THE SIGNATURE OF A TORIC VARIETY

NAICHUNG CONAN LEUNG AND VICTOR REINER

Abstract. We identify a combinatorial quantity (the alternating sum of the h-vector) defined for any simple polytope as the signature of a toric variety. This quantity was introduced by Charney and Davis in their work, which in particular showed that its non-negativity is closely related to a conjecture of Hopf on the Euler characteristic of a non-positively curved manifold.

We prove positive (or non-negative) lower bounds for this quantity under geometric hypotheses on the polytope, and in particular, resolve a special case of their conjecture. These hypotheses lead to ampleness (or weaker conditions) for certain line bundles on toric divisors, and then the lower bounds follow from calculations using the Hirzebruch Signature Formula.

Moreoever, we show that under these hypotheses on the polytope, the i^{th} L-class of the corresponding toric variety is (−1)^{i} times an effective class for any i.

1. Introduction

Much attention in combinatorial geometry has centered on the problem of characterizing which non-negative integer sequences (f_{0}, f_{1}, …, f_{d}) can be the f-vector f(P) of a d-dimensional convex polytope P, that is, f_{i} is the number of i-dimensional faces of P; see [3] for a nice survey.

For the class of simple polytopes, this problem was completely solved by the combined work of Billera and Lee [4] and of Stanley [33]. A simple d-dimensional polytope is one in which every vertex lies on exactly d edges. McMullen’s gconjecture (now the g-theorem) gives necessary [33] and sufficient [4] conditions for (f_{0}, f_{1}, …, f_{d}) to be the f-vector of a simple d-dimensional polytope. Stanley’s proof of the necessity of these conditions showed that they have a very natural phrasing in terms of the cohomology of the toric variety X_{Δ} associated to the (inner) normal fan Δ of P, and then the Hard Lefschetz Theorem for X_{Δ} played a crucial role. This construction of X_{Δ} from Δ requires that P be rational, i.e. that its vertices all have rational coordinates with respect to some lattice, which can be achieved by a small perturbation that does not affect f(P). Later, McMullen [29] demonstrated that one can construct a ring Π(P), isomorphic (with a doubling of the grading) to the cohomology ring of X_{Δ} if P is rational, and proved that Π(P) formally satisfies the Hard Lefschetz Theorem, using only tools from convex geometry. In particular, he recovered the necessity of the conditions of the g-theorem in this way.

This paper shares a similar spirit with Stanley’s proof. We attempt to use further facts about the geometry of X_{Δ} to deduce information about the f-vector f(P) under certain hypotheses on P. The starting point of our investigation is an interpretation of the alternating sum of the h-vector which follows from the Hard Lefschetz Theorem. Recall that for a simple polytope P, the h-vector is the sequence h(P) = (h_{0}, h_{1}, …, h_{d}) defined as follows. If we let f(P, t) := ∑_{i=0}^{d} f_{i}(P)t^{i}, then

First, second authors partially supported by NSF grants DMS-9803616 and DMS-9877047 respectively.

h(P, t) := ∑_{i=0}^{d} h_{i}(P)t^{i} = f(P, t − 1).

The h-vector has a topological interpretation: h_{i} is the 2i^{th} Betti number for X_{Δ}, or the dimension of the i^{th}-graded component in McMullen’s ring Π(P). Part of the conditions of the g-theorem are the Dehn-Sommerville equations h_{i} = h_{d−i}, which reflect Poincaré duality for X_{Δ}.

Define the alternating sum

\begin{aligned}σ(P) : & = ∑_{i=0}^{d}(−1)^{i}h_{i}(P) \\ [ & = h(P, −1) = f(P, −2) = ∑_{i=0}^{d} f_{i}(P)(−2)^{i} ], \end{aligned}

a quantity which is (essentially) equivalent to one arising in a conjecture of Charney and Davis [6], related to a conjecture of Hopf (see Section 5 below). Note that when d is odd, σ(P) vanishes by the Dehn-Sommerville equations. When d is even, we have the following result (see Section 2).

Theorem 1.1. Let P be a simple d-dimensional polytope, with d even. Then σ(P) is the signature of the quadratic form Q(x) = x^{2} defined on the \frac{d}{2}^{th}-graded component of McMullen’s ring Π(P).

In particular, when P has rational vertices, σ(P) is the signature or index σ(X_{Δ}) of the associated toric variety X_{Δ}.

An important special case of the previously mentioned Charney-Davis conjecture asserts that a certain combinatorial condition on P (namely that of Δ being a flag complex; see Section 5) implies (−1)^{d/2}σ(P) ≥ 0. In this paper, we prove this conjecture when P satisfies certain stronger geometric conditions. We also give further conditions which give lower bounds on (−1)^{d/2}σ(P). In order to state these results, we give rough definitions of some of these conditions here (see Section 3 for the actual definitions).

Say that the fan Δ is locally convex (resp. locally pointed convex, locally strongly convex) if every 1-dimensional cone in Δ has the property that the union of all cones of Δ containing it is convex (resp. pointed convex, strongly convex). For example (see Propositions 4.1 and 4.10 below), if each angle in every 2-dimensional face of P is non-acute (resp. obtuse) then Δ will be locally convex (resp. locally strongly convex). It turns out that Δ being locally convex implies that it is flag (Proposition 5.3).

For a simple polytope P with rational vertices, we define an integer m(P) which measures how singular X_{Δ} is. To be precise, let P in ℝ^{d} be rational with respect to some lattice M, and then m(P) is defined to be the least common multiple over all d-dimensional cones σ of the normal fan Δ of the index [N : N_{σ}], where N is the lattice dual to M and N_{σ} is the sublattice spanned by the lattice vectors on the extremal rays of σ. Note that the condition m(P) = 1 is equivalent to the smoothness of the toric variety X_{Δ}, and such polytopes P are called Delzant in the symplectic geometry literature (e.g. [21]).

Now we can state Theorem 1.2. Let P be a rational simple d-dimensional polytope with d even, and Δ its normal fan.

(i) If Δ is locally convex, then

(−1)^{\frac{d}{2}} σ(P) ≥ 0.

(ii) If Δ is locally pointed convex, then

(−1)^{\frac{d}{2}} σ(P) ≥ \frac{f_{d−1}(P)}{3m(P)^{d−1}} .

(iii) If Δ is locally strongly convex, then

(−1)^{\frac{d}{2}} σ(P) ≥ coefficient of x^{d} in [\frac{t^{d}}{m(P)^{d−1}} f(P, t^{−1})]_{t↦1− \frac{x}{tan(x)}} .

We defer a discussion of the relation between Theorem 1.2 (i) and the Charney-Davis conjecture to Section 5. It is amusing to see what Theorem 1.2 says beyond the g-theorem, in the special case where d = 2, that is, when P is a (rational) polygon. The g-theorem says exactly that

f_{1} = f_{0} ≥ 3,

or in other words, every polygon has the same number of edges as vertices, and this number is at least 3. Since

(−1)^{\frac{d}{2}} σ(P) = f_{0}(P) − 4,

Theorem 1.2 (i) tells us that when Δ is locally convex, we must have f_{0} ≥ 4. In other words, triangles cannot have normal fan Δ which is locally convex, as one can easily check. For d = 2, the conditions that Δ is locally pointed convex or locally strongly convex coincide, and Theorem 1.2 (ii),(iii) both assert that under these hypotheses, a (rational) polygon P must have

f_{0}(P) − 4 ≥ \frac{f_{1}(P)}{3m(P)} = \frac{f_{0}(P)}{3m(P)}

or after a little algebra,

(1)  f_{0}(P) ≥ \frac{12}{3 − \frac{1}{m(P)}} .

Since the right-hand side is strictly greater than 4, we conclude that a quadrilateral P cannot have Δ locally pointed convex nor locally strongly convex. This agrees with an easily-checked fact: a quadrilateral P satisfies the weaker condition of having Δ locally convex if and only if P is a rectangle, and rectangles fail to have Δ locally pointed convex. On the other hand, the inequality (1) also implies a not-quite-obvious fact: even though a (rational) pentagon can easily have Δ locally strongly convex, this is impossible if m(P) = 1, i.e. there are no Delzant pentagons with this property. It is a fun exercise to show directly that no such pentagon exists, and to construct a Delzant hexagon with this property.

In fact, in the context of algebraic geometry, the proof of Theorem 1.2 gives the following stronger assertion, valid for rational simple polytopes of any dimension d (not necessarily even) about the expansion of the total L-class

L(X) = L_{0}(X) + L_{1}(X) + ⋯ + L_{\frac{d}{2}} (X)

where L_{i}(X) is a cycle in CH^{i}(X)_{ℚ}, the Chow ring of X. Theorem 1.3. Let X = X_{Δ} be a complete toric variety X associated to a simplicial fan Δ. If Δ is locally strongly convex (resp. locally convex), then for each i we have that (−1)^{i}L_{i}(X) is effective (resp. either effective or 0).

For instance, when i = 1 this implies that if Δ is locally convex, then

∫_{X}(c_{1}^{2}(X) − 2c_{2}(X)) · H_{1} · . . . · H_{d−2} ≤ 0

where {H_{i}} are any ample divisor classes. This is reminiscent of the Chern number inequality for the complex spinor bundle of X when this bundle is stable with respect to all polarizations; see e.g. [27].

Notice that if Δ is not locally convex, (−1)^{i}L_{i}(X) need not be effective. For example, if Δ is the normal fan of the standard 2-dimensional simplex having vertices at (0, 0), (1, 0), (0, 1), then X is the complex projective plane, and −L_{1} (X) is represented by the negative of the Poincaré dual of a point.

2. The alternating sum as signature

We wish to prove Theorem 1.1, whose statement we recall here.

Theorem 1.1. Let P be a simple d-dimensional polytope, with d even. Then σ(P) is the signature of the quadratic form Q(x) = x^{2}, defined on the \frac{d}{2}^{th}-graded component of McMullen’s ring Π(P).

In particular, when P has rational vertices, σ(P) is the signature or index σ(X_{Δ}) of the associated toric variety X_{Δ}.

Proof. Taking r = \frac{d}{2} in a result of McMullen [29, Theorem 8.6], we find that the

quadratic form (−1)^{\frac{d}{2}} Q(x) on the \frac{d}{2}^{th}-graded component of Π(P) has

∑_{i=0}^{\frac{d}{2}}(−1)^{i}h_{\frac{d}{2} −i}(P) positive eigenvalues, and

∑_{i=0}^{\frac{d}{2} −1}(−1)^{i}h_{\frac{d}{2} −i−1}(P) negative eigenvalues.

Consequently, the signature σ(Q) of Q is

\begin{aligned}σ(Q) & = (−1)^{\frac{d}{2}} [∑_{i=0}^{\frac{d}{2}}(−1)^{i}h_{\frac{d}{2} −i}(P) − ∑_{i=0}^{\frac{d}{2} −1}(−1)^{i}h_{\frac{d}{2} −i−1}(P)] \\ & = ∑_{i=0}^{d}(−1)^{i}h_{i}(P) \end{aligned}

where the second equality uses the Dehn-Sommerville equations [29, §4]:

h_{i}(P) = h_{d−i}(P).

The second assertion of the theorem follows immediately from McMullen’s identification of the ring Π(P) with the quotient of the Stanley-Reisner ring of Δ by a certain linear system of parameters [29, §14], which is known to be isomorphic (after a doubling of the grading) with the cohomology of X_{Δ} [18, §5.2]. □

Remark 2.1. Starting from any complete rational simplicial fan Δ, one can construct a toric variety X_{Δ} which will be complete, but not necessarily projective, and satisfies Poincarè duality. The h-vector for Δ can still be defined, and again has an interpretation as the even-dimensional Betti numbers of X_{Δ} (see [18, §5.2]). We suspect that the alternating sum of the h-vector is still the signature of this complete toric variety.

Generalizing in a different direction, to any polytope P which is not necessarily simple, one can associate the normal fan Δ and a projective toric variety X_{Δ}. Although the (singular) cohomology of X_{Δ} does not satisfy Poincaré duality, its intersection cohomology (in middle perversity) IH^{·}(X_{Δ}) will. There is a combinatorially-defined generalized h-vector which computes these IH^{·} Betti numbers (see [34]). Moreover, using the Hard Lefschetz Theorem for intersection cohomology and the fact that X_{Δ} is a finite union of affine subvarieties, the alternating sum of the generalized h-vector equals the signature of the quadratic form on IH^{·}(X_{Δ}) defined by the intersection product. Remark 2.2. The special case of the second assertion in Theorem 1.1 is known when X_{Δ} is smooth (i.e. P is a Delzant polytope); see [31, Theorem 3.12 (3)].

3. Lower bounds derived from the signature theorem

The goal of this section is to explain the various notions used in Theorem 1.2, and to prove this theorem.

We begin with a d-dimensional lattice M ≅ ℤ^{d} and its associated real vector space M_{ℝ} = M ⊗_{ℤ} ℝ. A polytope P in M_{ℝ} is the convex hull of a finite set of points in M_{ℝ}. We say that P is rational if these points can be chosen to be rational with respect to the lattice M. The dimension of P is the dimension of the smallest affine subspace containing it. A face of P is the intersection of P with one of its supporting hyperplanes, and a face is always a polytope in its own right. Vertices and edges of P are 0-dimensional and 1-dimensional faces, respectively. Every vertex of a d-dimensional polytope lies on at least d edges, and P is called simple if every vertex lies on exactly d edges.

Let N = Hom(M, ℤ) be the dual lattice to M and N_{ℝ} = N ⊗_{ℤ} ℝ be the dual vector space to M_{ℝ}, with the natural pairing M_{ℝ} ⊗ N_{ℝ} → ℝ denoted by ⟨·, ·⟩. For a polytope P in M_{ℝ}, the normal fan Δ is the following collection of polyhedral cones in N_{ℝ}:

Δ = {σ_{F} : F a face of P },

where

σ_{F} := {v ∈ N_{ℝ} : ⟨u, v⟩ ≤ ⟨u′, v⟩ for all u ∈ F, u′ ∈ P }

Note that • the normal fan Δ is a complete fan, that is, it exhausts N_{ℝ},

• Δ is a rational fan, in the sense that its rays all have rational slopes, if P is rational, • if P is d-dimensional, then every cone σ_{F} in Δ will be pointed, that is, it

will contain no proper subspaces of N_{ℝ},

• P is a simple polytope if and only if Δ is a simplicial fan, that is, every cone σ in Δ is simplicial in the sense that its extremal rays are linearly independent.

We next define several affinely invariant conditions on a complete simplicial fan Δ in N_{ℝ} (and hence on simple polytopes P in M_{ℝ}) that appear in Theorem 1.2. For any collection of polyhedral cones Δ in N_{ℝ}, let |Δ| denote the support of Δ, that is the union of all of its cones as a point set. Define the star and link of one of the cones σ in Δ similarly to the analogous notions in simplicial complexes: star_{Δ}(σ) is the subfan consisting of those cones τ in Δ such that σ, τ lie in some common cone of Δ, while link_{Δ}(σ) is the subfan of star_{Δ}(σ) consisting of those cones which intersect σ only at the origin. For a ray (i.e. a 1-dimensional cone) ρ of Δ, say that the fan star_{Δ}(ρ) is

• convex if its support |star_{Δ}(ρ)| is a convex set in the usual sense,

• pointed convex if |star_{Δ}(ρ)| is convex and contains no proper subspace of N_{ℝ},

• strongly convex if furthermore for every cone σ in link_{Δ}(ρ), there exists a linear hyperplane H in N_{ℝ} which supports star_{Δ}(ρ) and whose intersection with star_{Δ}(ρ) is exactly σ.

Say that Δ is locally convex (resp. locally pointed convex, locally strongly convex) if every ray ρ of Δ has star_{Δ}(ρ) convex (resp. pointed convex, strongly convex). One has the easy implications

locally strongly convex ⇒ locally pointed convex ⇒ locally convex .

We recall here that the affine-lattice invariant m(P) for a rational polytope P was defined (in the introduction) to be the least common multiple of the positive integers [N : N_{σ}] as σ runs over all d-dimensional cones in Δ. Here N_{σ} is the d-dimensional sublattice of N generated by the lattice vectors on the d extremal rays of σ. In a sense m(P) measures how singular X_{Δ} is [18, §2.6], with m(P) = 1 if and only if X_{Δ} is smooth, in which case we say that P is Delzant.

We can now recall the statements of Theorems 1.2 and 1.3.

Theorem 1.2. Let P be a simple d-dimensional polytope in M_{ℝ}, which is rational with respect to M, and Δ its normal fan in N_{ℝ}. Assume d is even.

(i) If Δ is locally convex, then

(−1)^{\frac{d}{2}} σ(P) ≥ 0.

(ii) If Δ is locally pointed convex, then

(−1)^{\frac{d}{2}} σ(P) ≥ \frac{f_{d−1}(P)}{3m(P)^{d−1}} .

(iii) If Δ is locally strongly convex, then

(−1)^{\frac{d}{2}} σ(P) ≥ coefficient of x^{d} in [\frac{t^{d}}{m(P)^{d−1}} f(P, t^{−1})]_{t↦1− \frac{x}{tan(x)}} .

Theorem 1.3. Let X = X_{Δ} be a complete toric variety X associated to a simplicial fan Δ, and let the expansion of the total L-class be

L(X) = L_{0}(X) + L_{1}(X) + ⋯ + L_{\frac{d}{2}} (X)

where L_{i}(X) is a cycle in CH^{i}(X)_{ℚ}, the Chow ring of X.

If Δ is locally strongly convex (resp. locally convex), then for each i we have that (−1)^{i}L_{i}(X) is effective (resp. either effective or 0).

The remainder of this section is devoted to the proof of these theorems. We begin by recalling some toric geometry. As a general reference for toric varieties, we rely on Fulton [18], although many of the facts we will use can also be found in Oda’s book [31] or Danilov’s survey article [9].

Let X denote the toric variety X_{Δ}. Simpleness of P implies that X is an orbifold [18, §2.2]. Recall that irreducible toric divisors^{1} on X correspond in a one-to-one fashion with the codimension 1 faces of P, or to 1-dimensional rays in the normal fan Δ. Number these toric divisors on X as D_{1}, ..., D_{m}. Intersection theory for these D_{i}’s is studied in Chapter 5 of [18]. Every D_{i} is a toric variety in its own right with at worst orbifold singularities. Moreover D = ⋃_{i=1}^{m} D_{i} is a simple normal crossing divisor on X [18, §4.3].

Next we want to express the signature of X in terms of these D_{i}’s. When X is a smooth variety, a consequence of the hard Lefschetz Theorem is that its signature σ (X) can be expressed in terms of the Hodge numbers of X as follows [23, Theorem 15.8.2]:

σ (X) = Σ_{p,q=0}^{d} (−1)^{q} h^{p,q} (X) .

By the Dolbeault Theorem, h^{p,q} (X) = dim H^{q}(X, Ω_{X}^{p}), and hence the signature can be expressed in terms of twisted holomorphic Euler characteristics

σ (X) = ∑_{p=0}^{d} χ (X, Ω_{X}^{p})

where χ(X, E) := ∑_{q=0}^{d}(−1)^{q} dim H^{q}(X, E). Using the Riemann-Roch formula, we can write

χ (X, Ω_{X}^{p}) = ∫_{X} ch (Ω_{X}^{p}) T d_{X},

where ch is the Chern character and T d_{X} is the Todd class of X. Therefore

σ (X) = ∫_{X} ∑_{p=0}^{d} ch (Ω_{X}^{p}) T d_{X}.

When X is smooth, ∑_{p=0}^{d} ch (Ω_{X}^{p}) T d_{X} equals the Hirzebruch L-class L (X) of X (see page 16 in [23, Theorem 15.8.2] for example) and we recover the Hirzebruch Signature Formula

(2)  σ (X) = ∫_{X} L (X) .

If X is a projective variety with at worst orbifold singularities, the hard Lefschetz, Dolbeault, and Riemann-Roch Theorems continue to hold, and we can take the sum ∑_{p=0}^{d} ch (Ω_{X}^{p}) T d_{X} as a definition of L (X). Since we can express L (X) in terms of Chern roots of the orbi-bundle Ω_{X}^{1} and Chern classes for orbi-bundles satisfy the same functorial properties as for the usual Chern classes, the same holds true for L (X). For example we will use the splitting principle in the proof of the next lemma, where we write L (X) in terms of toric data.^{2} Lemma 3.1.

1Actually these are ℚ-Cartier divisors on the orbifold X.

\begin{aligned}(−1)^{\frac{d}{2}} σ (X) & = (−1)^{\frac{d}{2}} ∫_{X} L (X) \\ & = ∑_{p=1}^{d/2} ∑_{\begin{matrix}n_{1}+...+n_{p}=d/2 \\ n_{i}>0;i_{1}<...<i_{p} \end{matrix}} b_{n_{1}} ⋯ b_{n_{p}} (−1)^{p} D_{i_{1}}^{2n_{1}} · . . . · D_{i_{p}}^{2n_{p}} \end{aligned}

where D_{j_{1}} · . . . · D_{j_{d}} denotes the intersection number for the d divisors D_{j_{1}}, …, D_{j_{d}} on X, and b_{n} are the coefficients in the expansion

\begin{matrix}√ & \overline{x} \\ tanh & \sqrt{x} \end{matrix} = 1 − ∑_{n=1}^{∞} (−1)^{n} b_{n}x^{n}.

That is, b_{n} = \frac{2^{2n}B_{n}}{(2n)!} where B_{n} is the n^{th} Bernoulli number.

Proof. Recall [30] that the L-class is a multiplicative characteristic class corresponding to the power series \begin{matrix}√ & \overline{x} \\ tanh & \sqrt{x} \end{matrix}, as we are about to explain. In our situation, we need to compute the L-class of a holomorphic orbi-bundle E, namely the tangent orbi-bundle of X. For the purposes of this computation, we can treat E like a genuine vector bundle (see e.g. [8, Appendix A]). We will assume that E can be stably split into a direct sum of line bundles, that is

E ⊕ O_{X}^{⊕(m−d)} ≅ ⨁_{i=1}^{m} L_{i}.

Then c(E) = ∏_{i=1}^{m}(1 + x_{i}) where x_{i} is the first Chern class c_{1}(L_{i}), i.e. the x_{i}’s are stable Chern roots of E. We then have

c(E ⊗_{ℝ} ℂ) = c(E ⊕ \overline{E}) = ∏_{i=1}^{m}(1 + x_{i}) ∏_{j=1}^{m}(1 − x_{j}) = ∏_{i=1}^{m}(1 − x_{i}^{2}).

The L-class is then computed by the formula

L(E) = ∏_{i=1}^{m}(1 − ∑_{n≥1}(−1)^{n}b_{n}x_{i}^{2n})

where b_{n} is the positive number defined in the Lemma.

For example, in terms of Pontrjagin classes of X we have

\begin{aligned}L_{1} (X) & = \frac{1}{3}p_{1} \\ L_{2} (X) & = \frac{1}{45} (7p_{2} − p_{1}^{2}) \\ L_{3} (X) & = \frac{1}{945} (62p_{3} − 13p_{2}p_{1} + 2p_{1}^{3}) . \end{aligned}

2For a general orbifold X, not necessary an algebraic variety, Kawasaki [26] expressed the signature of X in terms of integral of certain curvature forms, thus generalizing the Hirzebruch signature formula in a different way.

To use the Hirzebruch Signature Formula (2), we need to express L (X) in terms of toric data, specifically intersection numbers of the toric divisors D_{1}, ..., D_{m} on X discussed above. To relate these divisors with characteristic classes of X, we consider the exact sequence of sheaves:

0 → Ω_{X}^{1} → Ω_{X}^{1} (log D) ⟶^{Res} ⨁_{i=1}^{m}O_{D_{i}} → 0

where Ω_{X}^{1} (resp. Ω_{X}^{1} (log D)) is the sheaf of differentials on X (resp. differentials on X with logarithmic poles along D). Notice that Ω_{X}^{1} (log D) is a trivial sheaf of rank d [18, §4.3]. On the other hand, there is an exact sequence of sheaves

0 → O_{X} (−D_{i}) → O_{X} → O_{D_{i}} → 0

for each toric divisor D_{i}. From the functorial properties of Chern classes, we have

c (Ω_{X}^{1}) = Π_{i=1}^{m}c (O_{X} (−D_{i})) = Π_{i=1}^{m} (1 − D_{i}) .

Here we have identified a divisor D_{i} with the Poincaré dual of the first Chern class of its associated line bundle. Since Ω_{X}^{1} is the sheaf of sections of the cotangent bundle on X, we can write the total Chern class of (the tangent bundle on) X as

c (E) = Π_{i=1}^{m} (1 + D_{i}) .

Namely these D_{i}’s behave as stable Chern roots of the tangent bundle of X. Therefore, by the multiplicative property of the L-class, we have

L (X) = ∏_{i=1}^{m} (1 − ∑_{n=1}^{∞} (−1)^{n} b_{n}D_{i}^{2n}) .

Expanding the right-hand side of the above equality gives the equality asserted in the lemma. □

To prove Theorem 1.2 we need to give lower bounds on the intersection numbers

(−1)^{p} D_{i_{1}}^{2n_{1}} ⋯ D_{i_{p}}^{2n_{p}}

that appear in the right-hand-side of Lemma 3.1, under our various hypotheses on the fan Δ. We begin by rewriting

(3)  (−1)^{p} D_{i_{1}}^{2n_{1}} ⋯ D_{i_{p}}^{2n_{p}} = ∫_{D_{i_{1}} ∩...∩D_{ip}} (−D_{i_{1}})^{2n_{1}−1} ⋯ (−D_{i_{p}})^{2n_{p}−1} .

This expression leads us to consider the restriction of the line bundles O_{X}(−D_{i}) to the subvarieties D_{i_{1}} ∩ ... ∩ D_{i_{p}}. For any of the irreducible toric divisors D_{i} on X, let O_{D_{i}}(−D_{i}) denote the restriction of O_{X}(−D_{i}) to the toric subvariety D_{i} (this is the conormal bundle of D_{i} in X). Recall that for an invertible sheaf O(E) on a d-dimensional orbifold X, one says that O(E) is big if the corresponding divisor E satisfies E^{d} > 0. The key observation in obtaining the desired lower bounds is then Lemma 3.2. Let P be a simple d-dimensional polytope in M_{ℝ}, which is rational with respect to M, and Δ its normal fan in N_{ℝ}. Let D_{i} be any of the irreducible toric divisors on X = X_{Δ}.

(i) If Δ is locally convex, then O_{D_{i}}(−D_{i}) is generated by global sections. (ii) If Δ is locally pointed convex, then O_{D_{i}}(−D_{i}) is generated by global sections

and big. (iii) If Δ is locally strongly convex, then O_{D_{i}}(−D_{i}) is ample.

Assuming Lemma 3.2 for the moment, we finish the proof of Theorem 1.2.

Proof of Theorem 1.2 (i). Under the assumption that Δ is locally convex, we know that the restriction of O_{X} (−D_{i_{j}}) to D_{i_{1}} ∩...∩D_{i_{p}} is generated by global sections for 1 ≤ j ≤ p by Lemma 3.2. This implies that the integral (3) equals the intersection number of such divisors on the toric subvariety D_{i_{1}} ∩ ... ∩ D_{i_{p}} and therefore it is nonnegative ^{3} , that is

(−1)^{p} D_{i_{1}}^{2n_{1}} ⋯ D_{i_{p}}^{2n_{p}} ≥ 0.

The non-negativity asserted in Theorem 1.2 (i) now follows term-by-term from the sum in Lemma 3.1. □

Proof of Theorem 1.2 (ii). If Δ is locally pointed convex, then O_{D_{i}} (−D_{i}) is generated by global sections and big. The bigness of O_{D_{i}} (−D_{i}) on D_{i} implies that

−D_{i}^{d} = ∫_{D_{i}} (−D_{i})^{d−1}

is strictly positive. Claim. −D_{i}^{d} ≥ \frac{1}{m(P)^{d−1}} . To prove this, we proceed as in the algebraic moving lemma [18, §5.2, p. 107], making repeated use of the fact that if n_{j} is the first non-zero lattice point on the ray of Δ corresponding to D_{j}, then for any u in M, one has

(4)  ∑_{j}⟨u, n_{j}⟩D_{j} = 0

in the Chow ring [18, Proposition, Part (ii), §5.2, p. 106]. This allows one to take intersection monomials that contain some divisor D_{j_{0}} raised to a power greater than 1, and replace one factor of D_{j_{0}} by a sum of other divisors. By repeating this process for all of the monomials in a total of d − 1 stages, one can replace D_{i}^{d} by a sum of the form ∑ a_{j_{1},...,j_{d}}D_{j_{1}} ⋯ D_{j_{d}} in which each term has D_{j_{1}}, …, D_{j_{d}} distinct divisors which intersect at an isolated point of X, and each a_{j_{1},...,j_{d}} is a rational number. We must keep careful track of the denominators of the coefficients introduced at each stage.

At the first stage, by choosing any u in M with ⟨u, n_{i}⟩ = 1, we can use (4) to replace one factor of D_{i} in D_{i}^{d} by a sum of other divisors D_{j} with integer coefficients (that is, introducing no denominators). However, in each of the next d − 2 stages, when one wishes to use (4) to substitute for a divisor D_{j_{0}}, one must choose u in M constrained to vanish on normal vectors n_{j} for other divisors D_{j} in the monomial, and this may force the coefficient ⟨u, n_{j_{0}}⟩ of D_{j_{0}} to be larger than 1 in (4), although it will always be an integer factor of m(P). Consequently, at each stage after the first, we may introduce factors into the denominators that divide into m(P). Since there are d − 2 stages after the first, we conclude that each a_{j_{1},...,j_{d}} can be written with the denominator m(P)^{d−2}. Finally, each intersection product D_{j_{1}} ⋯ D_{j_{d}} is the reciprocal of the multiplicity at the corresponding point of X, which is the index [N : N_{σ}] where σ is the d-dimensional cone of Δ corresponding to that point [18, §2.6]. Since each [N : N_{σ}] divides m(P), we conclude that −D_{i}^{d} lies in \frac{1}{m(P)^{d−1}} ℤ, and since it is positive, it is at least \frac{1}{m(P)^{d−1}} .

3This follows from the fact that the divisor class of a line bundle which is generated by global sections is a limit of ℚ-divisors which are ample. Positivity of intersection numbers of ample divisors is well-known; see e.g. [19, Chapter 12]

We have shown then that each term with p = 1 on the right-hand side of Lemma 3.1 is at least \frac{1}{m(P)^{d−1}} , and the number of such terms is the number of codimension one faces of P, i.e. f_{d−1}(P). Moreover we still have nonnegativity of the other terms (−1)^{p} D_{i_{1}}^{2n_{1}} ⋯ D_{i_{p}}^{2n_{p}} because O_{D_{i}} (−D_{i}) is generated by global sections. Therefore, since b_{1} = \frac{1}{3}, we conclude from Lemma 3.1 that

(−1)^{d/2} σ (Δ) ≥ \frac{f_{d−1}}{3m(P)^{d−1}} . □

Proof of Theorem 1.2 (iii). If Δ is locally strongly convex, then O_{D_{i}} (−D_{i}) is ample. By similar arguments as in assertions (i) and (ii), we have

(−1)^{p} D_{i_{1}}^{2n_{1}} ⋯ D_{i_{p}}^{2n_{p}} ≥ \frac{1}{m(P)^{d−1}}

provided that D_{i_{1}} ∩ ... ∩ D_{i_{p}} is non-empty. By the simplicity of P, each of its codimension p faces can be expressed uniquely as the intersection of distinct codimension one faces, corresponding to the non-empty intersection of divisors D_{i_{1}}, …, D_{i_{p}}. Therefore, after choosing positive integers n_{1}, …, n_{p}, the number of non-vanishing terms of the form (−1)^{p} D_{i_{1}}^{2n_{1}} ⋯ D_{i_{p}}^{2n_{p}} in the expansion of Lemma 3.1 is f_{d−p}(P). Hence

(−1)^{\frac{d}{2}} σ (X) = (−1)^{\frac{d}{2}} ∫_{X} L (X)

= ∑_{p=1}^{d/2} (−1)^{p} ∑_{\begin{matrix}n_{1}+...+n_{p}=d/2 \\ n_{i}>0;i_{1}<...<i_{p} \end{matrix}} b_{n_{1}} ⋯ b_{n_{p}}D_{i_{1}}^{2n_{1}} ⋯ D_{i_{p}}^{2n_{p}}

≥ ∑_{p=1}^{d/2} ∑_{\begin{matrix}n_{1}+...+n_{p}=d/2 \\ n_{i}>0 \end{matrix}} b_{n_{1}} ⋯ b_{n_{p}} \frac{f_{d−p}(P)}{m(P)^{d−1}}

= ∑_{p=1}^{d/2} \frac{f_{d−p}(P)}{m(P)^{d−1}} [ coefficient of x^{d} in (∑_{n≥1} b_{n}x^{2n})^{p} ] .

Note that

\begin{matrix}√ & \overline{x} \\ tanh & \sqrt{x} \end{matrix} = 1 − ∑_{n≥1}(−1)^{n}b_{n}x^{n}

implies

∑_{n≥1} b_{n}x^{2n} = 1 − \frac{x}{tan(x)},

and note also that

∑_{p≥1} f_{d−p}(P) t^{p} = t^{d}f(P, t^{−1}).

This allows us to rewrite the above inequality as in the assertion of Theorem 1.2 (iii). □

Proof of Theorem 1.3 . Recall from the proof of Lemma 3.1 that the total L-class has expansion

L(X) = ∑_{p≥1} (−1)^{p} ∑_{\begin{matrix}(n_{1},...,n_{p}) \\ n_{i}>0;i_{1}<...<i_{p} \end{matrix}}(−1)^{∑ n_{i}}b_{n_{1}} ⋯ b_{n_{p}}D_{i_{1}}^{2n_{1}} ⋯ D_{i_{p}}^{2n_{p}}.

Consequently,

(−1)^{j}L_{j}(X) = ∑_{p≥1} (−1)^{p} ∑_{\begin{matrix}n_{1}+···+n_{p}=j \\ n_{i}>0;i_{1}<...<i_{p} \end{matrix}} b_{n_{1}} ⋯ b_{n_{p}}D_{i_{1}}^{2n_{1}} ⋯ D_{i_{p}}^{2n_{p}}.

Therefore it suffices to show that each term

(−1)^{p} D_{i_{1}}^{2n_{1}} ⋯ D_{i_{p}}^{2n_{p}}

is effective if Δ is locally strongly convex (the case where Δ is locally convex is similar). Here we use the fact from Lemma 3.2 that restriction of O(−D_{i_{k}}) to D_{i_{k}} is ample, and therefore also ample when further restricted to the transverse intersection V = D_{i_{1}} ∩ ⋯ ∩ D_{i_{p}}. Consequently, the cycle class

(−D_{i_{1}})^{2n_{1}−1} ⋯ (−D_{i_{p}})^{2n_{p}−1}

is effective in the Chow ring CH(V)_{ℚ} by Bertini’s Theorem. Therefore (−1)^{p} D_{i_{1}}^{2n_{1}} · · · D_{i_{p}}^{2n_{p}} is effective in CH(X)_{ℚ}. □

Proof of Lemma 3.2 . We recall some facts about toric divisors contained generally in [18, §3.3, 3.4]. In general, any divisor E on X defines a continuous piecewise linear function Ψ_{E}^{X} on the support |Δ| = N_{ℝ}. Every divisor E on X is linearly equivalent to a linear combination of irreducible toric divisors. If we write E = ∑_{i=1}^{m} a_{i}D_{i}, then Ψ_{E}^{X} is determined by the property that Ψ_{E}^{X} (n_{i}) = −a_{i} where n_{i} is the first nonzero lattice point of N along ρ_{i}. In particular, Ψ_{−D_{i}}^{X} : N_{ℝ} → ℝ is determined by Ψ_{−D_{i}}^{X} (n_{j}) = δ_{ij}. The ampleness of the line bundle O_{X} (E) can be measured by the convexity of the piecewise linear function Ψ_{E}^{X}. More precisely, O_{X} (E) is ample (resp. generated by global sections) if and only if Ψ_{E}^{X} is strictly convex (resp. convex).

We now discuss assertions (i), (iii) of the lemma, leaving (ii) for later. First we examine the particular case of the discussion in the previous paragraph where the bundle is O_{D_{i}}(−D_{i}) on the toric subvariety D_{i}. Assume the divisor D_{i} corresponds to a ray ρ_{i} in the normal fan Δ. The fan Δ^{D_{i}} associated to D_{i} naturally lives in the quotient space N_{ℝ}/ρ_{i} (here we are abusing notation by letting ρ_{i} denote both a ray and also the 1-dimensional subspace spanned by this ray) [18, §3.1]. Then every cone in Δ^{D_{i}} corresponds to a cone in Δ containing ρ_{i} as a face (and vice-versa), that is, a cone in star_{Δ}(ρ_{i}). The boundary of star_{Δ}(ρ_{i}) is link_{Δ}(ρ_{i}), and here we use the fact, proven in the Appendix, that link_{Δ}(ρ_{i}) is affinely equivalent to the graph of the continuous piecewise linear function Ψ_{−D_{i}}^{D_{i}} : N_{ℝ}/ρ_{i} → ℝ. From the discussion in the previous paragraph, we conclude that O_{D_{i}}(−D_{i}) is generated by global sections (resp. ample) if Δ is locally convex (resp. locally strongly convex).

Lastly we discuss asertion (ii) of the lemma. We want to prove that O_{D_{i}} (−D_{i}) is big for every irreducible toric divisor D_{i} under the assumption that Δ is locally pointed convex. The fact that Δ is locally pointed convex says that the space |star_{Δ}(ρ_{i})| is a pointed convex polyhedral cone. There is then a unique fan Σ having the following properties:

• Σ is refined by star_{Δ}(ρ_{i}), and they have the same support, that is,

|Σ| = |star_{Δ}(ρ_{i})|,

• ρ_{i} is the only ray in the interior of Σ, and

• Σ is strongly convex in the sense that that every ray of Σ except for ρ_{i} is

the intersection of |Σ| with some supporting hyperplane. This cone Σ projects to a complete fan Δ̄^{D_{i}} in N_{ℝ}/ρ_{i}, which is refined by Δ^{D_{i}}.

Therefore we obtain a birational morphism [18, §1.4].

π : D_{i} = X_{Δ^{D_{i}}} → XΔ̄^{D_{i}} .

Moreover Ψ_{−D_{i}}^{D_{i}} still defines a continuous piecewise linear function on N_{ℝ}/ρ_{i} , the support of Δ̄^{D_{i}}, which is now strongly convex. Therefore it defines an ample Cartier divisor on XΔ̄^{D_{i}}. Call this divisor C. Then it is not difficult to see that O_{D_{i}} (−D_{i}) is just the pullback of O_{XΔ̄^{D_{i}}} (C). Moreover, their self-intersection numbers are equal, that is

(−D_{i})^{d−1} = C^{d−1}.

Now C is an ample divisor on XΔ̄^{D_{i}} and therefore C^{d−1} is strictly positive. Hence the same is true for −D_{i}. That is, O_{D_{i}}(−D_{i}) is a big line bundle on the toric subvariety D_{i}. This completes the proof of Lemma 3.2. □

The previous proof raises the following question: is the assumption of rationality for the simple polytope P really necessary in Theorem 1.2? In approaching this problem, it would be interesting if Lemma 3.1 and the intersection numbers that appear within it have some interpretation purely within the convexity framework used by McMullen [29].

4. Examples

In this section we discuss examples of simple polytopes P whose normal fans Δ satisfy the hypotheses of Theorem 1.2.

We begin with some properties of P that are Euclidean invariants, so we will assume that M_{ℝ} is endowed with a (positive definite) inner product ⟨·, ·⟩ which identifies M_{ℝ} with its dual space N_{ℝ}. Thus we can think of both P and its normal fan Δ as living in M_{ℝ}.

Say that a polytope P is non-acute in codimension 1 (resp. obtuse in codimension 1) if every codimension 2 face of P has the property that the dihedral angle between the two codimension 1 faces meeting there is non-acute (resp. obtuse), that is, at least (resp. greater than) \frac{π}{2} . Say that P is non-acute (resp. obtuse) if P and every one of its faces of each dimension considered as polytopes in their own right are non-acute (resp. obtuse) in codimension 1. We have the following obvious implications:

obtuse ⇒

obtuse in codimension 1 and non-acute ⇒ non-acute .

The next proposition shows that these Euclideanly invariant conditions on P imply the affinely invariant conditions on Δ defined in the Section 3. Proposition 4.1. Let P be a simple d-dimensional polytope in M_{ℝ}, with normal fan Δ.

(i) If P is non-acute, then Δ is locally convex.

(ii) If P is obtuse in codimension 1 and non-acute, then Δ is locally pointed

convex. (iii) If P is obtuse, then Δ is locally strongly convex. The next corollary then follows immediately from the previous proposition and

Theorem 1.2. Corollary 4.2. Let P be a simple, rational d-dimensional polytope in M_{ℝ}, with normal fan Δ.

(i) If P is non-acute, then

(−1)^{\frac{d}{2}} σ(P) ≥ 0.

(ii) If P is obtuse in codimension 1, and non-acute, then

(−1)^{\frac{d}{2}} σ(P) ≥ \frac{f_{d−1}(P)}{3m(P)^{d−1}} .

(iii) If P is obtuse, then

(−1)^{\frac{d}{2}} σ(P) ≥ coefficient of x^{d} in [\frac{t^{d}}{m(P)^{d−1}} f(P, t^{−1})]_{t↦1− \frac{x}{tan(x)}} .

Remark 4.3. It is easy to see that obtuse simple polytopes can always be made rational without changing their facial structure by a slight perturbation of their facets, so that one might think of removing the rationality assumption from part (iii) of the previous corollary. However, after this perturbation it is not clear what the lattice-invariant m(P) will be, i.e. it could be any positive integer.

It is not obvious whether a non-acute, simple polytope always has the same facial structure as a rational, non-acute, simple polytope. This would follow if every nonacute, simple polytope had the same facial structure as an obtuse, simple polytope, but this is false. For example, a regular 3-dimensional cube is non-acute and simple, but no obtuse polytope can have the facial structure of a 3-cube. Remark 4.4. M. Davis has pointed out to us that the first assertion of Corollary 4.2 can be proven using facts from [6] and the mirror construction M(P) of the next section, without any assumption that P is rational. We defer a sketch of this proof until the description of M(P) at the end of that section.

Proof of Proposition 4.1. We begin by rephrasing some of our definitions about non-acuteness and obtuseness in terms of Δ. Obtuseness (resp. non-acuteness) in codimension 1 for P corresponds to the following property of Δ: any two vectors n, n′ spanning the extremal rays of a 2-dimensional cone of Δ must have

⟨n, n′⟩ > 0 ( resp. ⟨n, n′⟩ ≥ 0).

Similarly, obtuseness (resp. non-acuteness) for P corresponds to the following property of Δ: any vectors n_{1}, …, n_{t} spanning the extremal rays of a t-dimensional (simplicial) cone of Δ must have

⟨π(n_{1}), π(n_{2})⟩ > 0 ( resp. ⟨π(n_{1}), π(n_{2})⟩ ≥ 0)

where π is the orthogonal projection onto the space perpendicular to the span of the vectors n_{3}, n_{4}, …, n_{t}.

Having said this, observe that if P is non-acute in codimension 1, for any ray ρ in Δ (spanned by a vector which we name n), the hyperplane ρ^{⊥} normal to ρ supports star_{Δ}(ρ): if P is non-acute in codimension 1, we must have ⟨n, n′⟩ ≥ 0 for each vector n′ spanning a ray in star_{Δ}(ρ), and hence for every vector in star_{Δ}(ρ). Similarly, if P is obtuse in codimension 1 then this hyperplane ρ^{⊥} not only supports star_{Δ}(ρ), but also intersects it only in the origin. Consequently, assertion (ii) of the lemma follows once we prove assertion (i).

For assertions (i), (iii), we make use of the fact that strong or weak convexity of star_{Δ}(ρ) can be checked locally in a certain way, similar to checking regularity of triangulations (see e.g. [16, §1.3]). Roughly speaking, each cone σ in the link of ρ must have the property that the union of cones containing σ within link_{Δ}(n) “bend outwards” at σ away from ρ, rather than “bending inward” toward ρ. To be more formal, consider every minimal dependence of the form

(5)  ∑_{i∈F} α_{i}n_{i} = βn + ∑_{j∈G} β_{j}m_{j}

where - {n_{i}}_{i∈F} are vectors spanning the extremal rays of some cone σ in link_{Δ}(n),

- each m_{j} for f in G spans a ray in link_{Δ}(σ),

- the coefficients α_{i}, β_{j} are all strictly positive. Then star_{Δ}(ρ) is strictly convex if and only in every such dependence we have

β < 0. It is weakly convex if and only if in every such dependence we have β ≤ 0.

As a step toward proving assertions (i), (iii), given a dependence as in (5) we apply the orthogonal projection π onto the space perpendicular to all of the {n_{i}}_{i∈F}, yielding the following equation

0 = β π(n) + ∑_{j∈G} β_{j} π(m_{j}),

and then taking the inner product with π(n) on both sides yields

(6)  0 = β ⟨π(n), π(n)⟩ + ∑_{j∈G} β_{j} ⟨π(m_{j}), π(n)⟩.

To prove (iii), we assume P is obtuse and that there is some choice of a dependence as in (5) such that β ≥ 0. But then we reach a contradiction in Equation (6), because we assumed β_{j} > 0, we have ⟨π(m_{j}), π(n)⟩ > 0 by virtue of the obtuseness of P, and ⟨π(n), π(n)⟩ is always non-negative.

To prove (i), we assume P is non-acute and that there is some choice of a dependence as in (5) such that β > 0. Then similar considerations in equation (6) imply that we must have ⟨π(n), π(n)⟩ = 0, i.e. π(n) = 0. This would imply ⟨n_{i}, n⟩ = 0 for each i in F. To reach a contradiction from this, take the inner product with n on both sides of equation (5), to obtain

0 = β ⟨n, n⟩ + ∑_{j∈G} β_{j} ⟨m_{j}, n⟩.

Non-acuteness (even in codimension 1) of P implies ⟨m_{j}, n⟩ ≥ 0, and ⟨n, n⟩ is always positive, so this last equation is a contradiction to β > 0. □

One source of non-acute simple polytopes are finite Coxeter groups (see [24, Chapter 1] for background). Recall that a finite Coxeter group is a finite group W acting on a Euclidean space and generated by reflections. Given a finite Coxeter group W, there is associated a set of (normalized) roots Φ by taking all the unit normals of reflecting hyperplanes. Let Z be the zonotope ( [36, §7.3]) associated with Φ, that is,

Z = {∑_{α∈Φ} c_{α}α : 0 ≤ c_{α} ≤ 1} .

Proposition 4.5. The zonotope Z associated to any finite Coxeter group W is nonacute and simple. Furthermore Z is obtuse in codimension 1 if W is irreducible.

Proof. We refer to [24] for all facts about Coxeter groups used in this proof.

By general facts about zonotopes [36, §7.3], the normal fan Δ of Z is the complete fan cut out by the hyperplanes associated with reflections in W. The maximal cones in this fan are the Weyl chambers of W, which are all simplicial cones. Hence Z is a simple polytope.

To show that Z is non-acute, we must show that each of its faces is non-acute in codimension 1. However, these faces are always affine translations of Coxeter zonotopes corresponding to standard parabolic subgroups of W. So we only need to show Z itself is non-acute in codimension 1. This is equivalent to showing that every pair of rays in Δ which span a 2-dimensional cone form a non-obtuse angle.

Because W acts transitively on the Weyl chambers in Δ, we may assume that this pair of rays lie in the fundamental Weyl chamber, that is, we may assume that these rays come from the dual basis to some choice of simple roots α_{1}, …, α_{d}. Since every choice of simple roots has the property that ⟨α_{i}, α_{j}⟩ ≤ 0 for all i ≠ j, the first assertion follows from the first part of Lemma 4.6 below.

The second assertion follows from Lemma 4.6 (ii) below. This is because the obtuseness graph for any choice of simple roots associated with a Coxeter group W is isomorphic to the (unlabelled) Coxeter graph, and the Coxeter graph is connected exactly when W is irreducible. □

The following lemma was used in the preceding proof.

Lemma 4.6. Let {α_{i}}_{i=1}^{d} be a basis for ℝ^{d} with ⟨α_{i}, α_{j}⟩ ≤ 0 for all i ≠ j. Then the dual basis {α_{i}^{∨}}_{i=1}^{d} satisfies

(i) ⟨α_{i}^{∨}, α_{j}^{∨}⟩ ≥ 0 for all i ≠ j, and

(ii) ⟨α_{i}^{∨}, α_{j}^{∨}⟩ > 0 for all i ≠ j if the “obtuseness graph” on {1, 2, …, d}, having

an edge {i, j} whenever ⟨α_{i}, α_{j}⟩ < 0, is connected.

Proof. We prove assertion (i) by induction on d, with the cases d = 1, 2 being trivial. In the inductive step, assume d ≥ 3. Without loss of generality, we must show ⟨α_{1}, α_{2}⟩ ≥ 0.

Let π : ℝ^{d} → α_{d}^{⊥} be orthogonal projection. Write

α_{i} = π(α_{i}) + c_{i}α_{d}

for each i ≤ d − 1.

Our first claim is that c_{i} ≤ 0 for each i ≤ d − 1. To see this, note that

0 ≥ ⟨α_{i}, α_{d}⟩
= ⟨π(α_{i}), α_{d}⟩ + c_{i}⟨α_{d}, α_{d}⟩
= c_{i}⟨α_{d}, α_{d}⟩.

Our second claim is that ⟨π(α_{i}), π(α_{j})⟩ ≤ 0 for 1 ≤ i ≠ j ≤ d − 1. To see this, note that

0 ≥ ⟨α_{i}, α_{j}⟩
= ⟨π(α_{i}), π(α_{j})⟩ + c_{j}⟨α_{d}, π(α_{j})⟩ + c_{i}⟨π(α_{i}), α_{d}⟩ + c_{i}c_{j}⟨α_{d}, α_{d}⟩
= ⟨π(α_{i}), π(α_{j})⟩ + c_{i}c_{j}⟨α_{d}, α_{d}⟩.

and the last term in the last sum is non-negative by our first claim.

Our third claim is that {π(α_{i})}_{i=1}^{d−1} and {α_{i}^{∨}}_{i=1}^{d−1} are dual bases inside α_{d}^{⊥}. To see this, note that

δ_{ij} = ⟨α_{i}, α_{j}^{∨}⟩
= ⟨π(α_{i}), α_{j}^{∨}⟩ + c_{i}⟨α_{d}, α_{j}^{∨}⟩
= ⟨π(α_{i}), α_{j}^{∨}⟩.

From the second and third claims, we can apply induction to conclude that ⟨α_{i}^{∨}, α_{j}^{∨}⟩ ≥ 0 for 1 ≤ i ≠ j ≤ d − 1, and in particular this holds for i = 1, j = 2 as desired.

To prove assertion (ii), we use the same induction on d, with the cases d = 1, 2 still being trivial. We must in addition show that if {α_{i}}_{i=1}^{d} have connected obtuseness graph, then there is a re-indexing (that is a choice of α_{d}) so that {π(α_{i})}_{i=1}^{d−1} also satisfies this hypothesis. To achieve this, let α_{d} correspond to a node d in the obtuseness graph whose removal does not disconnect it, e.g. choose d to be a leaf in some spanning tree for the graph. Then for i ≠ j with i, j ≤ d − 1 we have

⟨π(α_{i}), π(α_{j})⟩ = ⟨α_{i}, α_{j}⟩ − c_{i}c_{j}⟨α_{d}, α_{d}⟩.

This implies π(α_{i}), π(α_{j}) were obtuse whenever α_{i}, α_{j} were, so the obtuseness graph remains connected. □

Remark 4.7. If the finite Coxeter group W is crystallographic (or a Weyl group) then a crystallographic root system associated with W gives a more natural choice of hyperplane normals to use than the unit normals in defining the Coxeter zonotope Z. With this choice, the normal fan Δ is not only rational with respect to the weight lattice N, but also m(Z) = 1 with respect to the dual lattice M. Hence Z is Delzant, so that the toric variety X_{Δ} is smooth.

For the classical Weyl groups W of types A, B(= C), D, there are known generating functions for the h-vectors of the associated Coxeter zonotopes Z, which specialize to give explicit generating functions for the signature σ(Z). The h-vector in this case turns out to give the distribution of the elements of the Weyl group W according to their descents, i.e. the number of simple roots which they send to negative roots (see [5]). Generating functions for the descent distribution of all classical Weyl groups may be found in [32]. For example, it follows from these that if Z_{A_{n−1}} is the Coxeter zonotope of type A_{n−1}, then we have the formula

∑_{n≥0} σ(Z_{A_{n−1}})\frac{x^{n}}{n!} = tanh(x)

which was computed in [17, Example p. 52] for somewhat different reasons.

The fact that Coxeter zonotopes have locally convex normal fans also follows because these normal fans come from simplicial hyperplane arrangements (we thank M. Davis for suggesting this). Say that an arrangement of linear hyperplanes in ℝ^{d} is simplicial if it decomposes ℝ^{d} into a simplicial fan. Proposition 4.8. The fan Δ associated to a simplicial hyperplane arrangement is locally convex.

Proof. For each ray ρ of Δ, we will express star_{Δ}(ρ) as an intersection of closed half-spaces defined by a subset of the hyperplanes of A, thereby showing that it is convex. To describe this intersection, note that since Δ is simplicial, given any chamber (d-dimensional cone) σ of Δ that contains ρ, there is a unique hyperplane H_{σ} bounding σ which does not contain ρ. Choose a linear functional u_{σ} which vanishes on H_{σ} and is positive on ρ, and then we claim that

|star_{Δ}(ρ)| = ⋂_{chambers σ⊃ρ}{u_{σ} ≥ 0}.

To see that the left-hand side is contained in the right, note that for any chamber σ containing ρ and any hyperplane H in A not containing ρ, we must have σ and ρ on the same side of H. Consequently, for every pair of chambers σ, σ′ containing ρ we have u_{σ} ≥ 0 on σ′ (and symmetrically u_{σ′} ≥ 0 on σ). This implies the desired inclusion.

To show that the right-hand side is contained in the left, since both sets are closed and d-dimensional, it suffices to show that every chamber in the left is contained in the right, or contrapositively, that every chamber not contained in the right is not in the left. Given a chamber σ not in the right, consider the unique chamber σ′ containing ρ which is “perturbed in the direction of σ”. In other words, σ′ is chosen so that it contains a vector v + ϵw where v is any non-zero vector in ρ, w is any vector in the interior of σ, and ϵ is a very small positive number. Since σ does not contain ρ, we know σ ≠ σ′, and hence there is at least one hyperplane of A separating them. Since Δ is simplicial, every bounding hyperplane of σ′ except for H_{σ′} will contain r, and hence have σ and σ′ on the same side (by construction of σ′). This means H_{σ′} must separate σ and σ′, so u_{σ′} < 0 on σ, implying σ is not in the left-hand side. □

The Coxeter zonotopes of type A are related to another infinite family of simple polytopes, the associahedra, which turn out to have locally convex normal fans. Recall [28] that the associahedron A_{n} is an (n − 3)-dimensional polytope whose vertices correspond to all possible parenthesizations of a product a_{1}a_{2} ⋯ a_{n−1}, and having an edge between two parenthesizations if they differ by a single “rebracketing”. Equivalently, vertices of A_{n} correspond to triangulations of a convex n-gon, and there is an edge between two triangulations if they differ only by a ”diagonal flip” within a single quadrilateral. Proposition 4.9. The associahedron A_{n} has a realization as a simple convex polytope whose normal fan Δ_{n} is locally convex.

Proof. In [28, §3], the normal fan Δ_{n} is thought of as a simplicial complex, and more precisely, as the boundary of a simplicial polytope Q_{n} having the origin in its interior. There Q_{n} is constructed by a sequence of stellar subdivisions of certain faces of an (n − 3)-simplex having vertices labelled 1, 2, …, n − 2. Since the normal fan Δ_{n} is simplicial, the associahedron is simple (as is well-known).

Our strategy for showing Δ_{n} is locally convex is to relate it to the Weyl chambers of type A_{n−3}. If we assume that the (n − 3)-simplex above is regular, and take its barycenter as the origin in ℝ^{n−3}, then the barycentric subdivision of its boundary is a simplicial polytope isomorphic to the Coxeter complex for type A_{n−3}. Hence the normal fan Δ_{n} of A_{n} refines the fan of Weyl chambers for type A_{n−3}. Note that an alternate description of this Weyl chamber fan is that it is the set of all chambers cut out by the hyperplanes x_{i} = x_{j}, that is, its (open) chambers are defined by inequalities of the form x_{π_{1}} > x_{π_{2}} > ⋯ > x_{π_{n−2}} for permutations π of {1, 2, …, n − 2}.

To show Δ_{n} is locally convex, we must first identify the rays ρ of Δ_{n}, and then show that star_{Δ_{n}}(ρ) is a pointed convex cone. According to the construction of [28, §3], a ray ρ_{ij} of Δ_{n} corresponds to the barycenter of a face of the (n − 3)- simplex which is spanned by a set of vertices labelled by a contiguous sequence i, i + 1, …, j − 1, j with 1 ≤ i ≤ j ≤ n − 2, with (i, j) ≠ (1, n − 2). It is then not hard to check from the construction that star_{Δ}(ρ_{ij}) consists of the union of all (closed) chambers for type A_{n−3} which satisfy the inequalities

x_{i}, x_{i+1}, …, x_{j−1}, x_{j} ≥ x_{i−1}, x_{j+1}

(where here we omit the inequalities involving x_{i−1} if i = 1, and similarly for x_{j+1} if j = n + 2). It is clear that these inequalities describe a convex cone, and hence Δ_{n} is locally convex. □

It follows then from this Proposition and Theorem 1.2(ii) that (−1)^{\frac{n−3}{2}} σ(A_{n}) ≥ 0 for n odd (and of course, σ(A_{n}) = 0 for n even). However, as in the case of Coxeter zonotopes of type A, we can compute σ(A_{n}) explicitly using the formulas for the f-vector or h-vector of A_{n} given in [28, Theorem 3]. Specifically, these formulas imply that for n ≥ 3 we have

σ(A_{n}) = ∑_{i=0}^{n−3}(−1)^{i} \frac{1}{n − 1}\binom{n − 3}{i}\binom{n − 1}{i + 1}

= _{2}F_{1} (\begin{matrix}3 − n & 2 − n \\ & 2 \end{matrix}| − 1)

= \begin{cases}(−1)^{\frac{n−3}{2}} C_{\frac{n−1}{2}} & if n is odd \\ 0 & if n is even \end{cases}

where C_{n} denotes the Catalan number \frac{1}{n} \binom{2n−2}{n−1}. Here the _{2}F_{1} is hypergeometric series notation, and the last equality uses Kummer’s summation of a well-poised 2F_{1} at −1 (see e.g. [2, p. 9]).

Returning to the discusion of non-acute and obtuse polytopes, it is worth noting the following facts, pointed out to us by M. Davis. Recall that a simplicial complex K is called flag if every set of vertices v_{1}, …, v_{r} which pairwise span edges of K also jointly span an (r − 1)-simplex of K. Proposition 4.10. A polytope P is non-acute (resp. obtuse) if and only if each of its 2-dimensional faces are non-acute (resp. obtuse).

Furthermore, non-acuteness of any polytope P implies that P is simple

Proof. The first assertion for non-acute polytopes follows from an easy lemma due to Moussong [6, Lemma 2.4.1]. In the notation of [6], saying that every 2dimensional face of P is non-acute (in codimension 1) is equivalent to saying that for every vertex v of P, the spherical simplex σ = Lk(v, P) has size ≥ \frac{π}{2} . Then [6, Lemma 2.4.1] asserts that every link Lk(τ, σ) of a face of this spherical simplex also has size ≥ \frac{π}{2} . But a face F of P containing v has Lk(v, F) of the form Lk(τ, σ) for some τ, and hence F is non-acute in codimension 1 when considered as a polytope in its own right. That is, P is non-acute. An easy adaptation of this argument to the obtuse case proves the first assertion of the proposition for obtuse polytopes.

The fact that non-acuteness implies simplicity again comes from considering the spherical simplex σ = Lk(v, P) for any vertex v, which will have size ≥ \frac{π}{2} . Then its polar dual spherical convex polytope σ^{∗} will have all of its dihedral angles less than or equal to \frac{π}{2} . This forces σ^{∗} to be a spherical simplex, by [35, p. 44], and hence σ itself must be a spherical simplex. This implies v has exactly d neighbors, so P is simple. □

Obtuse polytopes turn out to be relatively scarce in comparison with non-acute polytopes, For example, it is easily seen that Coxeter zonotopes, although always non-acute by Proposition 4.5, are not in general obtuse in dimensions 3 and higher. It is easy to find obtuse polytopes in dimensions up to 4:

• in dimension 2, the regular n-gons for n ≥ 5,

• in dimension 3, the dodecahedron,

• in dimension 4 the “120-cell” (see [7, pp. 292-293]) However M. Davis has pointed out to us that in dimensions 5 higher, there are

no obtuse polytopes, due to a result of Kalai [25, Theorem 1] (see also [35, p. 68] for the case of simple polytopes): every d-dimensional convex polytope for d ≥ 5 contains either a triangular or quadrangular 2-dimensional face.

5. Relation to conjectures of Hopf and of Charney and Davis

In this section we discuss the relation of Theorem 1.2(i) to the conjectures of Hopf and of Charney and Davis mentioned in the Introduction.

Let M^{d} be a compact d-dimensional closed Riemannian manifold. When d is odd, Poincaré duality implies that the Euler characteristic χ(M) vanishes. When d is even, a conjecture of H. Hopf (see e.g. [6]) asserts that if M^{d} has non-positive sectional curvature, the Euler characteristic χ(M^{d}) satsfies

(−1)^{\frac{d}{2}} χ(M^{d}) ≥ 0.

This result is known for d = 2, 4 by Chern’s Gauss-Bonnet formula, but open for general d; see [6, §0] for some history.

Charney and Davis [6] explored a combinatorial analogue of this conjecture, and we refer the reader to their paper for terms which are not defined precisely here. Let M^{d} be a compact d-dimensional closed manifold which has the structure of a (locally finite) Euclidean cell complex, that is, it is formed by gluing together convex polytopes via isometries of their faces. One can endow such a cell complex with a metric space structure that is Euclidean within each polytopal cell, making it a geodesic space. Gromov has defined a notion of when a geodesic space is nonpositively curved, and Charney and Davis made the following conjecture: Conjecture 5.1. [6, Conjecture A] If M^{d} is a non-positively curved, piecewise Euclidean, closed manifold with d even, then

(−1)^{\frac{d}{2}} χ(M^{d}) ≥ 0.

For piecewise Euclidean cell complexes, nonpositive curvature turns out to be equivalent to a local condition at each vertex. Specifically, at each vertex v of M^{d}, one has a piecewise spherical cell complex Lk(v, M^{d}) called the link of v in M^{d}, which is homeomorphic to a generalized homology (d − 1)-sphere, and inherits its own geodesic space structure. Nonpositive curvature of M^{d} turns out to be a metric condition on each of these complexes Lk(v, M^{d}). Charney and Davis show [6, (3.4.3)] that the Euler characteristic χ(M^{d}) can be written as the sum of certain local quantities κ(Lk(v, M^{d})) defined in terms of the metric structure on Lk(v, M^{d}):

(7)  χ(M^{d}) = ∑_{v} κ(Lk(v, M^{d})).

In the special case where the polytopes in the Euclidean cell decomposition of M^{d} are all right-angled cubes, the links Lk(v, M^{d}) are all simplicial complexes, and the quantity κ(Lk(v, M^{d})) has a simple combinatorial expression purely in terms of the numbers of simplices of each dimension in these complexes (that is, independent of their metric structure). Furthermore, in this case, non-positive curvature corresponds to the combinatorial condition that each link is a flag complex, that is, the minimal subsets of vertices in Lk(v, M^{d}) which do not span a simplex always have cardinality two. They then noted that in this special case, Conjecture 5.1 would follow via Equation (7) from Conjecture 5.2. [6, Conjecture D] If Δ is a flag simplicial complex triangulating a generalized homology (d − 1)-sphere with d even, then

(−1)^{\frac{d}{2}} κ(Δ) ≥ 0.

This Charney-Davis conjecture is trivial for d = 2, has recently been proven by Davis and Okun [15] for d = 4 using L_{2}-homology of Coxeter groups, and is also known (by an observation of Babson and a result of Stanley; see [6, §7]) for the special class of flag simplicial complexes which are barycentric subdivisions of boundaries of convex polytopes.

Local convexity of simplicial fans turns out to be stronger than flagness: Proposition 5.3. A locally convex complete simplicial fan Δ in ℝ^{d} is flag, when considered as a simplicial complex triangulating a (d − 1)-sphere.

Proof. Assume that Δ is not flag, so that there exist rays ρ_{1}, …, ρ_{k} whose convex hull σ := conv (ρ_{1}, …, ρ_{k}) is not a cone of Δ, but conv (ρ_{i}, ρ_{j}) is a cone of Δ for each i, j. Choose such a collection of rays of minimum cardinality k, so that conv (ρ_{1}, …, ρ̂_{i}, …, ρ_{k}) is a cone of Δ for each i (in other words, the boundary complex ∂σ is a subcomplex of Δ).

We wish to show that star_{Δ}(ρ_{1}) is not convex. To see this, consider σ∩star_{Δ}(ρ_{1}), that is, the collection of cones

{σ′ ∩ σ : σ′ ∈ star_{Δ}(ρ_{1})}.

Since σ is not a cone of Δ but |Δ| = ℝ^{d}, this collection must contain at least one 2-dimensional cone of the form σ′ ∩ σ = conv (ρ_{1}, ρ), where ρ is a ray of σ but ρ ∉ {ρ_{2}, …, ρ_{k}}. Since ρ lies inside σ and ∂σ is a subcomplex of Δ, ρ cannot lie in ∂σ (else some cone of ∂Δ would be further subdivided, and not be a cone of Δ). Consequently ρ lies in the interior of σ. Then the ray ρ′ := ρ − ϵρ_{1} for very small ϵ > 0 has the following properties:

• ρ′ lies in σ, because ρ was in the interior of σ, • ρ′ therefore lies in the convex hull of star_{Δ}(ρ_{1}), since σ does (as its extreme

rays ρ_{1}, …, ρ_{k} of σ are all in star_{Δ}(ρ_{1})),

• ρ′ does not lie in star_{Δ}(ρ_{1}), else it would lie in a cone σ′ of Δ containing ρ_{1}, and then σ′ would contain ρ in the relative interior of one of its faces, a contradiction.

Therefore star_{Δ}(ρ_{1}) is not convex. □

In light of the preceeding proposition, one might ask if every flag simplicial sphere has a realization as a locally convex complete simplicial fan. We thank X. Dong for the following argument showing that an even weaker statement is false. One can show that complete simplicial fans always give rise to P L-spheres. Therefore if one takes the barycentric subdivision of any regular cellular sphere which is not P L (such as the double suspension of Poincaré’s famous homology sphere), this will give a flag simplicial sphere which is not P L and therefore has no realization as a complete simplicial fan (let alone one which is locally convex).

Our results were motivated by the Charney-Davis conjecture and the following fact: when P is a simple d-polytope and Δ is its normal fan considered as a (d−1)- dimensional simplicial complex, one can check that

(8)  σ(P) = 2^{d}κ(Δ).

As a consequence, we deduce the following from Proposition 5.3 and Theorem 1.2 (i). Corollary 5.4. Let P be rational simple polytope, Δ its normal fan. If Δ is locally convex, then it is flag and satisfies the Charney-Davis conjecture.

In particular by Corollary 4.2, if P is a non-acute simple rational polytope then its normal fan Δ is flag and satisfies the Charney-Davis conjecture.

It is worth mentioning that the special case of Conjecture 5.1 considered in [6] where M^{d} is decomposed into right-angled cubes is “polar dual” to another special case that fits nicely with our results. Say that M^{d} has a corner decomposition if the local structure at every vertex in the decomposition is combinatorially isomorphic to the coordinate orthants in ℝ^{d}, that is, each link Lk(v, M^{d}) has the combinatorial structure of the boundary complex of a d-dimensional cross-polytope or hyperoctahedron. (Note that this condition immediately implies that each of the d-dimensional polytopes in the decomposition must be simple). A straightforward counting argument (essentially equivalent to the calculation proving [6, (3.5.2)]) shows that for a manifold M^{d} with corner decomposition into simple polytopes P_{1}, …, P_{N} one has

(9)  χ(M^{d}) = \frac{1}{2^{d}} ∑_{i=1}^{N} σ(P_{i}).

The following corollary is then immediate from this relation and Theorem 1.2. Corollary 5.5. Let M^{d} be an d-dimensional manifold with d even, having a corner decomposition.

If each of the simple d-polytopes in the corner decomposition is rational and has normal fan which is locally convex, then

(−1)^{\frac{d}{2}} χ(M^{d}) ≥ 0.

In particular, this holds if each of the simple d-polytopes is non-acute.

Several interesting examples of manifolds with corner decompositions into simple polytopes that are either Coxeter zonotopes (hence non-acute) or associahedra (hence locally convex) may be found in [14].

There is also an important general construction of such manifolds called mirroring which we now discuss. This construction (or its polar dual) appears repeatedly in the work of Davis [10, 11, 13, 14], and was used in [6, §6] to show that the case of their Conjecture 5.1 for manifolds decomposed into right-angled cubes is equivalent to their Conjecture 5.2. In a special case, this construction begins with a generalized homology (d − 1)-sphere L with n vertices and produces a cubical orientable generalized homology d-manifold ML having 2^{n} vertices, with the link at each of these vertices isomorphic to L. Hence we have

χ(ML) = 2^{n} · κ(L).

We wish to make use of the polar dual of this construction, which applies to an arbitrary simple d-dimensional polytope P, yielding an orientable d-manifold M(P) with a corner decomposition having every d-dimensional cell isometric to P. The construction is as follows: denote the (d − 1)-dimensional faces of P by F_{1}, F_{2}, …, F_{n}, and let M(P) be the quotient of 2^{n} disjoint copies {P_{ϵ}}_{ϵ∈{+,−}^{n}} of P, in which two copies P_{ϵ}, P_{ϵ′} are identified along their face F_{i} whenever ϵ, ϵ′ differ in the i^{th} coordinate and nowhere else. As a consequence of equation (9) we have

(10)  χ(M(P)) = 2^{n−d} · σ(P),

which shows that the “non-acute” assertions in Corollaries 5.5 and 5.4 are equivalent.

We can now use the mirror construction to complete the proof of an assertion from the previous section. We are indebted to M. Davis for the statement and proof of this assertion.

Proof of Corollary 4.2 (i) without assuming rationality of P (as referred to in Remark 4.4): Assume that P is a simple non-acute d-dimensional polytope with d even. We wish to show that (−1)^{\frac{d}{2}} σ(P) ≥ 0.

Construct M(P) as above, a manifold with corner decomposition into non-acute simple polytopes having χ(M(P)) = 2^{n−d}σ(P) if P had n codimension 1 faces. In the notation of [6], this means that all the links Lk(v, M(P)) have size ≥ \frac{π}{2} and are combinatorially isomorphic to boundaries of cross-polytopes. This implies that these links’ underlying simplicial complexes are flag complexes satisfying [6, Conjecture D’], and then [6, Proposition 5.7] implies that each of these links Lk(v, M(P)) satisfies [6, Conjecture C’]. This implies that (−1)^{\frac{d}{2}} κ(Lk(v, M(P))) ≥ 0. Combining this with Equation (7), we conclude that (−1)^{\frac{d}{2}} χ(M(P)) ≥ 0, and finally via Equation (10), that (−1)^{\frac{d}{2}} σ(P) ≥ 0.

We note that a similar argument (involving an adaptation of [6, Lemma 2.4.1]) proves (−1)^{\frac{d}{2}} σ(P) > 0 when P is obtuse, but does not yield in any obvious way the stronger assertion of Corollary 4.2 (iii). □

6. Appendix: the conormal bundle of a toric divisor

In this appendix we describe the conormal bundle of a toric divisor on X = X_{Δ} when the fan Δ is complete and simplicial. Denote the collection of toric divisors on X by D_{1}, …, D_{m}. As mentioned in Section 3, the conormal bundle of a divisor, say D_{1}, can be identified as the restriction of O_{X}(−D_{1}) to D_{1}, which we renamed O_{D_{1}} (−D_{1}). It corresponds to a continuous piecewise linear function:

Ψ_{−D_{1}}^{D_{1}} : N_{ℝ}/ρ_{1} → ℝ

as in the discussion of Section 3. Here ρ_{1} is the ray in the fan Δ corresponding to the divisor D_{1}. We wish to identify the graph of Ψ_{−D_{1}}^{D_{1}} with link_{Δ} (ρ_{1}), which we recall is the boundary of star_{Δ}(ρ_{1}), the latter being the union of all cones of Δ containing ρ_{1}. Proposition 6.1. Let D_{1} be a toric divisor of a toric variety X = X_{Δ} with Δ simplicial. Then the graph of the piecewise linear function for O_{D_{1}}(D_{1}) is affinely equivalent to the boundary link_{Δ} (ρ_{1}) of star_{Δ} (ρ_{1}), where ρ_{1} is the ray corresponding to D_{1}.

Proof. We can index the toric divisors D_{1}, …, D_{m} of X in such a way that D = D_{1} and D_{2}, ..., D_{l} are those which are adjacent to D_{1}. Let n_{i} be the first nonzero lattice point along the ray ρ_{i} corresponding to D_{i}. We choose a decomposition of N into a direct sum of ℤn_{1} with another lattice N′ which is isomorphic to N/ρ_{1} (here we are abusing notation by referring to the quotient lattice N/ℤn_{1} as N/ρ_{1}). Then we can write n_{i} = b_{i}n_{1} + c_{i}n′_{i} where n′_{i} ∈ N′ ≅ N/ρ_{1} is indecomposable (i.e. not of the form k n′′_{i} for some integer k with |k| ≥ 2 and n′′_{i} ∈ N′ ), and c_{i} is some nonnegative integer.

Now we choose the linear functional u on N such that ⟨u, n_{1}⟩ = 1 and its restriction to N′ is zero. Then in the Chow group of X we have the following relation (see [18, p.106]):

∑_{i=1}^{m} ⟨u, n_{i}⟩ D_{i} = 0.

When we restrict this relation to the toric subvariety D_{1} then those terms involving D_{i} with i > l will disappear because they are disjoint from D_{1}, and using the formula on [18, p. 108], we have

⨂_{i=1}^{l} O_{D_{1}} (\frac{⟨u, n_{i}⟩}{c_{i}}D_{i}) = O_{D_{1}}.

Or equivalently, since ⟨u, n_{1}⟩ = 1 and ⟨u_{i}, n_{1}⟩ = b_{i}, we have

O_{D_{1}} (−D_{1}) = ⨂_{i=2}^{l} O_{D_{1}} (\frac{b_{i}}{c_{i}} D_{i}) .

Now under the identification N′ ≅ N/ρ_{1}, the restriction of the divisor D_{i} to D_{1} corresponds to the ray in N/ρ_{1} spanned by n′_{i} when 2 ≤ i ≤ l. Therefore the piecewise linear function Ψ_{−D_{1}}^{D_{1}} : N_{ℝ}/ρ_{1} → ℝ is determined by Ψ_{−D_{1}}^{D_{1}} (n′_{i}) = \frac{b_{i}}{c_{i}}. This implies the assertion of the proposition. □

7. Acknowledgements

The authors are grateful to Michael Davis for several very helpful comments, proofs, references, and the permission to include them here. They also thank Hugh Thomas for pointing out an error in an earlier version, and Xun Dong, Paul Edelman, Kefeng Liu, William Messing, and Dennis Stanton for helpful conversations.

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School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA E-mail address, N.C. Leung: leung@math.umn.edu

School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA E-mail address, V. Reiner: reiner@math.umn.edu