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\begin{document}
\title{The signature of a toric variety}
\author{Naichung Conan Leung}
\address{School of Mathematics\\
University of Minnesota\\
Minneapolis, MN 55455 USA}
\email[N.C. Leung]{leung@math.umn.edu}
\author{Victor Reiner}
\address{School of Mathematics\\
University of Minnesota\\
Minneapolis, MN 55455 USA}
\email[V. Reiner]{reiner@math.umn.edu}
\thanks{First, second authors partially supported by NSF grants DMS-9803616 and
DMS-9877047 respectively.}
\begin{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 $i$.
\end{abstract}
\maketitle
\section{Introduction}
Much attention in combinatorial geometry has centered on the problem of
characterizing which non-negative integer sequences $(f_0,f_1,\ldots,f_d)$
can be the \textit{$f$-vector} $f(P)$ of a $d$-dimensional convex polytope $%
P $, that is, $f_i$ is the number of $i$-dimensional faces of $P$; see \cite
{BilleraBjorner} for a nice survey.
For the class of \textit{simple polytopes}, this problem was completely
solved by the combined work of Billera and Lee \cite{BilleraLee} and of
Stanley \cite{Stanley-g-theorem}. A simple $d$-dimensional polytope is one
in which every vertex lies on exactly $d$ edges. McMullen's \textit{$g$%
-conjecture} (now the \textit{$g$-theorem}) gives necessary \cite
{Stanley-g-theorem} and sufficient \cite{BilleraLee} conditions for $%
(f_0,f_1,\ldots,f_d)$ 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 $X_\Delta$ associated to the \textit{(inner) normal fan} $\Delta$ of
$P$, and then the Hard Lefschetz Theorem for $X_\Delta$ played a crucial
role. This construction of $X_\Delta$ from $\Delta$ requires that $P$ be
\textit{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 \cite{McMullen} demonstrated that
one can construct a ring $\Pi(P)$, isomorphic (with a doubling of the
grading) to the cohomology ring of $X_\Delta$ if $P$ is rational, and proved
that $\Pi(P)$ 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 $X_\Delta$ 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 \textit{%
$h$-vector} which follows from the Hard Lefschetz Theorem. Recall that for a
simple polytope $P$, the $h$-vector is the sequence $h(P)=(h_0,h_1,%
\ldots,h_d)$ defined as follows. If we let $f(P,t) := \sum_{i=0}^d f_i(P)
t^i $, then
\begin{equation*}
h(P,t) := \sum_{i=0}^d h_i(P) t^i = f(P,t-1).
\end{equation*}
The $h$-vector has a topological interpretation: $h_i$ is the $2i^{th}$
Betti number for $X_\Delta$, or the dimension of the $i^{th}$-graded
component in McMullen's ring $\Pi(P)$. Part of the conditions of the $g$%
-theorem are the Dehn-Sommerville equations $h_i = h_{d-i}$, which reflect
Poincar\'e duality for $X_\Delta$.
Define the alternating sum
\begin{equation*}
\begin{aligned} \sigma(P): & = \sum_{i=0}^d (-1)^i h_i(P) \\ [& = h(P,-1) =
f(P,-2) = \sum_{i=0}^d f_i(P) (-2)^i \,\, ], \end{aligned}
\end{equation*}
a quantity which is (essentially) equivalent to one arising in a conjecture
of Charney and Davis \cite{CharneyDavis}, related to a conjecture of Hopf
(see Section \ref{Hopf} below). Note that when $d$ is odd, $\sigma(P)$
vanishes by the Dehn-Sommerville equations. When $d$ is even, we have the
following result (see Section \ref{signature}).
\begin{theorem}
\label{identify-signature} Let $P$ be a simple $d$-dimensional polytope,
with $d$ even. Then $\sigma (P)$ is the signature of the quadratic form $%
Q(x)=x^{2}$ defined on the $\frac{d}{2}^{th}$-graded component of McMullen's
ring $\Pi (P)$.
In particular, when $P$ has rational vertices, $\sigma (P)$ is the signature
or index $\sigma (X_{\Delta })$ of the associated toric variety $X_{\Delta }$%
.
\end{theorem}
An important special case of the previously mentioned Charney-Davis
conjecture asserts that a certain combinatorial condition on $P$ (namely
that of $\Delta$ being a \textit{flag} complex; see Section \ref{Hopf})
implies $(-1)^{d/2}\sigma(P) \geq 0$. In this paper, we prove this
conjecture when $P$ satisfies certain stronger geometric conditions. We also
give further conditions which give lower bounds on $(-1)^{d/2}\sigma(P)$. In
order to state these results, we give rough definitions of some of these
conditions here (see Section \ref{Hirzebruch} for the actual definitions).
Say that the fan $\Delta$ is \textit{locally convex} (resp. \textit{locally
pointed convex, locally strongly convex}) if every $1$-dimensional cone in $%
\Delta$ has the property that the union of all cones of $\Delta$ containing
it is convex (resp. pointed convex, strongly convex). For example (see
Propositions \ref{euclidean-imply-affine} and \ref{non-acute-by-polygons}
below), if each angle in every $2$-dimensional face of $P$ is non-acute
(resp. obtuse) then $\Delta$ will be locally convex (resp. locally strongly
convex). It turns out that $\Delta$ being locally convex implies that it is
flag (Proposition \ref{locally-convex-implies-flag}).
For a simple polytope $P$ with rational vertices, we define an integer $m(P)$
which measures how singular $X_\Delta$ is. To be precise, let $P$ in $%
\mathbb{R}^d$ 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 $%
\sigma$ of the normal fan $\Delta$ of the index $[N:N_\sigma]$, where $N$ is
the lattice dual to $M$ and $N_\sigma$ is the sublattice spanned by the
lattice vectors on the extremal rays of $\sigma$. Note that the condition $%
m(P)=1$ is equivalent to the smoothness of the toric variety $X_\Delta$, and
such polytopes $P$ are called \textit{Delzant} in the symplectic geometry
literature (e.g. \cite{Guillemin}).
Now we can state
\begin{theorem}
\label{lower-bounds} Let $P$ be a rational simple $d$-dimensional polytope
with $d$ even, and $\Delta $ its normal fan.
\begin{enumerate}
\item[(i)] If $\Delta $ is locally convex, then
\begin{equation*}
(-1)^{\frac{d}{2}}\sigma (P)\geq 0.
\end{equation*}
\item[(ii)] If $\Delta $ is locally pointed convex, then
\begin{equation*}
(-1)^{\frac{d}{2}}\sigma (P)\geq \frac{f_{d-1}(P)}{3m(P)^{d-1}}.
\end{equation*}
\item[(iii)] If $\Delta $ is locally strongly convex, then
\begin{equation*}
(-1)^{\frac{d}{2}}\sigma (P)\geq \text{ coefficient of }x^{d}\text{ in }%
\left[ \frac{t^{d}}{m(P)^{d-1}}\,f(P,t^{-1})\right]_{t\mapsto 1-\frac{x}{%
\tan (x)}}.
\end{equation*}
\end{enumerate}
\end{theorem}
\noindent We defer a discussion of the relation between Theorem \ref
{lower-bounds} (i) and the Charney-Davis conjecture to Section \ref{Hopf}.
It is amusing to see what Theorem \ref{lower-bounds} 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
\begin{equation*}
f_1=f_0 \geq 3,
\end{equation*}
or in other words, every polygon has the same number of edges as vertices,
and this number is at least $3$. Since
\begin{equation*}
(-1)^{\frac{d}{2}} \sigma(P) = f_0(P) - 4,
\end{equation*}
Theorem \ref{lower-bounds} (i) tells us that when $\Delta$ is locally
convex, we must have $f_0 \geq 4$. In other words, triangles cannot have
normal fan $\Delta$ which is locally convex, as one can easily check. For $%
d=2$, the conditions that $\Delta$ is locally pointed convex or locally
strongly convex coincide, and Theorem \ref{lower-bounds} (ii),(iii) both
assert that under these hypotheses, a (rational) polygon $P$ must have
\begin{equation*}
f_0(P) - 4 \,\, \geq \,\,\frac{f_1(P)}{3 m(P)} \,\,=\,\, \frac{f_0(P)}{3 m(P)%
}
\end{equation*}
or after a little algebra,
\begin{equation} \label{polygon-inequality}
f_0(P) \,\, \geq \,\, \frac{12}{3-\frac{1}{m(P)}}.
\end{equation}
Since the right-hand side is strictly greater than $4$, we conclude that a
quadrilateral $P$ cannot have $\Delta$ locally pointed convex nor locally
strongly convex. This agrees with an easily-checked fact: a quadrilateral $P$
satisfies the weaker condition of having $\Delta$ locally convex if and only
if $P$ is a rectangle, and rectangles fail to have $\Delta$ locally pointed
convex. On the other hand, the inequality (\ref{polygon-inequality}) also
implies a not-quite-obvious fact: even though a (rational) pentagon can
easily have $\Delta$ 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 \ref
{lower-bounds} 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
\begin{equation*}
L(X) = L_0(X) + L_1(X) + \cdots + L_{\frac{d}{2}}(X)
\end{equation*}
where $L_i(X)$ is a cycle in $CH^{i}(X)_\rationals$, the Chow ring of $X$.
\begin{theorem}
\label{L-positive} Let $X=X_{\Delta }$ be a complete toric variety $X$
associated to a simplicial fan $\Delta $. If $\Delta $ 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 $0$).
\end{theorem}
For instance, when $i=1$ this implies that if $\Delta $ is locally convex,
then
\begin{equation*}
\int_{X}(c_{1}^{2}(X)-2c_{2}(X))\cdot H_{1}\cdot \ldots \cdot H_{d-2}\leq 0
\end{equation*}
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. \cite{Kobayashi}%
.
Notice that if $\Delta $ is not locally convex, $(-1)^{i}L_{i}(X)$ need not
be effective. For example, if $\Delta $ 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}\left( X\right) $ is represented by the
negative of the Poincar\'{e} dual of a point.
\section{The alternating sum as signature}
\label{signature}
We wish to prove Theorem \ref{identify-signature}, whose statement we recall
here.
\vskip .1in \noindent \textbf{Theorem \ref{identify-signature}.} \textit{Let
$P$ be a simple $d$-dimensional polytope, with $d$ even. Then $\sigma(P)$ is
the signature of the quadratic form $Q(x)=x^2$, defined on the $\frac{d}{2}%
^{th}$-graded component of McMullen's ring $\Pi(P)$. }
\textit{In particular, when $P$ has rational vertices, $\sigma(P)$ is the
signature or index $\sigma(X_\Delta)$ of the associated toric variety $%
X_\Delta$. } \vskip .1in
\begin{proof}
Taking $r=\frac{d}{2}$ in a result of McMullen
\cite[Theorem 8.6]{McMullen}, we find
that the quadratic form $(-1)^{\frac{d}{2}} Q(x)$ on the
$\frac{d}{2}^{th}$-graded component of $\Pi(P)$ has
$$
\begin{aligned}
\sum_{i=0}^{\frac{d}{2}} (-1)^i h_{\frac{d}{2}-i}(P)
&\text{ positive eigenvalues, and} \\
\sum_{i=0}^{\frac{d}{2}-1} (-1)^i h_{\frac{d}{2}-i-1}(P)
&\text{ negative eigenvalues.}
\end{aligned}
$$
Consequently, the signature $\sigma(Q)$ of $Q$ is
$$
\begin{aligned}
\sigma(Q)
&= (-1)^{\frac{d}{2}} \left[
\sum_{i=0}^{\frac{d}{2}} (-1)^i h_{\frac{d}{2}-i}(P)-
\sum_{i=0}^{\frac{d}{2}-1} (-1)^i h_{\frac{d}{2}-i-1}(P)
\right] \\
&= \sum_{i=0}^d (-1)^i h_i(P)
\end{aligned}
$$
where the second equality uses the {\it Dehn-Sommerville}
equations \cite[\S 4]{McMullen}:
$$
h_i(P) = h_{d-i}(P).
$$
The second assertion of the theorem follows immediately
from McMullen's identification of the ring $\Pi(P)$ with
the quotient of the Stanley-Reisner ring of $\Delta$ by a certain
linear system of parameters \cite[\S 14]{McMullen},
which is known to be isomorphic (after a doubling of the grading)
with the cohomology of $X_\Delta$ \cite[\S 5.2]{Fulton}.
\end{proof}
\begin{remark}
\textrm{\ \newline
Starting from any complete rational simplicial fan $\Delta $, one can
construct a toric variety $X_{\Delta }$ which will be complete, but not
necessarily projective, and satisfies Poincar\`{e} duality. The $h$-vector
for $\Delta $ can still be defined, and again has an interpretation as the
even-dimensional Betti numbers of $X_{\Delta }$ (see \cite[\S 5.2]{Fulton}).
We suspect that the alternating sum of the $h$-vector is still the signature
of this complete toric variety. }
\textrm{Generalizing in a different direction, to \textit{any} polytope $P$
which is not necessarily simple, one can associate the normal fan $\Delta $
and a projective toric variety $X_{\Delta }$. Although the (singular)
cohomology of $X_{\Delta }$ does not satisfy Poincar\'{e} duality, its
\textit{intersection cohomology} (in middle perversity) $IH^{\cdot
}(X_{\Delta })$ will. There is a combinatorially-defined \textit{generalized
$h$-vector} which computes these $IH^{\cdot }$ Betti numbers (see \cite
{Stanley-generalized-h-vector}). Moreover, using the Hard Lefschetz Theorem
for intersection cohomology and the fact that $X_{\Delta }$ is a finite
union of affine subvarieties, the alternating sum of the generalized $h$%
-vector equals the signature of the quadratic form on $IH^{\cdot }(X_{\Delta
})$ defined by the intersection product. }
\end{remark}
\begin{remark}
\textrm{\ \newline
The special case of the second assertion in Theorem \ref{identify-signature}
is known when $X_{\Delta }$ is smooth (i.e. $P$ is a Delzant polytope); see
\cite[Theorem 3.12 (3)]{Oda}. }
\end{remark}
\section{Lower bounds derived from the signature theorem}
\label{Hirzebruch}
The goal of this section is to explain the various notions used in Theorem
\ref{lower-bounds}, and to prove this theorem.
We begin with a $d$-dimensional lattice $M \cong \mathbb{Z}^d$ and its
associated real vector space $M_\reals=M\otimes_\integers \mathbb{R}$. A
\textit{polytope} $P$ in $M_\reals$ is the convex hull of a finite set of
points in $M_\reals$. We say that $P$ is \textit{rational} if these points
can be chosen to be rational with respect to the lattice $M$. The \textit{%
dimension} of $P$ is the dimension of the smallest affine subspace
containing it. A \textit{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. \textit{Vertices} and \textit{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 \textit{simple} if
every vertex lies on exactly $d$ edges.
Let $N=Hom(M,\mathbb{Z})$ be the dual lattice to $M$ and $%
N_\reals=N\otimes_\integers \mathbb{R}$ be the dual vector space to $%
M_\reals $, with the natural pairing $M_\reals \otimes N_\reals \rightarrow
\mathbb{R} $ denoted by $\langle \cdot, \cdot \rangle$. For a polytope $P$
in $M_\reals$, the \textit{normal fan} $\Delta$ is the following collection
of polyhedral cones in $N_\reals$:
\begin{equation*}
\Delta = \{\sigma_F: F \text{ a face of }P\},
\end{equation*}
where
\begin{equation*}
\sigma_F:=\{ v \in N_\reals: \langle u, v \rangle \leq \langle u^{\prime}, v
\rangle \text{ for all }u \in F, u^{\prime}\in P \}
\end{equation*}
Note that
\begin{enumerate}
\item[$\bullet $] the normal fan $\Delta $ is a \textit{complete} fan, that
is, it exhausts $N_{\mathbb{R}}$,
\item[$\bullet $] $\Delta $ is a \textit{rational} fan, in the sense that
its rays all have rational slopes, if $P$ is rational,
\item[$\bullet $] if $P$ is $d$-dimensional, then every cone $\sigma _{F}$
in $\Delta $ will be \textit{pointed}, that is, it will contain no proper
subspaces of $N_{\mathbb{R}}$,
\item[$\bullet $] $P$ is a simple polytope if and only if $\Delta $ is a
\textit{simplicial fan}, that is, every cone $\sigma $ in $\Delta $ is
simplicial in the sense that its extremal rays are linearly independent.
\end{enumerate}
We next define several affinely invariant conditions on a complete
simplicial fan $\Delta$ in $N_\reals$ (and hence on simple polytopes $P$ in $%
M_\reals$) that appear in Theorem \ref{lower-bounds}. For any collection of
polyhedral cones $\Delta$ in $N_\reals$, let $|\Delta|$ denote the support
of $\Delta$, that is the union of all of its cones as a point set. Define
the \textit{star} and \textit{link} of one of the cones $\sigma$ in $\Delta$
similarly to the analogous notions in simplicial complexes: $\mathrm{star}%
_\Delta(\sigma)$ is the subfan consisting of those cones $\tau$ in $\Delta$
such that $\sigma, \tau$ lie in some common cone of $\Delta$, while $\mathrm{%
link}_\Delta(\sigma)$ is the subfan of $\mathrm{star}_\Delta(\sigma)$
consisting of those cones which intersect $\sigma$ only at the origin. For a
\textit{ray} (i.e. a $1$-dimensional cone) $\rho$ of $\Delta$, say that the
fan $\mathrm{star}_\Delta(\rho)$ is
\begin{enumerate}
\item[$\bullet $] \textit{convex} if its support $|\mathrm{star}_{\Delta
}(\rho )|$ is a convex set in the usual sense,
\item[$\bullet $] \textit{pointed convex} if $|\mathrm{star}_{\Delta }(\rho
)|$ is convex and contains no proper subspace of $N_{\mathbb{R}}$,
\item[$\bullet $] \textit{strongly convex} if furthermore for every cone $%
\sigma $ in $\mathrm{link}_{\Delta }(\rho )$, there exists a linear
hyperplane $H$ in $N_{\mathbb{R}}$ which supports $\mathrm{star}_{\Delta
}(\rho )$ and whose intersection with $\mathrm{star}_{\Delta }(\rho )$ is
exactly $\sigma $.
\end{enumerate}
Say that $\Delta$ is \textit{locally convex} (resp. \textit{locally pointed
convex}, \textit{locally strongly convex}) if every ray $\rho$ of $\Delta$
has $\mathrm{star}_\Delta(\rho)$ convex (resp. pointed convex, strongly
convex). One has the easy implications
\begin{equation*}
\text{ locally strongly convex } \Rightarrow \text{ locally pointed convex }
\Rightarrow \text{ locally convex }.
\end{equation*}
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 $[N:N_\sigma]$ as $\sigma$ runs over all $%
d $-dimensional cones in $\Delta$. Here $N_\sigma$ is the $d$-dimensional
sublattice of $N$ generated by the lattice vectors on the $d$ extremal rays
of $\sigma$. In a sense $m(P)$ measures how singular $X_\Delta$ is \cite[\S
2.6]{Fulton}, with $m(P)=1$ if and only if $X_\Delta$ is smooth, in which
case we say that $P$ is \textit{Delzant}.
We can now recall the statements of Theorems \ref{lower-bounds} and \ref
{L-positive}.
\vskip .1in \noindent \textbf{Theorem \ref{lower-bounds}.} \textit{Let $P$
be a simple $d$-dimensional polytope in $M_\reals$, which is rational with
respect to $M$, and $\Delta$ its normal fan in $N_\reals$. Assume $d$ is
even. }
\begin{enumerate}
\item[(i)] \textit{If $\Delta $ is locally convex, then
\begin{equation*}
(-1)^{\frac{d}{2}}\sigma (P)\geq 0.
\end{equation*}
}
\item[(ii)] \textit{If $\Delta $ is locally pointed convex, then
\begin{equation*}
(-1)^{\frac{d}{2}}\sigma (P)\geq \frac{f_{d-1}(P)}{3m(P)^{d-1}}.
\end{equation*}
}
\item[(iii)] \textit{If $\Delta $ is locally strongly convex, then
\begin{equation*}
(-1)^{\frac{d}{2}}\sigma (P)\geq \text{ coefficient of }x^{d}\text{ in }%
\left[ \frac{t^{d}}{m(P)^{d-1}}\,f(P,t^{-1})\right] _{t\mapsto 1-\frac{x}{%
\tan (x)}}.
\end{equation*}
}
\end{enumerate}
\vskip .1in \noindent \textbf{Theorem \ref{L-positive}.} \textit{Let $X =
X_\Delta$ be a complete toric variety $X$ associated to a simplicial fan $%
\Delta$, and let the expansion of the total $L$-class be
\begin{equation*}
L(X) = L_0(X) + L_1(X) + \cdots + L_{\frac{d}{2}}(X)
\end{equation*}
where $L_i(X)$ is a cycle in $CH^{i}(X)_\rationals$, the Chow ring of $X$. }
\textit{If $\Delta$ 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 $0$). } \vskip .1in
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 \cite{Fulton}, although many of the facts we
will use can also be found in Oda's book \cite{Oda} or Danilov's survey
article \cite{Danilov}.
Let $X$ denote the toric variety $X_{\Delta }$. Simpleness of $P$ implies
that $X$ is an orbifold \cite[\S 2.2]{Fulton}. Recall that irreducible toric
divisors\footnote{%
Actually these are $\mathbb{Q}$-Cartier divisors on the orbifold $X$.} 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 $\Delta $. Number these toric
divisors on $X$ as $D_{1},...,D_{m}$. Intersection theory for these $D_{i}$%
's is studied in Chapter 5 of \cite{Fulton}. Every $D_{i}$ is a toric
variety in its own right with at worst orbifold singularities. Moreover $D=%
\textstyle\bigcup_{i=1}^{m}D_{i}$ is a simple normal crossing divisor on $X$
\cite[\S 4.3]{Fulton}.
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 $\sigma \left( X\right) $ can be expressed in terms of
the Hodge numbers of $X$ as follows \cite[Theorem 15.8.2]{Hirzebruch}:
\begin{equation*}
\sigma \left( X\right) = \Sigma _{p, q=0}^{d}\left( -1\right)
^{q}h^{p,q}\left( X\right).
\end{equation*}
By the Dolbeault Theorem, $h^{p,q}\left( X\right) = \dim H^q(X,\Omega_X^p)$,
and hence the signature can be expressed in terms of twisted holomorphic
Euler characteristics
\begin{equation*}
\sigma \left( X\right) =\sum_{p=0}^{d}\chi \left( X,\Omega _{X}^{p}\right)
\end{equation*}
where $\chi(X,E):=\sum_{q=0}^d(-1)^q \dim H^q(X,E)$. Using the Riemann-Roch
formula, we can write
\begin{equation*}
\chi \left( X,\Omega _{X}^{p}\right) =\int_{X}ch\left( \Omega
_{X}^{p}\right) Td_{X}\text{,}
\end{equation*}
where $ch$ is the Chern character and $Td_X$ is the Todd class of $X$.
Therefore
\begin{equation*}
\sigma \left( X\right) =\int_{X}\textstyle\sum_{p=0}^{d}ch\left( \Omega
_{X}^{p}\right) Td_{X}.
\end{equation*}
When $X$ is smooth, $\sum_{p=0}^{d}ch\left( \Omega _{X}^{p}\right) Td_{X}$
equals the Hirzebruch L-class $L\left( X\right) $ of $X$ (see page 16 in
\cite[Theorem 15.8.2]{Hirzebruch} for example) and we recover the Hirzebruch
Signature Formula
\begin{equation}
\sigma \left( X\right) =\int_{X}L\left( X\right) \text{.} \label{HSF}
\end{equation}
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 $\sum_{p=0}^{d}ch\left( \Omega _{X}^{p}\right) Td_{X}$
as a definition of $L\left( X\right) $. Since we can express $L\left(
X\right) $ in terms of Chern roots of the orbi-bundle $\Omega _{X}^{1}$ and
Chern classes for orbi-bundles satisfy the same functorial properties as for
the usual Chern classes, the same holds true for $L\left( X\right) $. For
example we will use the splitting principle in the proof of the next lemma,
where we write $L\left( X\right) $ in terms of toric data.\footnote{%
For a general orbifold $X$, not necessary an algebraic variety, Kawasaki
\cite{Kawasaki} expressed the signature of $X$ in terms of integral of
certain curvature forms, thus generalizing the Hirzebruch signature formula
in a different way.}
%Next we want to express the signature of $X$ in terms of these $D_{i}$'s.
%When $X$ is a smooth variety, its signature $\sigma \left( X\right) $ can be
%expressed in terms of the Hodge numbers of twisted holomorphic Euler characteristics,
%\begin{equation*}
%\sigma \left( X\right) =\sum_{p=0}^{d}\chi \left( X,\Omega _{X}^{p}\right) .
%\end{equation*}
%Here $\chi \left( X,\Omega _{X}^{p}\right) :=
%\Sigma _{q=0}^{d}\left( -1\right) ^{q}h^{p,q}\left( X\right).$
%This formula follows from the hard
%Lefschetz theorem and therefore it continues to hold true for orbifolds.
%Using Riemann-Roch formula, we can write
%\begin{equation*}
%\chi \left( X,\Omega _{X}^{p}\right) =\int_{X}ch\left( \Omega
%_{X}^{p}\right) Td_{X}\text{,}
%\end{equation*}
%and therefore
%\begin{equation*}
%\sigma \left( X\right) =\int_{X}\textstyle\sum_{p=0}^{d}ch\left( \Omega
%_{X}^{p}\right) Td_{X}.
%\end{equation*}
%
%When $X$ is smooth, $\Sigma ch\left( \Omega _{X}^{p}\right) Td_{X}$ equals
%to the Hirzebruch L-class $L\left( X\right) $ of $X$ and we recover the
%Hirzebruch Signature Formula
%\begin{equation}
%\label{HSF}
%\sigma \left( X\right) =\int_{X}L\left( X\right) \text{.}
%\end{equation}
%If $X$ is a projective variety with at worst orbifold singularities, we can
%take the sum $\Sigma ch\left( \Omega _{X}^{p}\right) Td_{X}$ as a definition of $%
%L\left( X\right) $. When we express $L\left( X\right) $ in terms of Chern
%roots of the orbi-bundle $\Omega _{X}$, they satisfy the same functorial
%properties for the usual L-class.
\begin{lemma}
\label{crucial-calculation}
\begin{equation*}
\begin{aligned} (-1)^{\frac{d}{2}} \sigma\left( X\right) &=
(-1)^{\frac{d}{2}} \int_{X}L\left( X\right) \\ & =\sum_{p=1}^{d/2} \,\,
\sum_{\substack{n_{1}+...+n_{p}=d/2\\n_{i}>0;i_{1}<... 0$. The key
observation in obtaining the desired lower bounds is then
\begin{lemma}
\label{ample-big-nef} Let $P$ be a simple $d$-dimensional polytope in $M_{%
\mathbb{R}}$, which is rational with respect to $M$, and $\Delta $ its
normal fan in $N_{\mathbb{R}}$. Let $D_{i}$ be any of the irreducible toric
divisors on $X=X_{\Delta }$.
\begin{enumerate}
\item[(i)] If $\Delta $ is locally convex, then $O_{D_{i}}(-D_{i})$ is
generated by global sections.
\item[(ii)] If $\Delta $ is locally pointed convex, then $O_{D_{i}}(-D_{i})$
is generated by global sections and big.
\item[(iii)] If $\Delta $ is locally strongly convex, then $%
O_{D_{i}}(-D_{i})$ is ample.
\end{enumerate}
\end{lemma}
Assuming Lemma \ref{ample-big-nef} for the moment, we finish the proof of
Theorem \ref{lower-bounds}.
\vskip .1in \noindent \textit{Proof of Theorem \ref{lower-bounds} (i)}.
Under the assumption that $\Delta $ is locally convex, we know that the
restriction of $O_{X}\left( -D_{i_{j}}\right) $ to $D_{i_{1}}\cap ...\cap
D_{i_{p}}$ is generated by global sections for $1\leq j\leq p$ by Lemma \ref
{ample-big-nef}. This implies that the integral (\ref{integral}) equals the
intersection number of such divisors on the toric subvariety $D_{i_{1}}\cap
...\cap D_{i_{p}}$ and therefore it is nonnegative \footnote{%
This follows from the fact that the divisor class of a line bundle which is
generated by global sections is a limit of $\mathbb{Q}$-divisors which are
ample. Positivity of intersection numbers of ample divisors is well-known;
see e.g. \cite[Chapter 12]{Fulton-intersection-theory}} , that is
\begin{equation*}
\left( -1\right) ^{p}D_{i_{1}}^{2n_{1}}\cdot \cdot \cdot
D_{i_{p}}^{2n_{p}}\geq 0.
\end{equation*}
The non-negativity asserted in Theorem \ref{lower-bounds} (i) now follows
term-by-term from the sum in Lemma \ref{crucial-calculation}.$\qed$
\vskip .1in \noindent \textit{Proof of Theorem \ref{lower-bounds} (ii)}. If $%
\Delta $ is locally pointed convex, then $O_{D_i}\left( -D_i\right) $ is
generated by global sections and big. The bigness of $O_{D_i}\left(
-D_i\right) $ on $D_i$ implies that
\begin{equation*}
-D_i^{d}=\int_{D_i}\left( -D_i\right) ^{d-1}
\end{equation*}
is strictly positive.
\noindent \textbf{Claim.} $-D_i^{d} \geq \frac{1}{m(P)^{d-1}}$.
\noindent To prove this, we proceed as in the \textit{algebraic moving lemma}
\cite[\S 5.2, p. 107]{Fulton}, making repeated use of the fact that if $%
n_{j} $ is the first non-zero lattice point on the ray of $\Delta $
corresponding to $D_{j}$, then for any $u$ in $M$, one has
\begin{equation}
\sum_{j}\langle u,n_{j}\rangle D_{j}=0 \label{moving-equation}
\end{equation}
in the Chow ring \cite[Proposition, Part (ii), \S 5.2, p. 106]{Fulton}. 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 $\sum a_{j_{1},\ldots ,j_{d}}D_{j_{1}}\cdots D_{j_{d}}$ in
which each term has $D_{j_{1}},\ldots ,D_{j_{d}}$ \textit{distinct} divisors
which intersect at an isolated point of $X$, and each $a_{j_{1},\ldots
,j_{d}}$ 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 $\langle u,n_{i}\rangle
=1$, we can use \eqref{moving-equation} to replace one factor of $D_{i}$ in $%
D_{i}^{d}$ by a sum of other divisors $D_{j}$ with \textit{integer
coefficients} (that is, introducing \textit{no} denominators). However, in
each of the next $d-2$ stages, when one wishes to use \eqref{moving-equation}
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 $\langle
u,n_{j_{0}}\rangle $ of $D_{j_{0}}$ to be larger than $1$ in %
\eqref{moving-equation}, 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 $a_{j_{1},\ldots ,j_{d}}$ can be
written with the denominator $m(P)^{d-2}$. Finally, each intersection
product $D_{j_{1}}\cdots D_{j_{d}}$ is the reciprocal of the multiplicity at
the corresponding point of $X$, which is the index $[N:N_{\sigma }]$ where $%
\sigma $ is the $d$-dimensional cone of $\Delta $ corresponding to that
point \cite[\S 2.6]{Fulton}. Since each $[N:N_{\sigma }]$ divides $m(P)$, we
conclude that $-D_{i}^{d}$ lies in $\frac{1}{m(P)^{d-1}}\mathbb{Z}$, and
since it is positive, it is at least $\frac{1}{m(P)^{d-1}}$.
We have shown then that each term with $p=1$ on the right-hand side of Lemma
\ref{crucial-calculation} 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 $%
\left( -1\right) ^{p}D_{i_{1}}^{2n_{1}}\cdot \cdot \cdot D_{i_{p}}^{2n_{p}}$
because $O_{D_i}\left( -D_i\right) $ is generated by global sections.
Therefore, since $b_1=\frac{1}{3}$, we conclude from Lemma \ref
{crucial-calculation} that
\begin{equation*}
\left( -1\right) ^{d/2}\sigma \left( \Delta \right) \geq \frac{f_{d-1}}{3
m(P)^{d-1}}. \qed
\end{equation*}
\vskip .1in \noindent \textit{Proof of Theorem \ref{lower-bounds} (iii)}. If
$\Delta $ is locally strongly convex, then $O_{D_i}\left( -D_i\right) $ is
ample. By similar arguments as in assertions (i) and (ii), we have
\begin{equation*}
\left( -1\right) ^{p}D_{i_{1}}^{2n_{1}}\cdot \cdot \cdot
D_{i_{p}}^{2n_{p}}\geq \frac{1}{m(P)^{d-1}}
\end{equation*}
provided that $D_{i_{1}}\cap ...\cap D_{i_{p}}$ 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 $D_{i_{1}}, \ldots,
D_{i_{p}}.$ Therefore, after choosing positive integers $n_1,\ldots, n_p$,
the number of non-vanishing terms of the form $\left( -1\right)
^{p}D_{i_{1}}^{2n_{1}}\cdot \cdot \cdot D_{i_{p}}^{2n_{p}}$ in the expansion
of Lemma \ref{crucial-calculation} is $f_{d-p}(P)$. Hence
\begin{equation*}
\begin{aligned} (-1)^{\frac{d}{2}} \sigma\left( X\right) &=
(-1)^{\frac{d}{2}} \int_{X}L\left( X\right) \\ & =\sum_{p=1}^{d/2}\left(
-1\right) ^{p}\sum_{\substack{n_{1}+...+n_{p}=d/2\\n_{i}>0;i_{1}<...0}}b_{n_{1}}\cdot \cdot \cdot
b_{n_{p}} \frac{f_{d-p}(P)}{m(P)^{d-1}}\\ & = \sum_{p=1}^{d/2}
\frac{f_{d-p}(P)}{m(P)^{d-1}} \left[ \,\,\text{ coefficient of }x^d\text{ in
} \left( \sum_{n \geq 1} b_n x^{2n} \right )^p \,\, \right]. \end{aligned}
\end{equation*}
Note that
\begin{equation*}
\frac{\sqrt x}{\tanh\sqrt x} = 1 - \sum_{n \geq 1} (-1)^n b_n x^{n}
\end{equation*}
implies
\begin{equation*}
\sum_{n \geq 1} b_n x^{2n} = 1-\frac{x}{\tan(x)},
\end{equation*}
and note also that
\begin{equation*}
\sum_{p \geq 1} f_{d-p}(P) \, t^p = t^d f(P,t^{-1}).
\end{equation*}
This allows us to rewrite the above inequality as in the assertion of
Theorem \ref{lower-bounds} (iii). $\qed$
\vskip .1in \noindent \textit{Proof of Theorem \ref{L-positive} }. Recall
from the proof of Lemma \ref{crucial-calculation} that the total $L$-class
has expansion
\begin{equation*}
L(X) = \sum_{p \geq 1} \left( -1\right) ^{p}\sum_{\substack{ %
(n_1,\ldots,n_p) \\ n_{i}>0;i_{1}<...0;i_{1}<... 0 \qquad ( \text{ resp. } \langle
n,n^{\prime}\rangle \geq 0 ).
\end{equation*}
Similarly, obtuseness (resp. non-acuteness) for $P$ corresponds to the
following property of $\Delta$: any vectors $n_1, \ldots, n_t$ spanning the
extremal rays of a $t$-dimensional (simplicial) cone of $\Delta$ must have
\begin{equation*}
\langle \pi(n_1),\pi(n_2) \rangle > 0 \qquad ( \text{ resp. } \langle
\pi(n_1),\pi(n_2) \rangle \geq 0 )
\end{equation*}
where $\pi$ is the orthogonal projection onto the space perpendicular to the
span of the vectors $n_3, n_4, \ldots, n_t$.
Having said this, observe that if $P$ is non-acute in codimension $1$, for
any ray $\rho$ in $\Delta$ (spanned by a vector which we name $n$), the
hyperplane $\rho^\perp$ normal to $\rho$ supports $\mathrm{star}%
_\Delta(\rho) $: if $P$ is non-acute in codimension $1$, we must have $%
\langle n,n^{\prime}\rangle \geq 0$ for each vector $n^{\prime}$ spanning a
ray in $\mathrm{star}_\Delta(\rho)$, and hence for every vector in $\mathrm{%
star}_\Delta(\rho)$. Similarly, if $P$ is obtuse in codimension $1$ then
this hyperplane $\rho^\perp$ not only supports $\mathrm{star}_\Delta(\rho)$,
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 $\mathrm{star}_\Delta(\rho)$ can be checked \textit{locally} in
a certain way, similar to checking regularity of triangulations (see e.g.
\cite[\S 1.3]{DeLoera}). Roughly speaking, each cone $\sigma$ in the link of
$\rho$ must have the property that the union of cones containing $\sigma$
within $\mathrm{link}_\Delta(n)$ ``bend outwards" at $\sigma$ away from $%
\rho $, rather than ``bending inward" toward $\rho$. To be more formal,
consider every minimal dependence of the form
\begin{equation} \label{minimal-dependence}
\sum_{i \in F} \alpha_i n_i = \beta n + \sum_{j \in G} \beta_j m_j
\end{equation}
where
\begin{enumerate}
\item[-] $\{n_i\}_{i\in F}$ are vectors spanning the extremal rays of some
cone $\sigma$ in $\mathrm{link}_\Delta(n)$,
\item[-] each $m_j$ for $f$ in $G$ spans a ray in $\mathrm{link}%
_\Delta(\sigma)$,
\item[-] the coefficients $\alpha_i, \beta_j$ are all strictly positive.
\end{enumerate}
Then $\mathrm{star}_\Delta(\rho)$ is strictly convex if and only in every
such dependence we have $\beta < 0$. It is weakly convex if and only if in
every such dependence we have $\beta \leq 0$.
As a step toward proving assertions (i), (iii), given a dependence as in (%
\ref{minimal-dependence}) we apply the orthogonal projection $\pi$ onto the
space perpendicular to all of the $\{n_i\}_{i \in F}$, yielding the
following equation
\begin{equation*}
0 = \beta \, \pi(n) + \sum_{j \in G} \beta_j \, \pi(m_j),
\end{equation*}
and then taking the inner product with $\pi(n)$ on both sides yields
\begin{equation} \label{contradiction-equation}
0 = \beta \, \langle \pi(n), \pi(n) \rangle + \sum_{j \in G} \beta_j \,
\langle \pi(m_j) , \pi(n) \rangle .
\end{equation}
To prove (iii), we assume $P$ is obtuse and that there is some choice of a
dependence as in (\ref{minimal-dependence}) such that $\beta \geq 0$. But
then we reach a contradiction in Equation (\ref{contradiction-equation}),
because we assumed $\beta_j > 0$, we have $\langle \pi(m_j) , \pi(n) \rangle
> 0$ by virtue of the obtuseness of $P$, and $\langle \pi(n), \pi(n) \rangle$
is always non-negative.
To prove (i), we assume $P$ is non-acute and that there is some choice of a
dependence as in (\ref{minimal-dependence}) such that $\beta > 0$. Then
similar considerations in equation (\ref{contradiction-equation}) imply that
we must have $\langle \pi(n), \pi(n) \rangle = 0$, i.e. $\pi(n) = 0$. This
would imply $\langle n_i, n \rangle =0$ for each $i$ in $F$. To reach a
contradiction from this, take the inner product with $n$ on both sides of
equation (\ref{minimal-dependence}), to obtain
\begin{equation*}
0= \beta \, \langle n, n \rangle + \sum_{j \in G} \beta_j \, \langle m_j, n
\rangle.
\end{equation*}
Non-acuteness (even in codimension 1) of $P$ implies $\langle m_j, n \rangle
\geq 0$, and $\langle n, n \rangle$ is always positive, so this last
equation is a contradiction to $\beta > 0$. $\qed$
One source of non-acute simple polytopes are finite Coxeter groups (see
\cite[Chapter 1]{Humphreys} for background). Recall that a finite \textit{%
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) \textit{roots} $\Phi$ by taking all the
unit normals of reflecting hyperplanes. Let $Z$ be the \textit{zonotope} (
\cite[\S 7.3]{Ziegler}) associated with $\Phi$, that is,
\begin{equation*}
Z= \left\{ \sum_{\alpha \in \Phi} c_\alpha \alpha: 0 \leq c_\alpha \leq
1\right \}.
\end{equation*}
\begin{proposition}
\label{Coxeter-zonotopes-non-acute} The zonotope $Z$ associated to any
finite Coxeter group $W$ is non-acute and simple. Furthermore $Z$ is obtuse
in codimension $1$ if $W$ is irreducible.
\end{proposition}
\begin{proof}
We refer to \cite{Humphreys} for all facts about Coxeter groups
used in this proof.
By general facts about zonotopes \cite[\S 7.3]{Ziegler},
the normal fan $\Delta$ of $Z$ is the complete fan cut out
by the hyperplanes associated with reflections in $W$.
The maximal cones in this fan are the {\it 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 $\Delta$
which span a $2$-dimensional cone form a non-obtuse angle.
Because $W$ acts transitively on the
Weyl chambers in $\Delta$, we may assume that this
pair of rays lie in the {\it fundamental Weyl chamber}, that is,
we may assume that these rays come from the dual basis
to some choice of simple roots $\alpha_1,\ldots,\alpha_d$.
Since every choice of simple roots has the property that
$\langle \alpha_i, \alpha_j \rangle \leq 0$ for all $i \neq j$,
the first assertion follows from the first part of
Lemma \ref{pairwise-non-acute} below.
The second assertion follows from Lemma \ref{pairwise-non-acute} (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.
\end{proof}
The following lemma was used in the preceding proof.
\begin{lemma}
\label{pairwise-non-acute} Let $\{\alpha_i\}_{i=1}^d$ be a basis for $%
\mathbb{R}^d$ with $\langle \alpha_i, \alpha_j \rangle \leq 0$ for all $i
\neq j$. Then the dual basis $\{\alpha^\vee_i\}_{i=1}^d$ satisfies
\begin{enumerate}
\item[(i)] $\langle \alpha^\vee_i, \alpha^\vee_j \rangle \geq 0$ for all $i
\neq j$, and
\item[(ii)] $\langle \alpha^\vee_i, \alpha^\vee_j \rangle > 0$ for all $i
\neq j$ if the ``obtuseness graph" on $\{1,2,\ldots,d\}$, having an edge $%
\{i,j\}$ whenever $\langle \alpha_i, \alpha_j \rangle < 0$, is connected.
\end{enumerate}
\end{lemma}
\begin{proof}
We prove assertion (i) by induction
on $d$, with the cases $d=1,2$ being trivial.
In the inductive step, assume $d \geq 3$. Without loss of
generality, we must show $\langle \alpha_1, \alpha_2 \rangle \geq 0$.
Let $\pi: \reals^d \rightarrow \alpha_d^\perp$ be
orthogonal projection. Write
$$
\alpha_i = \pi(\alpha_i) + c_i \alpha_d
$$
for each $i \leq d-1$.
Our first claim is that $c_i \leq 0$ for each $i \leq d-1$.
To see this, note that
$$
\begin{aligned}
0 &\geq \langle \alpha_i, \alpha_d \rangle \\
&= \langle \pi(\alpha_i), \alpha_d \rangle +
c_i \langle \alpha_d, \alpha_d \rangle \\
&= c_i \langle \alpha_d, \alpha_d \rangle.
\end{aligned}
$$
Our second claim is that
$\langle \pi(\alpha_i), \pi(\alpha_j) \rangle \leq 0$
for $1 \leq i \neq j \leq d-1$.
To see this, note that
$$
\begin{aligned}
0 &\geq \langle \alpha_i, \alpha_j \rangle \\
&= \langle \pi(\alpha_i), \pi(\alpha_j) \rangle +
c_j \langle \alpha_d, \pi(\alpha_j) \rangle +
c_i \langle \pi(\alpha_i), \alpha_d \rangle +
c_i c_j \langle \alpha_d, \alpha_d \rangle \\
&=\langle \pi(\alpha_i), \pi(\alpha_j) \rangle +
c_i c_j \langle \alpha_d, \alpha_d \rangle.
\end{aligned}
$$
and the last term in the last sum is non-negative by our
first claim.
Our third claim is that $\{\pi(\alpha_i)\}_{i=1}^{d-1}$
and $\{\alpha^\vee_i\}_{i=1}^{d-1}$ are dual bases inside
$\alpha_d^\perp$. To see this, note that
$$
\begin{aligned}
\delta_{ij} &= \langle \alpha_i, \alpha^\vee_j \rangle \\
&=\langle \pi(\alpha_i), \alpha^\vee_j \rangle +
c_i \langle \alpha_d, \alpha^\vee_j \rangle \\
&= \langle \pi(\alpha_i), \alpha^\vee_j \rangle.
\end{aligned}
$$
From the second and third claims, we can apply induction
to conclude that $\langle \alpha^\vee_i, \alpha^\vee_j \rangle \geq 0$
for $1 \leq i \neq j \leq d-1$, 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 $\{\alpha_i\}_{i=1}^d$ have connected obtuseness graph, then
there is a re-indexing (that is a choice of $\alpha_d$)
so that $\{\pi(\alpha_i)\}_{i=1}^{d-1}$ also satisfies this hypothesis.
To achieve this, let $\alpha_d$ 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 $i \neq j$ with $i,j \leq d-1$ we have
$$
\langle \pi(\alpha_i) , \pi(\alpha_j) \rangle =
\langle \alpha_i , \alpha_j \rangle -
c_i c_j \langle \alpha_d , \alpha_d \rangle.
$$
This implies $\pi(\alpha_i) , \pi(\alpha_j)$ were obtuse whenever
$\alpha_i, \alpha_j$ were, so the obtuseness graph remains connected.
\end{proof}
\begin{remark}
\textrm{\ \newline
If the finite Coxeter group $W$ is \textit{crystallographic} (or a \textit{%
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 $\Delta$
is not only rational with respect to the \textit{weight lattice} $N$, but
also $m(Z)=1$ with respect to the dual lattice $M$. Hence $Z$ is Delzant, so
that the toric variety $X_\Delta$ is smooth. }
\end{remark}
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 $\sigma(Z)$. The $h$-vector in this case turns out to give the
distribution of the elements of the Weyl group $W$ according to their
\textit{descents}, i.e. the number of simple roots which they send to
negative roots (see \cite{Bjorner-Coxetercomplexes}). Generating functions
for the descent distribution of all classical Weyl groups may be found in
\cite{Reiner}. 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
\begin{equation*}
\sum_{n \geq 0} \sigma(Z_{A_{n-1}}) \frac{x^n}{n!} = \tanh(x)
\end{equation*}
which was computed in \cite[Example p. 52]{EdelmanReiner} for somewhat
different reasons.
The fact that Coxeter zonotopes have locally convex normal fans also follows
because these normal fans come from \textit{simplicial hyperplane
arrangements} (we thank M. Davis for suggesting this). Say that an
arrangement of linear hyperplanes ${}$ in $\mathbb{R}^d$ is \textit{%
simplicial} if it decomposes $\mathbb{R}^d$ into a simplicial fan.
\begin{proposition}
\label{simple-zonotope-implies-locally-convex} The fan $\Delta$ associated
to a simplicial hyperplane arrangement ${}$ is locally convex.
\end{proposition}
\begin{proof}
For each ray $\rho$ of $\Delta$, we will express $\str_\Delta(\rho)$ as
an intersection of closed half-spaces defined by a subset of the
hyperplanes of ${\mathcal A}$, thereby showing that it is convex.
To describe this intersection, note that since $\Delta$ is simplicial, given
any chamber ($d$-dimensional cone) $\sigma$ of
$\Delta$ that contains $\rho$, there is a unique hyperplane
$H_\sigma$ bounding $\sigma$ which does not contain $\rho$. Choose
a linear functional $u_\sigma$ which vanishes on $H_\sigma$
and is positive on $\rho$, and then we claim that
$$
|\str_\Delta(\rho)| =
\bigcap_{\text{chambers }\sigma \supset \rho} \{u_\sigma \geq 0\}.
$$
To see that the left-hand side is contained in the right,
note that for any chamber $\sigma$ containing $\rho$ and any hyperplane
$H$ in ${\mathcal A}$ not containing $\rho$, we must have $\sigma$ and
$\rho$ on the same side of $H$. Consequently, for every pair of
chambers $\sigma, \sigma'$ containing $\rho$ we have $u_\sigma \geq 0$
on $\sigma'$ (and symmetrically $u_{\sigma'} \geq 0$ on $\sigma$).
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 $\sigma$ not in the right, consider the unique chamber $\sigma'$
containing $\rho$ which is ``perturbed in the direction of $\sigma$''. In other
words, $\sigma'$ is chosen so that it
contains a vector $v+\epsilon w$ where $v$ is any non-zero vector
in $\rho$, $w$ is any vector in the interior of $\sigma$, and $\epsilon$ is a
very small positive number. Since $\sigma$ does not contain $\rho$, we know
$\sigma \neq \sigma'$, and hence there is at least one hyperplane of ${\mathcal A}$
separating them. Since $\Delta$ is simplicial, every bounding hyperplane
of $\sigma'$ except for $H_{\sigma'}$ will contain $r$, and hence have
$\sigma$ and $\sigma'$ on the same side (by construction of $\sigma'$). This
means $H_{\sigma'}$ must separate $\sigma$ and $\sigma'$, so $u_{\sigma'} < 0$
on $\sigma$, implying $\sigma$ is not in the left-hand side.
\end{proof}
The Coxeter zonotopes of type $A$ are related to another infinite family of
simple polytopes, the \textit{associahedra}, which turn out to have locally
convex normal fans. Recall \cite{Lee} that the associahedron $\assoc_n$ is an $%
(n-3)$-dimensional polytope whose vertices correspond to all possible
parenthesizations of a product $a_1 a_2 \cdots a_{n-1}$, and having an edge
between two parenthesizations if they differ by a single ``rebracketing".
Equivalently, vertices of $\assoc_n$ 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.
\begin{proposition}
The associahedron $\assoc_n$ has a realization as a simple convex polytope whose
normal fan $\Delta_n$ is locally convex.
\end{proposition}
\begin{proof}
In \cite[\S 3]{Lee}, the normal fan $\Delta_n$ 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
$1,2,\ldots,n-2$. Since the normal fan $\Delta_n$ is simplicial,
the associahedron is simple (as is well-known).
Our strategy for showing $\Delta_n$ 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 $\reals^{n-3}$, 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
$\Delta_n$ of $\assoc_n$ 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
$x_{\pi_1} > x_{\pi_2} >\cdots >x_{\pi_{n-2}}$
for permutations $\pi$ of $\{1,2,\ldots,n-2\}$.
To show $\Delta_n$ is locally convex, we must first identify the rays
$\rho$ of $\Delta_n$, and then show that $\str_{\Delta_n}(\rho)$ is a pointed
convex cone. According to the construction of \cite[\S 3]{Lee},
a ray $\rho_{ij}$ of $\Delta_n$ corresponds to the barycenter of
a face of the $(n-3)$-simplex which
is spanned by a set of vertices labelled by a {\it contiguous}
sequence $i,i+1,\ldots,j-1,j$ with $1 \leq i \leq j \leq n-2$,
with $(i,j) \neq (1,n-2)$.
It is then not hard to check from the construction that
$\str_\Delta(\rho_{ij})$ consists of the union of all (closed) chambers
for type $A_{n-3}$ which satisfy the inequalities
$$
x_{i},x_{i+1},\ldots,x_{j-1},x_{j} \geq x_{i-1}, x_{j+1}
$$
(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 $\Delta_n$ is
locally convex.
\end{proof}
\noindent It follows then from this Proposition and Theorem \ref
{lower-bounds}(ii) that $(-1)^{\frac{n-3}{2}} \sigma(\assoc_n) \geq 0$ for $n$ odd
(and of course, $\sigma(\assoc_n) = 0$ for $n$ even). However, as in the case of
Coxeter zonotopes of type $A$, we can compute $\sigma(\assoc_n)$ explicitly using
the formulas for the $f$-vector or $h$-vector of $\assoc_n$ given in \cite[Theorem
3]{Lee}. Specifically, these formulas imply that for $n \geq 3$ we have
\begin{equation*}
\begin{aligned} \sigma(\mathcal A_n) &=\sum_{i = 0}^{n-3} (-1)^i
\frac{1}{n-1} \binom{n-3}{i} \binom{n-1}{i+1} \\ &={}_2F_1 \left( \left.
\begin{matrix} 3-n & 2-n \\ & 2 \end{matrix} \right| -1 \right)\\ &=\left\{
\begin{matrix} (-1)^{\frac{n-3}{2}}C_{\frac{n-1}{2}} & \text{ if }n\text{ is
odd}\\ 0 & \text{ if }n\text{ is even} \end{matrix} \right. \end{aligned}
\end{equation*}
where $C_n$ denotes the Catalan number $\frac{1}{n}\binom{2n-2}{n-1}$. Here
the ${}_2F_1$ is hypergeometric series notation, and the last equality uses
Kummer's summation of a well-poised $_2F_1$ at $-1$ (see e.g. \cite[p. 9]
{Bailey}).
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 \textit{flag} if every set of vertices $%
v_1,\ldots, v_r$ which pairwise span edges of $K$ also jointly span an $%
(r-1) $-simplex of $K$.
\begin{proposition}
\label{non-acute-by-polygons} 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
\end{proposition}
\begin{proof}
The first assertion for non-acute polytopes
follows from an easy lemma due to Moussong \cite[Lemma 2.4.1]{CharneyDavis}.
In the notation of \cite{CharneyDavis},
saying that every $2$-dimensional face of
$P$ is non-acute (in codimension $1$) is equivalent to saying that
for every vertex $v$ of $P$, the spherical
simplex $\sigma = Lk(v,P)$ has size $\geq \frac{\pi}{2}$. Then
\cite[Lemma 2.4.1]{CharneyDavis}
asserts that every link $Lk(\tau,\sigma)$ of a face of this spherical
simplex also has size $\geq \frac{\pi}{2}$. But a face $F$ of
$P$ containing $v$ has $Lk(v,F)$ of the form $Lk(\tau,\sigma)$
for some $\tau$, 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 $\sigma = Lk(v,P)$ for any vertex $v$, which will
have size $ \geq \frac{\pi}{2}$. Then its polar dual
spherical convex polytope $\sigma^*$ will have all of its dihedral
angles less than or equal to $\frac{\pi}{2}$. This forces $\sigma^*$
to be a spherical simplex, by \cite[p. 44]{Vinberg}, and hence $\sigma$
itself must be a spherical simplex. This implies $v$ has exactly $d$ neighbors,
so $P$ is simple.
\end{proof}
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 \ref{Coxeter-zonotopes-non-acute},
are not in general obtuse in dimensions $3$ and higher. It is easy to find
obtuse polytopes in dimensions up to $4$:
\begin{enumerate}
\item[$\bullet$] in dimension 2, the regular $n$-gons for $n \geq 5$,
\item[$\bullet$] in dimension 3, the dodecahedron,
\item[$\bullet$] in dimension 4 the ``$120$-cell" (see \cite[pp. 292-293]
{Coxeter})
\end{enumerate}
However M. Davis has pointed out to us that in dimensions $5$ higher, there
are no obtuse polytopes, due to a result of Kalai \cite[Theorem 1]{Kalai}
(see also \cite[p. 68]{Vinberg} for the case of simple polytopes): every $d$%
-dimensional convex polytope for $d \geq 5$ contains either a triangular or
quadrangular $2$-dimensional face.
\section{Relation to conjectures of Hopf and of Charney and Davis}
\label{Hopf}
In this section we discuss the relation of Theorem \ref{lower-bounds}(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, Poincar\'e duality implies that the Euler characteristic $\chi(M)$
vanishes. When $d$ is even, a conjecture of H. Hopf (see e.g. \cite
{CharneyDavis}) asserts that if $M^d$ has non-positive sectional curvature,
the Euler characteristic $\chi(M^d)$ satsfies
\begin{equation*}
(-1)^{\frac{d}{2}} \chi(M^d) \geq 0.
\end{equation*}
This result is known for $d=2,4$ by Chern's Gauss-Bonnet formula, but open
for general $d$; see \cite[\S 0]{CharneyDavis} for some history.
Charney and Davis \cite{CharneyDavis} 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) \textit{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
\textit{geodesic space}. Gromov has defined a notion of when a geodesic
space is \textit{nonpositively curved}, and Charney and Davis made the
following conjecture:
\begin{conjecture}
\cite[Conjecture A]{CharneyDavis} \label{conjecture-A} If $M^{d}$ is a
non-positively curved, piecewise Euclidean, closed manifold with $d$ even,
then
\begin{equation*}
(-1)^{\frac{d}{2}}\chi (M^{d})\geq 0.
\end{equation*}
\end{conjecture}
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 \textit{link} of $v$ in $M^d$, which is homeomorphic to a \textit{%
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 \cite[(3.4.3)]
{CharneyDavis} that the Euler characteristic $\chi(M^d)$ can be written as
the sum of certain local quantities $\kappa( Lk(v,M^d) )$ defined in terms
of the metric structure on $Lk(v,M^d)$:
\begin{equation} \label{combinatorial-Gauss-Bonnet}
\chi(M^d) = \sum_v \kappa( Lk(v,M^d) ).
\end{equation}
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 $\kappa( Lk(v,M^d) )$ 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 \textit{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 \ref{conjecture-A} would follow via Equation (\ref
{combinatorial-Gauss-Bonnet}) from
\begin{conjecture}
\cite[Conjecture D]{CharneyDavis} \label{conjecture-D} If $\Delta $ is a
flag simplicial complex triangulating a generalized homology $(d-1)$-sphere
with $d$ even, then
\begin{equation*}
(-1)^{\frac{d}{2}}\kappa (\Delta )\geq 0.
\end{equation*}
\end{conjecture}
\noindent This \textit{Charney-Davis conjecture} is trivial for $d=2$, has
recently been proven by Davis and Okun \cite{DavisOkun} 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 \cite[\S 7]{CharneyDavis}) 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:
\begin{proposition}
\label{locally-convex-implies-flag} A locally convex complete simplicial fan
$\Delta $ in $\mathbb{R}^{d}$ is flag, when considered as a simplicial
complex triangulating a $(d-1)$-sphere.
\end{proposition}
\begin{proof}
Assume that $\Delta$ is not flag, so that there exist rays
$\rho_1,\ldots,\rho_k$ whose convex hull $\sigma:=\conv(\rho_1,\ldots,\rho_k)$
is {\it not} a cone of $\Delta$, but $\conv(\rho_i,\rho_j)$ {\it is} a cone
of $\Delta$ for each $i,j$. Choose such a collection of rays
of minimum cardinality $k$, so that $\conv(\rho_1,\ldots,\hat{\rho_i},\ldots,\rho_k)$
is a cone of $\Delta$ for each $i$ (in other words, the boundary complex
$\partial\sigma$ is a subcomplex of $\Delta$).
We wish to show that $\str_\Delta(\rho_1)$ is not convex. To see this, consider
$\sigma \cap \str_\Delta(\rho_1)$, that is, the collection of cones
$$
\{ \sigma' \cap \sigma : \sigma' \in \str_\Delta(\rho_1)\}.
$$
Since $\sigma$ is not a cone of $\Delta$ but $|\Delta|=\reals^d$,
this collection must contain at least one $2$-dimensional cone
of the form $\sigma' \cap \sigma = \conv(\rho_1,\rho)$,
where $\rho$ is a ray of $\sigma$ but $\rho \not\in \{\rho_2,\ldots,\rho_k\}$.
Since $\rho$ lies inside $\sigma$ and $\partial\sigma$ is a
subcomplex of $\Delta$, $\rho$ cannot lie in $\partial\sigma$ (else
some cone of $\partial\Delta$ would be further
subdivided, and not be a cone of $\Delta$). Consequently $\rho$ lies
in the interior of $\sigma$. Then the ray $\rho': =\rho - \epsilon \rho_1$
for very small $\epsilon > 0$ has the following properties:
\begin{enumerate}
\item[$\bullet$] $\rho'$ lies in $\sigma$, because $\rho$ was in the interior of
$\sigma$,
\item[$\bullet$] $\rho'$ therefore lies in the convex hull of $\str_\Delta(\rho_1)$,
since $\sigma$ does (as its extreme rays $\rho_1,\ldots,\rho_k$ of $\sigma$
are all in $\str_\Delta(\rho_1)$),
\item[$\bullet$] $\rho'$ does not lie in $\str_\Delta(\rho_1)$, else it would lie
in a cone $\sigma'$ of $\Delta$ containing $\rho_1$, and then $\sigma'$ would
contain $\rho$ in the relative interior of one of its faces, a contradiction.
\end{enumerate}
Therefore $\str_\Delta(\rho_1)$ is not convex.
\end{proof}
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 $PL$-spheres. Therefore if one takes the barycentric subdivision of
any regular cellular sphere which is not $PL$ (such as the double suspension
of Poincar\'{e}'s famous homology sphere), this will give a flag simplicial
sphere which is not $PL$ 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 $\Delta$ is its normal fan
considered as a $(d-1)$-dimensional simplicial complex, one can check that
\begin{equation} \label{sigma-kappa-relation}
\sigma(P) = 2^d \kappa( \Delta ).
\end{equation}
As a consequence, we deduce the following from Proposition \ref
{locally-convex-implies-flag} and Theorem \ref{lower-bounds} (i).
\begin{corollary}
\label{non-acute-lower-bound} Let $P$ be rational simple polytope, $\Delta $
its normal fan. If $\Delta $ is locally convex, then it is flag and
satisfies the Charney-Davis conjecture.
In particular by Corollary \ref{euclidean-imply-bounds}, if $P$ is a
non-acute simple rational polytope then its normal fan $\Delta $ is flag and
satisfies the Charney-Davis conjecture.
\end{corollary}
It is worth mentioning that the special case of Conjecture \ref{conjecture-A}
considered in \cite{CharneyDavis} 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 \textit{corner decomposition} if the
local structure at every vertex in the decomposition is combinatorially
isomorphic to the coordinate orthants in $\mathbb{R}^d$, that is, each link $%
Lk(v,M^d)$ has the combinatorial structure of the boundary complex of a $d$%
-dimensional \textit{cross-polytope} or \textit{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 \cite[(3.5.2)]
{CharneyDavis}) shows that for a manifold $M^d$ with corner decomposition
into simple polytopes $P_1,\ldots,P_N$ one has
\begin{equation} \label{local-formula}
\chi(M^d) = \frac{1}{2^d} \sum_{i=1}^N \sigma(P_i).
\end{equation}
The following corollary is then immediate from this relation and Theorem \ref
{lower-bounds}.
\begin{corollary}
\label{corner-decomposition-lower-bound} 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
\begin{equation*}
(-1)^{\frac{d}{2}}\chi (M^{d})\geq 0.
\end{equation*}
In particular, this holds if each of the simple $d$-polytopes is non-acute.
\end{corollary}
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 \cite
{DavisJanuszkiewiczScott}.
There is also an important general construction of such manifolds called
\textit{mirroring} which we now discuss. This construction (or its polar
dual) appears repeatedly in the work of Davis \cite{Davis-Annals,
Davis-Duke, DavisJanuszkiewicz, DavisJanuszkiewiczScott}, and was used in
\cite[\S 6]{CharneyDavis} to show that the case of their Conjecture \ref
{conjecture-A} for manifolds decomposed into right-angled cubes is
equivalent to their Conjecture \ref{conjecture-D}. 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
\begin{equation*}
\chi(ML) = 2^n \cdot \kappa(L).
\end{equation*}
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 $F_1,F_2,\ldots,F_n$, and let $M(P)$ be the
quotient of $2^n$ disjoint copies $\{P_\epsilon\}_{\epsilon \in \{+,-\}^n}$
of $P$, in which two copies $P_\epsilon, P_{\epsilon^{\prime}}$ are
identified along their face $F_i$ whenever $\epsilon, \epsilon^{\prime}$
differ in the $i^{th}$ coordinate and nowhere else. As a consequence of
equation (\ref{local-formula}) we have
\begin{equation} \label{mirror-equation}
\chi(M(P)) = 2^{n-d} \cdot \sigma(P),
\end{equation}
which shows that the ``non-acute" assertions in Corollaries \ref
{corner-decomposition-lower-bound} and \ref{non-acute-lower-bound} 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.
\vskip .1in \noindent \textit{Proof of Corollary \ref{euclidean-imply-bounds}
(i) without assuming rationality of $P$ (as referred to in Remark \ref
{non-acute-bound-without-rationality})}: Assume that $P$ is a simple
non-acute $d$-dimensional polytope with $d$ even. We wish to show that $%
(-1)^{\frac{d}{2}} \sigma(P) \geq 0$.
Construct $M(P)$ as above, a manifold with corner decomposition into
non-acute simple polytopes having $\chi(M(P)) = 2^{n-d} \sigma(P)$ if $P$
had $n$ codimension $1$ faces. In the notation of \cite{CharneyDavis}, this
means that all the links $Lk(v,M(P))$ have size $\geq \frac{\pi}{2}$ and are
combinatorially isomorphic to boundaries of cross-polytopes. This implies
that these links' underlying simplicial complexes are flag complexes
satisfying \cite[Conjecture D']{CharneyDavis}, and then \cite[Proposition
5.7]{CharneyDavis} implies that each of these links $Lk(v,M(P))$ satisfies
\cite[Conjecture C']{CharneyDavis}. This implies that $(-1)^{\frac{d}{2}}
\kappa( Lk(v,M(P)) ) \geq 0$. Combining this with Equation (\ref
{combinatorial-Gauss-Bonnet}), we conclude that $(-1)^{\frac{d}{2}}
\chi(M(P)) \geq 0$, and finally via Equation (\ref{mirror-equation}), that $%
(-1)^{\frac{d}{2}} \sigma(P) \geq 0$.
We note that a similar argument (involving an adaptation of \cite[Lemma
2.4.1]{CharneyDavis}) proves $(-1)^{\frac{d}{2}} \sigma(P) > 0$ when $P$ is
obtuse, but does not yield in any obvious way the stronger assertion of
Corollary \ref{euclidean-imply-bounds} (iii). $\qed$
\section{Appendix: the conormal bundle of a toric divisor}
In this appendix we describe the conormal bundle of a toric divisor on $%
X=X_\Delta$ when the fan $\Delta$ is complete and simplicial. Denote the
collection of toric divisors on $X$ by $D_{1},\ldots,D_{m}.$ As mentioned in
Section \ref{Hirzebruch}, 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 $%
O_{D_{1}}\left( -D_{1}\right)$. It corresponds to a continuous piecewise
linear function:
\begin{equation*}
\Psi _{-D_{1}}^{D_{1}}:N_{\mathbb{R}}/\rho _{1}\rightarrow \mathbb{R}
\end{equation*}
as in the discussion of Section \ref{Hirzebruch}. Here $\rho _{1}$ is the
ray in the fan $\Delta $ corresponding to the divisor $D_{1}$. We wish to
identify the graph of $\Psi _{-D_{1}}^{D_{1}}$ with $\mathrm{link}_{\Delta
}\left( \rho _{1}\right) $, which we recall is the boundary of $\mathrm{star}%
_{\Delta}(\rho_1)$, the latter being the union of all cones of $\Delta$
containing $\rho_1$.
\begin{proposition}
Let $D_1$ be a toric divisor of a toric variety $X=X_\Delta$ with $\Delta$
simplicial. Then the graph of the piecewise linear function for $%
O_{D_1}(D_1) $ is affinely equivalent to the boundary $link_{\Delta }\left(
\rho _{1}\right) $ of $star_{\Delta }\left( \rho_1 \right)$, where $\rho_1$
is the ray corresponding to $D_1$.
\end{proposition}
\begin{proof} We can index the toric divisors $D_1,\ldots,D_m$
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 $\rho _{i}$
corresponding to $D_{i}$. We choose a decomposition of $N$ into a direct sum
of $\mathbb{Z}n_{1}$ with another lattice $N^{\prime }$ which is isomorphic
to $N/\rho _{1}$ (here we are abusing notation by referring to the quotient
lattice $N/\mathbb{Z}n_{1}$ as $N/\rho _{1}$).
Then we can write $n_{i}=b_{i}n_{1}+c_{i}n_{i}^{\prime }$
where $n_{i}^{\prime }\in N^{\prime }\cong N/\rho _{1}$ is indecomposable
(i.e. not of the form $k \, n_i^{\prime\prime}$
for some integer $k$ with $|k| \geq 2$ and
$n_i^{\prime\prime} \in N^{\prime }$ ),
and $c_{i}$ is some nonnegative integer.
Now we choose the linear functional $u$ on $N$ such that $\left\langle
u,n_{1}\right\rangle =1$ and its restriction to $N^{\prime }$ is zero. Then
in the Chow group of $X$ we have the following relation
(see \cite[p.106]{Fulton}):
\begin{equation*}
\sum_{i=1}^{m}\left\langle u,n_{i}\right\rangle D_{i}=0.
\end{equation*}
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 \cite[p. 108]{Fulton}, we have
\begin{equation*}
\bigotimes_{i=1}^{l} O_{D_{1}}
\left( \frac{\left\langle u,n_{i}\right\rangle}{c_i} D_{i}\right) =O_{D_{1}}.
\end{equation*}
Or equivalently, since $\left\langle u,n_1\right\rangle=1$ and
$\left\langle u_i,n_1\right\rangle=b_i$, we have
\begin{equation*}
O_{D_{1}}\left( -D_{1}\right)
=\bigotimes_{i=2}^{l}O_{D_{1}}\left(\frac{b_{i}}{c_i}D_{i}\right) .
\end{equation*}
Now under the identification $N^{\prime }\cong N/\rho _{1}$, the restriction
of the divisor $D_{i}$ to $D_{1}$ corresponds to the ray in $N/\rho _{1}$
spanned by $n_{i}^{\prime }$ when $2\leq i\leq l$. Therefore the piecewise
linear function $\Psi _{-D_{1}}^{D_{1}}:N_{\mathbb{R}}/\rho _{1}\rightarrow
\mathbb{R}$ is determined by $\Psi_{-D_{1}}^{D_{1}}\left( n_{i}^{\prime
}\right) =\frac{b_{i}}{c_i}$. This implies the assertion of the proposition.
\end{proof}
\section{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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\end{document}