime\prime}$ (see [Novik and Swartz2009a] and references therein).

### 8.5 General Situation

Let $\Delta$ be an arbitrary complete multi-fan. In the largest generality we do not have a combinatorial description for the dimensions of graded components of the multi-fan algebra.

###### Conjecture 8.5.

Let $w_{ch}$ be a simplicial cycle$,$ $\lambda\colon[m]\to V$ a characteristic function$,$ and $\Delta=(w_{ch},\lambda)$ the corresponding complete multi-fan. The numbers $d_{j}=\dim\mathcal{A}^{2j}(\Delta)$ do not depend on $\lambda$.

## 9 Geometry of Multi-Polytopes and Minkowski Relations

Here we give another proof of Theorem 8.4 which shows the geometrical nature of the elements lying in the socle of $\mathbb{R}[K]/\Theta$ when $K$ is an oriented homology manifold. It relates on explicit computations in coordinates but reveals an interesting connection with the Minkowski type relations, appearing in convex geometry. Recall the basic Minkowski theorem on convex polytopes.

###### Theorem 1.

(Minkowski)

1. 1.

$($Direct$)$ Let $P$ be a convex full-dimensional polytope in euclidian space $\mathbb{R}^{n}$. Let $V_{1},\ldots,V_{m}$ be the $(n-1)$-volumes of facets of $P$ and ${\mathbf{n}}_{1},\ldots,{\mathbf{n}}_{m}$ be the outward unit normal vectors to facets. Then $\sum_{i}V_{i}{\mathbf{n}}_{i}=0$ $($the Minkowski relation$).$

2. 2.

$($Inverse$)$ Let ${\mathbf{n}}_{1},\ldots,{\mathbf{n}}_{m}$ be the vectors of unit lengths$,$ spanning $\mathbb{R}^{n},$ and let $V_{1},\ldots,V_{m}$ be positive numbers satisfying the Minkowski relation. Then there exists a convex polytope $P$ whose facets have outward normal vectors ${\mathbf{n}}_{i}$ and volumes $V_{i}$. Such polytope is unique up to parallel shifts.

Usually only part (2) is called Minkowski theorem, since part (1) is fairly simple. The direct Minkowski theorem has a straightforward generalization.

###### Theorem 9.1.

Let $\sum_{s}a_{s}Q_{s}$ be a collection of $k$-dimensional multi-polytopes in euclidian space $\mathbb{R}^{n},$ forming a closed orientable cycle. Let $\operatorname{Vol}(Q_{s})$ be the $k$-volume$,$ and $\nu_{s}\in\Lambda^{n-k}\mathbb{R}^{n}$ be the unit normal skew form of the multi-polytope $Q_{s}$. Then there holds a relation $\sum_{s}a_{s}\operatorname{Vol}(Q_{s})\nu_{s}=0$ in $\Lambda^{n-k}\mathbb{R}^{n}$.

In the next subsection we explain the precise meaning of the terms used in the statement and give the proof.

### 9.1 Cycles of Multi-Polytopes

As before, $V^{*}\cong\mathbb{R}^{n}$ denotes the ambient affine space of $n$-dimensional polytopes, coming with fixed orientation. Let $k\leqslant n$ and let $\Pi$ be an oriented $k$-dimensional affine subspace of $V^{*}$. Let $Q$ be a $k$-dimensional multi-polytope in $\Pi$. Then $Q$ will be called a $k$-dimensional multi-polytope in $V^{*}$.

First let $k>0$. Denote by $G\mathcal{MP}_{k}$ the group (or a vector space over $\mathbb{R}$) freely generated by all $k$-dimensional multi-polytopes in $V^{*}$, where we identify the element $\overline{Q}$ (i.e. $Q$ with reversed orientation of the underlying subspace) and $-Q$. If $k=0$, the multi-polytope is just a point with weight. In this case let $G\mathcal{MP}_{0}$ denote the group of formal sums of points whose weights sum to zero. Formally set $G\mathcal{MP}_{-1}=0$. Define the differential $d\colon G\mathcal{MP}_{k}\to G\mathcal{MP}_{k-1}$ by setting

 $\displaystyle dQ:=\sum_{F_{i}:\,{\rm facet\,of\,}Q}F_{i},$

and extending by linearity. Note that each facet comes with the canonical orientation: we say that the hyperplane $H_{i}$ containing $F_{i}$ is positively oriented if

 $\displaystyle(\hbox{a positive basis of }H_{i},\lambda(i))$

is a positive basis of $V$. Thus the expression above is well defined.

###### Definition 9.2.

An element $A=\sum_{s}a_{s}Q_{s}\in G\mathcal{MP}_{k}$ which satisfies $dA=0$ is called a cycle of $k$-dimensional multi-polytopes.

As in Sect. 5, assume that there is a fixed inner product in $V$. This allows to define the inner product on the skew forms. In particular, if $\Pi$ is an oriented affine $k$-subspace in $V^{*}\cong V$, we may define its unit normal skew form $\nu_{\Pi}\in\Lambda^{n-k}V$ as the unique element of $\Lambda^{n-k}\Pi^{\bot}\cong\mathbb{R}$ which corresponds to the positive orientation of $\Pi^{\bot}$ and satisfies $\|\nu_{\Pi}\|=1$. It is easy to see that if $\dim\Pi=n-1$, the form $\nu_{\Pi}$ is just the positive unit normal vector to $\Pi$.

Let us prove Theorem 9.1.

###### Proof.

The idea of proof is straightforward and quite similar to the proof of classical Minkowski theorem: at first we prove the case $k=n$, then reduce the general case to the case $k=n$ by projecting $\sum_{s}a_{s}\operatorname{Vol}(Q_{s})\nu_{s}$ to all possible $k$-subspaces. Note that the case $n=0$ should be treated separately, but in this case the statement is trivial.

(1) Suppose $k=n$. Then all multi-polytopes $Q_{s}$ are full-dimensional. Their underlying subspaces $\Pi_{s}$ coincide with $V$ up to orientation. Without loss of generality assume that all orientations coincide with that of $V$. Normal skew forms lie in $\Lambda^{0}V\cong\mathbb{R}$ and are equal to 1. Hence we need to prove that $\sum a_{s}\operatorname{Vol}(Q_{s})=0$ for any cycle of $n$-multi-polytopes. Recall the wall-crossing formula ([Hattori and Masuda2003], Lemma 5.3):

###### Lemma 9.3.

Let $P$ be a multi-polytope and $H=H_{i}$ be one of the supporting hyperplanes: $H=H_{i}$. Let $u_{\alpha}$ and $u_{\beta}$ be elements in $V^{*}{\setminus}\bigcup_{i=1}^{m}H_{i}$ such that the segment from $u_{\alpha}$ to $u_{\beta}$ intersects the wall $H$ transversely at $\mu,$ and does not intersect any other $H_{j}\neq H$. Then

 $\displaystyle\operatorname{DH}_{P}(u_{\alpha})-\operatorname{DH}_{P}(u_{\beta}% )=\sum_{i:H_{i}=H}\operatorname{sgn}\langle u_{\beta}-u_{\alpha},\lambda(i)% \rangle\operatorname{DH}_{F_{i}}(\mu),$

where $F_{i}$ is the facet of $P,$ and $\operatorname{DH}_{F_{i}}\colon H_{i}\to\mathbb{R}$ is its Duistermaat–Heckman function.

Consider a cycle of multi-polytopes $A=\sum_{s=1}^{l}a_{s}Q_{s}$. Let $\mathcal{H}$ denote the set of all supporting hyperplanes of all polytopes $Q_{s}$, $s=1,\ldots,l$. We have a function

 $\displaystyle\operatorname{DH}_{A}\colon V^{*}{\setminus}\bigcup_{H\in\mathcal {H}}H\to\mathbb{R},\quad\operatorname{DH}_{A}:=\sum_{s=1}^{l}a_{s} \operatorname{DH}_{Q_{s}}.$

Let us choose a hyperplane $H\in\mathcal{H}$ and two points $u_{\alpha}$ and $u_{\beta}$ in $V^{*}{\setminus}\bigcup_{H\in\mathcal{H}}H$ such that the segment from $u_{\alpha}$ to $u_{\beta}$ intersects the wall $H$ transversely at $\mu$ and does not intersect any other wall from $\mathcal{H}$. Let us sum the differences $\operatorname{DH}_{P}(u_{\alpha})-\operatorname{DH}_{P}(u_{\beta})$ taken with coefficients $a_{s}$ over all multi-polytopes $Q_{s}$ for which $H$ is a supporting hyperplane. Since $dA=0$, Lemma 9.3 implies that this sum is zero. Obviously, this sum equals $\operatorname{DH}_{A}(u_{\alpha})-\operatorname{DH}_{A}(u_{\beta})$.

This argument shows that crossing of any wall does not change the value of $\operatorname{DH}_{A}$. Therefore, $\operatorname{DH}_{A}$ is constant (where it is defined). Since $\operatorname{DH}_{A}$ has compact support, it must be constantly zero. Thus

 $\displaystyle\sum a_{s}\operatorname{Vol}(Q_{s})=\int_{V^{*}}\operatorname{DH}% _{A}=0.$

(2) Let us prove the theorem for general $k$. Consider a generic oriented $k$-subspace $\Pi\subset V^{*}$ and let $\nu\in\Lambda^{n-k}V^{*}$ be its normal skew form. Let $\Gamma\colon V^{*}\to\Pi$ be the orthogonal projection. Then the image of $Q_{s}$ under $\Gamma$ is a full-dimensional multi-polytope in $\Pi$, which we denote by $\Gamma(Q_{s})$. The sum $\sum_{s=1}^{l}a_{s}\Gamma(Q_{s})$ is a cycle of $k$-dimensional multi-polytopes in $\Pi$. Therefore, step (1) implies

 $\displaystyle\sum_{s=1}^{l}a_{s}\operatorname{Vol}(\Gamma(Q_{s}))=0.$

By the standard property of orthogonal projections we have

 $\displaystyle\operatorname{Vol}(\Gamma(Q_{s}))=\operatorname{Vol}(Q_{s})\cdot% \langle\nu_{s},\nu\rangle.$

Hence

 $\displaystyle\left\langle\sum_{s=1}^{l}a_{s}\operatorname{Vol}(Q_{s})\nu_{s},% \nu\right\rangle=0,$

and this holds for any generic skew form $\nu$. Thus $\sum_{s=1}^{l}a_{s}\operatorname{Vol}(Q_{s})\nu_{s}=0$ which was to be proved.$\square$

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