# On Hypergeometric Duality Conjecture

Lev Borisov  Department of Mathematics
Rutgers University
Piscataway, NJ 08854<EMAIL_ADDRESS>and  Zengrui Han  Department of
Mathematics
Rutgers University
Piscataway, NJ 08854<EMAIL_ADDRESS>
###### Abstract.

We give an explicit formula for the duality, previously conjectured by Horja
and Borisov, of two systems of GKZ hypergeometric PDEs. We prove that in the
appropriate limit this duality can be identified with the inverse of the Euler
characteristics pairing on cohomology of certain toric Deligne-Mumford stacks,
by way of $\Gamma$-series cohomology valued solutions to the equations.

###### Contents

  1. 1 Introduction
  2. 2 Pairing of solutions
  3. 3 Pairing of the Gamma series
  4. 4 Euler characteristic pairing
  5. 5 Extensions and open questions

##  1\. Introduction

Let $C$ be a finite rational polyhedral cone in a lattice
$N={\mathbb{Z}}^{{\operatorname{rk}}N}$. We assume that all ray generators of
$C$ lie on a primitive hyperplane $\deg(\cdot)=1$ where
$\deg:N\to{\mathbb{Z}}$ is a linear function. This data encodes an affine
toric variety $X={\rm Spec}\,{\mathbb{C}}[N^{\vee}\cap C^{\vee}]$, with the
hyperplane condition equivalent to $X$ being Gorenstein, i.e. having trivial
dualizing sheaf.

Let $\\{v_{i}\\}_{i=1}^{n}$ be a set of $n$ lattice points in $C$ which
includes all of its ray generators, with $\deg(v_{i})=1$ for all $i$. One can
construct crepant resolutions ${\mathbb{P}}_{\Sigma}\to X$ by looking at
subdivisions $\Sigma$ of $C$ based on triangulations that involve some of the
points $v_{i}$. Typically, ${\mathbb{P}}_{\Sigma}$ is a smooth Deligne-Mumford
stack rather than a smooth variety, with the rare exception of when all cones
in $\Sigma$ are unimodular.

A particular case of Kawamata-Orlov $K\to D$ conjecture asserts that the
derived categories of coherent sheaves on ${\mathbb{P}}_{\Sigma}$ are
independent of the choice of $\Sigma$. In fact, it is expected that there is
an isotrivial family of triangulated categories which interpolates between the
categories in question. This rather mysterious family is well understood at
the level of complexified Grothendieck $K$-groups. Namely, these should
correspond to solutions of a certain version of the Gel’fand-Kapranov-
Zelevinsky system of hypergeometric PDEs. In fact, due to non-compactness of
$X$ and ${\mathbb{P}}_{\Sigma}$, there are two such systems, denoted by
$\mathrm{bbGKZ}(C,0)$ and $\mathrm{bbGKZ}(C^{\circ},0)$, conjecturally dual to
each other [1]. In the appropriate limit that corresponds to the triangulation
$\Sigma$, solutions to these systems can be identified with usual and
compactly supported orbifold cohomology of ${\mathbb{P}}_{\Sigma}$ by means of
two special $\Gamma$\- series. In this paper we settle positively the duality
conjecture of [1]. In fact, our duality formula is simple enough to hope that
it may provide hints as to how one could try to construct the aforementioned
triangulated categories.

We will now set up the notations and review the better-behaved GKZ
hypergeometric systems.

######  Definition 1.1.

Consider the system of partial differential equations on the collection of
functions $\\{\Phi_{c}(x_{1},\ldots,x_{n})\\}$ in complex variables
$x_{1},\ldots,x_{n}$, indexed by the lattice points in $C$:

$\displaystyle\partial_{i}\Phi_{c}=\Phi_{c+v_{i}},\quad\sum_{i=1}^{n}\langle\mu,v_{i}\rangle
x_{i}\partial_{i}\Phi_{c}+\langle\mu,c\rangle\Phi_{c}=0$

for all $\mu\in N^{\vee}$, $c\in C$ and $i=1,\ldots,n$. We denote this system
by $\mathrm{bbGKZ}(C,0)$. Similarly by considering lattice points in the
interior $C^{\circ}$ only, we can define $\mathrm{bbGKZ}(C^{\circ},0)$.

This system gives a holonomic system of PDEs. It follows from the general
theory of holonomic $D$-modules that its rank (i.e., the dimension of the
solution space) is finite. For more background on this, we refer to [9]. In
contrast to the usual GKZ system where rank jumps may occur at non-generic
parameters (see [10]), it is proved in [3] that the better-behaved GKZ systems
always have the expected rank which is equal to the normalized volume of the
convex hull of ray generators of the cone $C$.

It has been previously conjectured in [1] that the systems
$\mathrm{bbGKZ}(C,0)$ and $\mathrm{bbGKZ}(C^{\circ},0)$ are dual to each
other, in the sense that there is a pairing $\langle\cdot,\cdot\rangle$
between solutions $\Phi=(\Phi_{c})$ and $\Psi=(\Psi_{d})$ thereof in the form

$\langle\Phi,\Psi\rangle=\sum_{c,d}p_{c,d}({\bf x})\Phi_{c}\Psi_{d},$

where $p_{c,d}$ are polynomials in ${\bf x}$, with only finitely many of them
nonzero. This pairing should be constant in $\bf x$ and could be viewed as the
duality of the local systems of solutions. A nontrivial example of this
duality has been verified in [1] and the ${\operatorname{rk}}(N)=2$ case has
been settled affirmatively in [2]. Moreover, in certain regions of ${\bf x}$
that roughly correspond to the complexified Kähler cones of
${\mathbb{P}}_{\Sigma}$, one can construct solutions of $\mathrm{bbGKZ}(C,0)$
and $\mathrm{bbGKZ}(C^{\circ},0)$ with values in certain cohomology or
$K$-theory groups of ${\mathbb{P}}_{\Sigma}$. Then it was conjectured in [1]
that the above pairing should give (up to a constant) the inverse of a certain
Euler characteristic pairing on these spaces. In this paper we are able to
verify both statements and thus prove Conjecture 7.3 of [1] in full
generality.

Specifically, the following formula provides the pairing in question. Let
$v\in C^{\circ}$ be an element in general position. For a subset
$I\subseteq\\{1,\ldots,n\\}$ of size ${\operatorname{rk}}N$ we consider the
cone $\sigma_{I}=\sum_{i\in I}{\mathbb{R}}_{\geq 0}v_{i}$. We define the
coefficients $\xi_{c,d,I}$ for $c+d=v_{I}$ as

$\displaystyle\xi_{c,d,I}=\left\\{\begin{array}[]{ll}&(-1)^{\operatorname{deg}(c)},\text{
if }\dim\sigma_{I}=\operatorname{rk}N\text{ and both }c+\varepsilon v\text{
and }d-\varepsilon v\in\sigma_{I}^{\circ}\\\ &0,\text{
otherwise}.\end{array}\right.$

Here the condition has to hold for all sufficiently small $\varepsilon>0$. As
usual, we denote by $\operatorname{Vol}_{I}$ the absolute value of the
determinant of the matrix of coefficients of $v_{i},~{}i\in I$ in a basis of
$N$ (i.e., the normalized volume of $I$). We can now formulate the first
result of this paper.

Theorem 2.4. For any pair of solutions $(\Phi_{c})$ and $(\Psi_{d})$ of
$\operatorname{bbGKZ}(C,0)$ and $\operatorname{bbGKZ}(C^{\circ},0)$
respectively, the pairing

$\displaystyle\langle\Phi,\Psi\rangle=\sum_{c,d,I}\xi_{c,d,I}\operatorname{Vol}_{I}\left(\prod_{i\in
I}x_{i}\right)\Phi_{c}\Psi_{d}$

is a constant.

As was mentioned before, for a regular triangulation $\Sigma$ there is a
description of solutions to $\operatorname{bbGKZ}(C,0)$ and
$\operatorname{bbGKZ}(C^{\circ},0)$ in terms of the Gamma series
$\Gamma=(\Gamma_{c})$ and $\Gamma^{\circ}=(\Gamma^{\circ}_{d})$ with values in
certain orbifold cohomology spaces $H$ and $H^{c}$ associated to
${\mathbb{P}}_{\Sigma}$, considered in [1]. Then the second main result of the
paper is the following.

Theorem 4.2. The constant pairing $\langle\Gamma,\Gamma^{\circ}\rangle$ is
equal up to a constant factor to the inverse of the Euler characteristic
pairing $\chi(-,-):H\otimes H^{c}\rightarrow{\mathbb{C}}$.

The paper is organized as follows. In Section 2 we prove the above Theorem
2.4. In Section 3 we introduce the spaces $H$ and $H^{c}$, the solutions
$\Gamma$ and $\Gamma^{c}$ with values in them and compute the pairing of
Theorem 2.4 on them. We also calculate the asymptotic behavior of the series
and their pairing in the large Kähler limit, which is used in the next
section. In Section 4 we prove that this pairing is the inverse of the Euler
characteristic pairing between $H$ and $H^{c}$. This, in particular, implies
that the pairing of Theorem 2.4 is nondegenerate. Finally, in Section 5 we
explain some easy extensions of our results and state some open questions.

##  2\. Pairing of solutions

The goal of this section is to define a pairing between the solution spaces of
the better-behaved GKZ systems associated to $C$ and $C^{\circ}$. We first
study a particular class of pairings and find a sufficient condition to make
it give a constant for any pair of solutions of better-behaved GKZ systems.
Then we provide a special example of this pairing, inspired by the fan
displacement formula for the resolution of the diagonal in toric varieties,
due to Fulton and Sturmfels [6].

To state the first main result of this section, we first introduce some
notations. Suppose $J$ is a subset of $\\{1,2,\ldots,n\\}$ with
$|J|=\operatorname{rk}N+1$. We will call such subset _spanning_ if
$\\{v_{i},i\in J\\}$ spans $N_{\mathbb{R}}$ over ${\mathbb{R}}$. For a
spanning set $J$ there is a unique (up to multiplication by a constant factor)
linear relation among the vectors $\\{v_{i}\\}_{i\in J}$

$\displaystyle\sum_{i\in J}a_{i}v_{i}=0.$

We introduce $\operatorname{sgn}:J\rightarrow\\{0,\pm 1\\}$ by
$\operatorname{sgn}(j)$ being $-1$, $0$ or $1$ if $a_{i}$ is negative, zero or
positive, respectively. This gives a decomposition $J=J_{+}\sqcup J_{-}\sqcup
J_{0}$ of the spanning set $J$. Note that while $\operatorname{sgn}$ depends
on the choice of scaling of the above linear relation, the expressions
$\operatorname{sgn}(j_{1})\operatorname{sgn}(j_{2})$ are well-defined.

The following lemma will be used later in this section. For a subset
$I\subseteq\\{1,\ldots,n\\}$ of size ${\operatorname{rk}}\,N$ we denote by
${\rm Vol}_{I}$ the normalized volume of the convex hull of the origin and
$v_{i},{i\in I}$.

######  Lemma 2.1.

Let $I\subset\\{1,\ldots,n\\}$ be such that $\\{v_{i},i\in I\\}$ form a basis
of $N_{\mathbb{R}}$. Suppose that $I$ contains $1$ and consider $j\not\in I$.
Consider the spanning set $J=I\cup\\{j\\}$. Let $\mu$ denote the unique linear
function that takes value ${\rm Vol}_{I}$ on $v_{1}$ and $0$ on $v_{i},i\in
I\setminus 1$. Then
$\mu(v_{j})=-\operatorname{sgn}(1)\operatorname{sgn}(j){\rm Vol}_{J\setminus
1}$ for the $\operatorname{sgn}$ defined for $J$.

###### Proof.

Up to sign, we can think of the linear function $\mu$ as taking a wedge
product with $\Lambda_{i\in I\setminus 1}v_{i}$. Thus, $\mu(v_{j})=\pm{\rm
Vol}_{J\setminus 1}$ and we just need to determine the sign. Since
$\\{v_{i},i\in I\\}$ form a basis, the coefficient $a_{j}$ in the relation
$\sum_{i\in J}a_{i}v_{i}=0$ is nonzero and we may consider it to be $1$, which
ensures $\operatorname{sgn}(j)=1$. We apply $\mu$ to $\sum_{i\in
J}a_{i}v_{i}=0$ to get $a_{1}{\rm Vol}_{I}+\mu(v_{j})=0$. This implies that
$a_{1}$ and $\mu(v_{j})$ have opposite signs, and the definition of
$\operatorname{sgn}(1)$ finishes the argument. ∎

Motivated by our previous work [2], we will look at pairings
$\langle\cdot,\cdot\rangle$ that only have monomial terms $x_{I}=\prod_{i\in
I}x_{i}$ for subsets $I$ of $\\{1,\cdots,n\\}$ of size $\operatorname{rk}N$.
The following proposition provides a sufficient condition on the pairing being
a constant.

######  Proposition 2.2.

Let $\\{\xi_{c,d,I}\\}$ be a collection of complex numbers for all $c\in
C,\,d\in C^{\circ},\,I\subseteq\\{1,\ldots,n\\}$ such that $c+d=\sum_{i\in
I}v_{i}$ and $|I|={\operatorname{rk}}\,N$. Suppose that

$\displaystyle 0=\sum_{j\in
J}\operatorname{sgn}(j)\Big{(}\xi_{c-v_{j},d,J\setminus j}\chi(c-v_{j}\in
C)+\xi_{c,d-v_{j},J\setminus j}\chi(d-v_{j}\in C^{\circ})\Big{)}$

holds for all $c\in C,\,d\in C^{\circ}$ and all spanning subsets
$J\subseteq\\{1,2,\ldots,n\\}$ with $|J|=\operatorname{rk}N+1$ and $\sum_{i\in
J}v_{i}=c+d$. Here $\chi$ denotes the characteristic function ($1$ if the
statement is true and $0$ if it is false). Then

$\langle\Phi,\Psi\rangle=\sum_{|I|=\operatorname{rk}N}\sum_{c+d=v_{I}}\xi_{c,d,I}{\rm
Vol}_{I}x_{I}\Phi_{c}\Psi_{d}$

is a constant for any pair of solutions $(\Phi,\Psi)$.

###### Proof.

Without loss of generality, it suffices to show that
$\partial_{1}\langle\Phi,\Psi\rangle=0$. We compute it as follows

$\begin{split}\partial_{1}\langle\Phi,\Psi\rangle=\sum_{\stackrel{{\scriptstyle|I|=\operatorname{rk}N,}}{{c+d=v_{I}}}}&\xi_{c,d,I}{\rm
Vol}_{I}x_{I}(\Phi_{c+v_{1}}\Psi_{d}+\Phi_{c}\Psi_{d+v_{1}})\\\
&+\sum_{\stackrel{{\scriptstyle 1\in
I,|I|=\operatorname{rk}N,}}{{c+d=v_{I}}}}\xi_{c,d,I}{\rm Vol}_{I}x_{I\setminus
1}\Phi_{c}\Psi_{d}\end{split}$ (2.1)

and now use relations on $\Phi$ and $\Psi$ to manipulate the second sum. For
each term, let $\mu$ be the linear function given by

$\mu(v)=v\wedge(\Lambda_{j\in I\backslash 1}v_{j})$

under the standard identification of
$\Lambda^{{\operatorname{rk}}N}N_{\mathbb{R}}\cong{\mathbb{R}}$ where we
choose the order of $\\{v_{j}:j\in I\backslash 1\\}$ in the wedge product such
that $\mu(v_{1})=\operatorname{Vol}_{I}$ is positive. Note that $\mu(v_{j})=0$
for all $j\in I\setminus 1$.

We use $\mu(c)+\mu(d)=\mu(\sum_{i\in I}v_{i})={\rm Vol}_{I}$ and add
appropriate multiples of equations for $\Phi_{c}$ and $\Psi_{d}$ with this
$\mu$ to get

$\displaystyle\Phi_{c}\Psi_{d}{\rm Vol}_{I}$ $\displaystyle=-\sum_{j\notin
I\setminus 1}x_{j}\mu(v_{j})(\Phi_{c+v_{j}}\Psi_{d}+\Phi_{c}\Psi_{d+v_{j}}).$

Thus, (2.1) can be rewritten as

$\displaystyle\partial_{1}\langle\Phi,\Psi\rangle=$
$\displaystyle\sum_{\stackrel{{\scriptstyle|I|=\operatorname{rk}N,}}{{c+d=v_{I}}}}\xi_{c,d,I}{\rm
Vol}_{I}x_{I}(\Phi_{c+v_{1}}\Psi_{d}+\Phi_{c}\Psi_{d+v_{1}})$
$\displaystyle-\sum_{\stackrel{{\scriptstyle 1\in
I,|I|=\operatorname{rk}N,}}{{c+d=v_{I}}}}\sum_{j\notin I\setminus
1}\xi_{c,d,I}\mu(v_{j})x_{I_{1\to
j}}(\Phi_{c+v_{j}}\Psi_{d}+\Phi_{c}\Psi_{d+v_{j}})$ $\displaystyle=$
$\displaystyle\sum_{\stackrel{{\scriptstyle 1\not\in
I,|I|=\operatorname{rk}N,}}{{c+d=v_{I}}}}\xi_{c,d,I}{\rm
Vol}_{I}x_{I}(\Phi_{c+v_{1}}\Psi_{d}+\Phi_{c}\Psi_{d+v_{1}})$
$\displaystyle-\sum_{\stackrel{{\scriptstyle 1\in
I,|I|=\operatorname{rk}N,}}{{c+d=v_{I}}}}\sum_{j\notin
I}\xi_{c,d,I}\mu(v_{j})x_{I_{1\to
j}}(\Phi_{c+v_{j}}\Psi_{d}+\Phi_{c}\Psi_{d+v_{j}})$

where we canceled the terms with $1\in I$ in the first sum with $j=1$ in the
second sum. Here $I_{1\rightarrow j}=I\backslash\\{1\\}\cup\\{j\\}$. Note that
$\mu$ depends on the set $I$.

Let us now compute the coefficient at
$x_{\hat{I}}\Phi_{\hat{c}}\Psi_{\hat{d}}$ in the above expression. This
coefficient gets contributions from the first sum with $I=\hat{I}$ and from
the second sum with $I=\hat{I}\cup 1\setminus j$. We observe that $\hat{I}$
has size $\operatorname{rk}N$ and does not contain $1$. Also note that if
${\rm Vol}_{\hat{I}}=0$, then the coefficient is zero. Indeed, in the second
sum, $\mu(v_{j})=\pm{\rm Vol}_{I_{1\rightarrow j}}$. Finally, we must have

$\hat{c}+\hat{d}=v_{J},~{}J={\\{1\\}\sqcup\hat{I}}.$

We look at the set $J$ which we know to be spanning, since it contains
$\hat{I}$. By Lemma 2.1, we see that
$\mu(v_{j})=\operatorname{sgn}(1)\operatorname{sgn}(j){\rm Vol}_{\hat{I}}$.
Therefore, the first line contributes

$\xi_{\hat{c}-v_{1},\hat{d},\hat{I}}{\rm Vol}_{\hat{I}}\chi(\hat{c}-v_{1}\in
C)+\xi_{\hat{c},\hat{d}-v_{1},\hat{I}}{\rm Vol}_{\hat{I}}\chi(\hat{d}-v_{1}\in
C^{\circ})$

and the second line contributes

$\displaystyle\sum_{j\in J\setminus
1}\operatorname{sgn}(1)\operatorname{sgn}(j)\xi_{\hat{c}-v_{j},\hat{d},J\setminus
j}\operatorname{Vol}_{\hat{I}}\chi(\hat{c}-v_{j}\in C)$
$\displaystyle+\sum_{j\in J\setminus
1}\operatorname{sgn}(1)\operatorname{sgn}(j)\xi_{\hat{c},\hat{d}-v_{j},J\setminus
j}{\rm Vol}_{\hat{I}}\chi(\hat{d}-v_{j}\in C^{\circ}).$

We observe that if $\operatorname{sgn}(1)=0$, then $\\{v_{i},i\in J\setminus
1\\}$ do not span $N_{\mathbb{R}}$, so ${\rm Vol}_{\hat{I}}=0$ and the
statement trivially holds. Thus we can introduce $\operatorname{sgn}(1)^{2}$
into the first term to have the coefficient at
$x_{\hat{I}}\Phi_{\hat{c}}\Psi_{\hat{d}}$ equal

$\displaystyle\operatorname{sgn}(1)\operatorname{Vol}_{\hat{I}}\sum_{j\in
J}\operatorname{sgn}(j)\Big{(}\xi_{\hat{c}-v_{j},\hat{d},J\setminus
j}\chi(\hat{c}-v_{j}\in C)+\xi_{\hat{c},\hat{d}-v_{j},J\setminus
j}\chi(\hat{d}-v_{j}\in C^{\circ})\Big{)},$

and the claim follows. ∎

######  Remark 2.3.

After some sign changes, one can rephrase the condition of Proposition 2.2 as
$d\xi=0$ for an appropriate element
$\xi\in{\mathbb{C}}[C]\otimes{\mathbb{C}}[C^{\circ}]\otimes\Lambda^{{\operatorname{rk}}N}(\oplus_{i=1}^{n}{\mathbb{C}}e_{i})$
with the differential

$d=\sum_{i=1}^{n}[v_{i}]\otimes
1\otimes(e_{i}\wedge)+\sum_{j=1}^{n}1\otimes[v_{j}]\otimes(e_{j}\wedge)$

on
$\xi\in{\mathbb{C}}[C]\otimes{\mathbb{C}}[C^{\circ}]\otimes\Lambda^{\bullet}(\oplus_{i=1}^{n}{\mathbb{C}}e_{i})$.
We do not pursue this direction further in the paper.

Now we give an explicit formula of the pairing $\langle-,-\rangle$ between
solutions of the better-behaved GKZ systems $\mathrm{bbGKZ}(C,0)$ and
$\mathrm{bbGKZ}(C^{\circ},0)$. We prove that $\langle\Phi,\Psi\rangle$ is a
constant for any pair of solutions $\Phi$ and $\Psi$ by using Proposition 2.2.

Fix a choice of a generic vector $v\in C^{\circ}$. For a set $I$ of size
${\operatorname{rk}}N$ we consider the cone $\sigma_{I}=\sum_{i\in
I}{\mathbb{R}}_{\geq 0}v_{i}$. We define the coefficients $\xi_{c,d,I}$ for
$c+d=v_{I}$ as

$\displaystyle\xi_{c,d,I}=\left\\{\begin{array}[]{ll}&(-1)^{\operatorname{deg}(c)},\text{
if }\dim\sigma_{I}=\operatorname{rk}N\text{ and both }c+\varepsilon v\text{
and }d-\varepsilon v\in\sigma_{I}^{\circ}\\\ &0,\text{
otherwise}.\end{array}\right.$ (2.4)

Here the condition has to hold for all sufficiently small $\varepsilon>0$. It
is clear that $\xi$ is well-defined as long as the vector $v$ is chosen
sufficiently generic. Note that $\xi_{c,d,I}\not=0$ implies that both $c$ and
$d$ lie in the maximum-dimensional cone $\sigma_{I}$ (but not necessarily in
its interior).

We are now ready to tackle the main result of this section.

######  Theorem 2.4.

For any pair of solutions $(\Phi_{c})$ and $(\Psi_{d})$ of
$\operatorname{bbGKZ}(C,0)$ and $\operatorname{bbGKZ}(C^{\circ},0)$
respectively, the pairing

$\displaystyle\langle\Phi,\Psi\rangle=\sum_{c,d,I}\xi_{c,d,I}\operatorname{Vol}_{I}\left(\prod_{i\in
I}x_{i}\right)\Phi_{c}\Psi_{d}$

is a constant.

###### Proof.

We prove this theorem by showing that these coefficients $\xi_{c,d,I}$ satisfy
the conditions in Proposition 2.2, namely

$\displaystyle 0=\sum_{j\in J}\xi_{c-v_{j},d,J\backslash
j}\operatorname{sgn}(j)\chi\left(c-v_{j}\in C\right)+\sum_{j\in
J}\xi_{c,d-v_{j},J\backslash j}\operatorname{sgn}(j)\chi\left(d-v_{j}\in
C^{\circ}\right)$

for all spanning subsets $J\subseteq\\{1,2,\cdots,n\\}$ with
$|J|=\operatorname{rk}N+1$, and all $c\in C,~{}d\in C^{\circ}$ with
$c+d=\sum_{i\in J}v_{i}$.

We first observe that the conditions $c-v_{j}\in C$ and $d-v_{j}\in C^{\circ}$
in the equations above are redundant. Indeed, to ensure that
$\xi_{c-v_{j},d,J\backslash j}\not=0$ we must have $c-v_{j}+\varepsilon
v\in\sigma_{J\backslash j}^{\circ}$, which implies that
$c-v_{j}\in\sigma_{J\backslash j}\subseteq C$. For the second term, to ensure
that $\xi_{c,d-v_{j},J\backslash j}\not=0$ we must have $d-v_{j}-\varepsilon
v\in\sigma_{J\backslash j}^{\circ}\subseteq C^{\circ}$ (since $J\backslash j$
is a maximal cone), which implies $d\in C^{\circ}$. Thus, it suffices to
consider the equations

$\displaystyle 0=\sum_{j\in J_{+}\sqcup J_{-}}\xi_{c-v_{j},d,J\backslash
j}\operatorname{sgn}(j)+\sum_{j\in J_{+}\sqcup
J_{-}}\xi_{c,d-v_{j},J\backslash j}\operatorname{sgn}(j)$ (2.5)

for $\xi$ defined in (2.4). The nonzero terms occur for the indices $j$ such
that both $c-v_{j}+\varepsilon v$ and $d-\varepsilon v$ lie in
$\sigma_{J\backslash j}^{\circ}$, or both $c+\varepsilon v$ and
$d-v_{j}-\varepsilon v$ lie in $\sigma_{J\backslash j}^{\circ}$.

We consider the equation in the variables $a_{i}$

$\displaystyle\sum_{i\in J}a_{i}v_{i}=c+\varepsilon v.$

The solution set to this equation is an affine line $l_{c+\varepsilon v}$ in
the space ${\mathbb{R}}^{{\operatorname{rk}}N+1}$. A contribution to the first
term of (2.5) happens when there is a point on $l_{c+\varepsilon v}$ with
$a_{j}=1$ and all other $a_{i}$ lie in $(0,1)$ due to the definition of the
coefficient $\xi$. Similarly, a contribution to the second term happens for
$a_{j}=0$ and all other $a_{i}$ lie in $(0,1)$.

Recall from Lemma 2.1 that we have a decomposition $J=J_{+}\sqcup J_{-}\sqcup
J_{0}$. For $i\in J_{0}$, the value of $a_{i}$ on the line $l_{c+\varepsilon
v}$ is constant. Since $v$ is generic, we may assume it to be non-integer.
Thus, it either prohibits any contributions to (2.5) (if $a_{i}\not\in(0,1)$)
or provides no restrictions. Therefore, we may now assume that the latter
happens for all $i\in J_{0}$.

The key idea of the proof is to consider the line segments

$\displaystyle S_{i}=l_{c+\varepsilon v}\cap\\{0\leq a_{i}\leq 1\\}$

on $l_{c+\varepsilon v}$ for all $i\in J_{+}\sqcup J_{-}$. The nonzero
contributions to (2.5) happen exactly for the endpoints of a line segment
$S_{j}$ that lie strictly inside all other segments. The assumption that
$\varepsilon v$ is generic implies that the endpoints of different $S_{i}$ do
not coincide. Indeed, if it were the case, then $c+\varepsilon v$ would lie in
a shift of the span of ${{\operatorname{rk}}N}-1$ of $v$-s by a lattice
element, and we may assure that this does not happen. Consider now
$S=\bigcap_{i\in J_{+}\sqcup J_{-}}S_{i}$. If $S$ is empty then there are no
contribution, since this point would not lie in the interior of other $S_{i}$.
So it suffices to consider the case when $S$ is a segment $[p,q]$. It is clear
that the only points that could contribute to (2.5) are $p$ and $q$. In
particular, there are at most two nonzero terms in (2.5). We will show that
they always cancel each other.

We also note that the orientation of the segment $S_{i}$ (i.e., the direction
in which the parameter $a_{i}$ increases) on the line $l_{c+\varepsilon v}$ is
determined by $\operatorname{sgn}(i)$, since the vector along the line is
given by the nontrivial linear relation on $v_{k,k\in J}$. If both $p$ and $q$
are the $a_{i}=1$ and $a_{j}=1$ ends of the segments $S_{i}$ and $S_{j}$, then
the segments must have opposite orientations on $l_{c+\varepsilon v}$ (since
they both should point towards the other point). This means that
$\operatorname{sgn}(i)=-\operatorname{sgn}(j)$ and the two terms of (2.5)
cancel. Similarly, they cancel if $p$ and $q$ are the $a_{i}=0$ and $a_{j}=0$
ends of $S_{i}$ and $S_{j}$.

Now suppose that $p$ and $q$ correspond to $a_{i}=0$ and $a_{j}=1$ ends of
$S_{i}$ and $S_{j}$ (in this case it is possible to have $i=j$). In this case
the two segments must have the same orientation, and then the factor
$(-1)^{\deg c}$ in the definition of $\xi$ ensures that the two terms cancel
each other. ∎

######  Remark 2.5.

As $v$ varies, we get a finite number of different formulas for the pairing.
It is also possible to take a more uniform choice of the pairing by
integrating over $v$ of degree $1$ (ignoring the contributions of measure zero
set of nongeneric $v$). However, there does not appear to be any advantage in
doing so. We will later see that the pairing is in fact independent of the
choice of $v$.

##  3\. Pairing of the Gamma series

In this section we compute the pairing from the previous one on the
cohomology-valued solutions to the better-behaved GKZ systems provided by the
$\Gamma$ series. We will show in the next section that the result is the dual
of the intersection pairing which provides the proof of Conjecture 7.3 from
[1].

We consider a regular triangulation $\Sigma$ of the cone $C$ whose vertices
are among these vectors $\\{v_{i}\\}_{i=1}^{n}$ and its corresponding toric
Deligne-Mumford stack $\mathbb{P}_{\Sigma}$.

######  Remark 3.1.

It will be convenient for us to abuse notation and denote by $I$ both a subset
of $\\{1,\ldots,n\\}$ and the corresponding cone $\sum_{i\in
I}{\mathbb{R}}_{\geq 0}v_{i}$. Similarly, $\Sigma$ denotes both a simplicial
complex on $\\{1,\ldots,n\\}$ and the corresponding simplicial fan in
$N_{\mathbb{R}}$ which refines $C$ and its faces.

######  Definition 3.2.

For each cone $\sigma\in\Sigma$ we define $\operatorname{Box}(\sigma)$ to be
the set of lattice points $\gamma$ which can be written as
$\gamma=\sum_{i\in\sigma}\gamma_{i}v_{i}$ with $0\leq\gamma_{i}<1$. We denote
the union of all $\operatorname{Box}(\sigma)$ by $\operatorname{Box}(\Sigma)$.
To each element $\gamma\in\operatorname{Box}(\Sigma)$ we associate a _twisted
sector_ of $\mathbb{P}_{\Sigma}$ corresponding to the minimal cone
$\sigma(\gamma)$ in $\Sigma$ containing $\gamma$. We define the dual of a
twisted sector $\gamma=\sum\gamma_{i}v_{i}$ by

$\displaystyle\gamma^{\vee}=\sum_{\gamma_{i}\not=0}(1-\gamma_{i})v_{i}.$

or equivalently, the unique element in $\operatorname{Box}(\sigma(\gamma))$
that satisfies

$\displaystyle\gamma^{\vee}=-\gamma\hskip-3.0pt\mod\sum_{i\in\sigma}{\mathbb{Z}}v_{i}$

######  Remark 3.3.

The dual of $\gamma=0$ is itself. Clearly, we have
$\sigma(\gamma)=\sigma(\gamma^{\vee})$ and $(\gamma^{\vee})^{\vee}=\gamma$.

Twisted sectors are themselves smooth toric DM stacks and the following
propositions describe a Stanley-Reisner type presentation of the spaces of
cohomology and cohomology with compact support of their coarse moduli spaces,
see [1].

######  Proposition 3.4.

As usual, $\operatorname{Star}(\sigma(\gamma))$ denotes the set of cones in
$\Sigma$ that contain $\sigma(\gamma)$. Cohomology space $H_{\gamma}$ of the
twisted sector $\gamma$ is naturally isomorphic to the quotient of the
polynomial ring
${\mathbb{C}}[D_{i}:i\in\operatorname{Star}(\sigma(\gamma))\backslash\sigma(\gamma)]$
by the ideal generated by the relations

$\displaystyle\prod_{j\in J}D_{j},\
J\not\in\operatorname{Star}(\sigma(\gamma)),\quad\text{and
}\sum_{i\in\operatorname{Star}(\sigma(\gamma))\backslash\sigma(\gamma)}\mu(v_{i})D_{i},\
\mu\in\operatorname{Ann}(v_{i},i\in\sigma(\gamma)).$

We can also view $H_{\gamma}$ as a module over the polynomial ring
${\mathbb{C}}[D_{1},\ldots,D_{n}]$ by declaring $D_{i}=0$ for
$i\not\in\operatorname{Star}(\sigma(\gamma))$ and solving (uniquely) for
$D_{i},i\in\sigma(\gamma)$ to satisfy the linear relations
$\sum_{i=1}^{n}\mu(v_{i})D_{i}=0$ for all $\mu\in N^{\vee}$.

######  Proposition 3.5.

Cohomology space with compact support $H_{\gamma}^{c}$ (viewed as a module
over $H_{\gamma}$) is generated by $F_{I}$ for
$I\in\operatorname{Star}(\sigma(\gamma))$ such that
$\sigma_{I}^{\circ}\subseteq C^{\circ}$ with relations

$\displaystyle\quad D_{i}F_{I}-F_{I\cup\\{i\\}}\text{ for }i\not\in
I,I\cup\\{i\\}\in\operatorname{Star}(\sigma(\gamma))$ $\displaystyle\text{and
}D_{i}F_{I}\text{ for }i\not\in
I,I\cup\\{i\\}\not\in\operatorname{Star}(\sigma(\gamma))$

Similarly, it is given a structure of a module over
${\mathbb{C}}[D_{1},\ldots,D_{n}]$.

######  Definition 3.6.

The orbifold cohomology $H$ of the smooth toric DM stack $\mathbb{P}_{\Sigma}$
is defined as the direct sum $\bigoplus_{\gamma}H_{\gamma}$ over all twisted
sectors. Similarly, the orbifold cohomology with compact support $H^{c}$ is
defined as $\bigoplus_{\gamma}H_{\gamma}^{c}$. We denote by $1_{\gamma}$ the
generator of $H_{\gamma}$.

There is a natural perfect pairing between $H$ and $H^{c}$ called _Euler
characteristic pairing_. Its origin is the eponymous pairing on certain
Grothendick $K$-groups, which is then translated to the cohomology via the
Chern character, see [1]. We will not be using the original definition, but
rather the following formula for the Euler characteristic pairing, which is
proved in [2].

######  Proposition 3.7.

The Euler characteristic pairing $\chi:H\otimes H^{c}\rightarrow{\mathbb{C}}$
on the toric DM stack $\mathbb{P}_{\Sigma}$ is given by

$\displaystyle\chi(a,b)=\chi(\oplus_{\gamma}a_{\gamma},\oplus_{\gamma}b_{\gamma})=\sum_{\gamma}\frac{1}{|\operatorname{Box}(\sigma(\gamma))|}\int_{\gamma^{\vee}}\operatorname{Td}(\gamma^{\vee})a_{\gamma}^{*}b_{\gamma^{\vee}}$

Here $*:H\rightarrow H$ is the duality map given by
$(1_{\gamma})^{*}=1_{\gamma^{\vee}}$ and $(D_{i})^{*}=-D_{i}$, and
$\operatorname{Td}(\gamma)$ is the Todd class of the twisted sector $\gamma$
which is defined as

$\displaystyle\operatorname{Td}(\gamma)=\frac{\prod_{i\in\operatorname{Star}\sigma(\gamma)\backslash\sigma(\gamma)}D_{i}}{\prod_{i\in\operatorname{Star}\sigma(\gamma)}(1-e^{-D_{i}})}.$

The linear function $\int\colon H_{\gamma}^{c}\rightarrow{\mathbb{C}}$ takes
values $\frac{1}{{\rm Vol}_{\overline{I}}}$ on each generator $F_{I}$, where
${\rm Vol}_{\overline{I}}$ denotes the volume of the cone
$\overline{\sigma_{I}}$ in the quotient fan $\Sigma/\sigma(\gamma)$. It takes
value zero on all elements of $H_{\gamma}^{c}$ of lower degree.

Let $\Sigma$ be a regular (=projective) subdivision of $C$ based on some of
the $v_{i}$. Let $\psi_{i}$ be the real numbers such that $\Sigma$ reads off
the lower boundary of the convex hull of the origin and
$\\{(v_{i},\psi_{i}),1\leq i\leq n\\}$ in $N_{\mathbb{R}}\oplus{\mathbb{R}}$.
We assume that $\psi_{i}$ are generic so this convex hull is simplicial. We
denote by $\psi$ the strictly convex piecewise linear function on $C$ whose
graph is the aforementioned lower boundary. It takes values $\psi_{i}$ on all
$v_{i}$ which generate rays in $\Sigma$ and has lower values than $\psi_{i}$
on other $v_{i}$. Its key property is that for any finite collection $w_{i}\in
C$ and $\alpha_{i}\in{\mathbb{R}}_{>0}$ there holds

$\psi(\sum_{i}\alpha_{i}w_{i})\leq\sum_{i}\alpha_{i}\psi(w_{i})$

with equality if and only if there exists a cone in $\Sigma$ which contains
all of the $w_{i}$.

Recall from [1] the following solution to the equations $\mathrm{bbGKZ}(C,0)$
with values in $H=\bigoplus_{\gamma}H_{\gamma}$. We define

$\displaystyle\Gamma_{c}(x_{1},\ldots,x_{n})=\bigoplus_{\gamma}\sum_{l\in
L_{c,\gamma}}\prod_{i=1}^{n}\frac{x_{i}^{l_{i}+\frac{D_{i}}{2\pi{\rm
i}}}}{\Gamma(1+l_{i}+\frac{D_{i}}{2\pi{\rm i}})}$ (3.1)

where the direct sum is taken over twisted sectors
$\gamma=\sum_{j\in\sigma(\gamma)}\gamma_{j}v_{j}$ and the set $L_{c,\gamma}$
is the set of solutions to $\sum_{i=1}^{n}l_{i}v_{i}=-c$ with
$l_{i}-\gamma_{i}\in{\mathbb{Z}}$ for all $i$. The numerator is defined by
picking a branch of $\log(x_{i})$.

We will first prove that for each $c\in C\cap N$ the series for $\Gamma$
converges absolutely and uniformly on compacts for ${\bf x}$ such that the
$(-\log|x_{i}|)$ are in an appropriate shift of the cone of values on $v_{i}$
of convex $\Sigma$-piecewise linear functions. The proof was skipped in [1]
because it is essentially the same as that in [4], but we will present it
here, both for completeness and to facilitate arguments about the asymptotic
behavior of $\Gamma_{c}$.

######  Proposition 3.8.

We denote by $C_{\Sigma}$ the cone of the secondary fan that corresponds to
$\Sigma$, i.e. the cone of $(\psi_{i})\in{\mathbb{R}}^{n}$ that give rise to
$\Sigma$. For each $c\in C\cap N$ there exists $\hat{\psi}\in{\mathbb{R}}^{n}$
such that the series (3.1) converges absolutely and uniformly on compacts in
the region of ${\mathbb{C}}^{n}$

$\displaystyle\\{(-\log|x_{1}|,\ldots,-\log|x_{n}|)\in\hat{\psi}+C_{\Sigma},~{}\arg({\bf
x})\in(-\pi,\pi)^{n}\\}.$ (3.2)

###### Proof.

An immediate observation is that we can ignore the factor

$\prod_{i=1}^{n}x_{i}^{\frac{D_{i}}{2\pi{\rm i}}}=\prod_{i=1}^{n}{\rm
e}^{\frac{D_{i}\log x_{i}}{2\pi{\rm i}}}$

because it does not depend on $l$ and is bounded on compacts in the region
(3.2).

It suffices to understand what happens for a fixed $\gamma$. Note that while
the summation takes place over an affine lattice $L_{c,\gamma}$, the nonzero
contributions only occur for $(l_{1},\ldots,l_{n})$ such that the set

$I(l)=\\{i,l_{i}\in{\mathbb{Z}}_{<0}\\}\sqcup\sigma(\gamma)$

is a cone $\sigma$ in $\Sigma$, because each $l_{i}\in{\mathbb{Z}}_{<0}$
contributes a factor $D_{i}$ due to a pole of $\Gamma$ at a nonpositive
integer. Consequently, it suffices to bound the summation over the subset
$L_{c,\gamma,\sigma}$ of $L_{c,\gamma}$ with the additional property that the
above defined $I(l)$ is a subset of some fixed maximum-dimensional cone
$\sigma$ of $\Sigma$ that contains $\sigma(\gamma)$. For any such $l\in
L_{c,\gamma,\sigma}$ we have

$\sum_{i,l_{i}<0}(-l_{i})v_{i}=\sum_{i,l_{i}\geq 0}l_{i}v_{i}+c.$

Let us denote by $\psi$ the $\Sigma$-piecewise linear convex function that
corresponds to $(-\hat{\psi}_{i}-\log|x_{i}|)$ by the assumption on ${\bf x}$.
Since the $v_{i}$ on the left hand side of the above equation lie in
$\sigma\in\Sigma$, we have

$\displaystyle\sum_{i,l_{i}<0}(-l_{i})(-\hat{\psi}_{i}-\log|x_{i}|)$
$\displaystyle=\psi(\sum_{i,l_{i}<0}(-l_{i})v_{i})\leq\sum_{i,l_{i}\geq
0}l_{i}\psi(v_{i})+\psi(c)$ $\displaystyle=\sum_{i,l_{i}\geq
0}l_{i}(-\hat{\psi}_{i}-\log|x_{i}|)+\psi(c)$

and therefore

$\displaystyle\sum_{i=1}^{n}l_{i}\log|x_{i}|\leq-\sum_{i=1}^{n}l_{i}\hat{\psi}_{i}+\psi(c).$
(3.3)

This leads to an upper bound

$\displaystyle\Big{|}\prod_{i=1}x_{i}^{l_{i}}\Big{|}\leq{\rm e}^{\psi(c)}{\rm
e}^{-\sum_{i=1}^{n}l_{i}\hat{\psi}_{i}}.$ (3.4)

Crucially, since all $v_{i}$ have degree $1$, we see that $\sum_{i}l_{i}=-\deg
c$. Thus, we can apply the key estimate of [4, Lemma A.4] which states that
for any $\delta>0$ and any collection of real numbers $a_{i},b_{i}$ for
$i=1,\ldots,n$ with

$|\sum_{i}a_{i}|\leq\delta,~{}~{}\sum_{i}|b_{i}|\leq\delta$

there exists a constant $A$ such that

$\Big{|}\prod_{i=1}^{n}\frac{1}{\Gamma(a_{i}+{\rm i}b_{i})}\Big{|}\leq
A(4n)^{\sum_{i=1}^{n}|a_{i}|}.$

By the Cauchy’s formula for partial derivatives, this implies an upper bound
of the form $A_{1}(A_{2})^{\sum_{i=1}^{n}|l_{i}|}$ on the coefficients on all
monomials in $D_{i}$ of bounded degree of the function

$\prod_{i=1}^{n}\frac{1}{\Gamma(1+l_{i}+\frac{D_{i}}{2\pi{\rm i}})}.$

Together with (3.4), we conclude that in any Euclidean norm on $H_{\gamma}$
the absolute value of each term of the series is bounded by

$\displaystyle\Big{|}\prod_{i=1}^{n}\frac{x_{i}^{l_{i}}}{\Gamma(1+l_{i}+\frac{D_{i}}{2\pi{\rm
i}})}\Big{|}\leq
A_{1}(A_{2})^{\sum_{i=1}^{n}|l_{i}|}\Big{|}\prod_{i=1}x_{i}^{l_{i}}\Big{|}\leq
A_{3}(A_{2})^{\sum_{i=1}^{n}|l_{i}|}{\rm
e}^{-\sum_{i=1}^{n}l_{i}\hat{\psi}_{i}}.$ (3.5)

We observe that the set $L_{c,\gamma,\sigma}$ is the set of lattice points in
a shift of a (lower-dimensional) polyhedral cone $C_{\sigma}$ in
${\mathbb{R}}^{n}$ given by the equality $\sum_{i=1}^{n}l_{i}v_{i}=0$ and
inequalities $l_{i}\geq 0$ for all $i\not\in\sigma$. We may assume
$\hat{\psi}$ to give a strictly $\Sigma$-convex function. It then follows that
for any ray generator $l$ of $C_{\sigma}$ there holds

$\sum_{i}l_{i}\hat{\psi}_{i}>0.$

Indeed, by convexity for $\hat{\psi}$ for any $l\in C_{\sigma}$ we have the
inequality $\sum_{i}l_{i}\hat{\psi}_{i}\geq 0$ (the proof is the same as that
of (3.3)) which holds even if $\hat{\psi}$ is deformed slightly, so it can
only be equality for $l=0$. As a consequence, there is a constant $r$ such
that

$\sum_{i=1}^{n}|l_{i}|\leq r(\sum_{i}l_{i}\hat{\psi}_{i})$

on $C_{\sigma}$.

Therefore, we can replace $\hat{\psi}$ by a large enough multiple of itself
and use (3.5) to get on any compact subset of the region (3.2)

$\displaystyle\Big{|}\prod_{i=1}^{n}\frac{x_{i}^{l_{i}}}{\Gamma(1+l_{i}+\frac{D_{i}}{2\pi{\rm
i}})}\Big{|}\leq A_{4}{\rm e}^{-A_{5}\sum_{i=1}^{n}l_{i}\hat{\psi}_{i}}$

for some $A_{5}>0$. Since the number of terms in $L_{c,\gamma,\sigma}$ with
$\sum_{i=1}^{n}l_{i}\hat{\psi}_{i}\in[m,m+1)$ is bounded by a polynomial in
$m$, we get the desired convergence. ∎

There is a similarly defined $\Gamma$-series solution $\Gamma^{\circ}$ of
$\mathrm{bbGKZ}(C^{\circ},0)$, with values in
$H^{c}=\bigoplus_{\gamma}H_{\gamma}^{c}$. We define

$\displaystyle\Gamma_{c}^{\circ}(x_{1},\ldots,x_{n})=\bigoplus_{\gamma}\sum_{l\in
L_{c,\gamma}}\prod_{i=1}^{n}\frac{x_{i}^{l_{i}+\frac{D_{i}}{2\pi{\rm
i}}}}{\Gamma(1+l_{i}+\frac{D_{i}}{2\pi{\rm
i}})}\left(\prod_{i\in\sigma}D_{i}^{-1}\right)F_{\sigma}$

where $\sigma$ is the set of $i$ with $l_{i}\in{\mathbb{Z}}_{<0}$.

######  Proposition 3.9.

The series $\Gamma^{\circ}$ converges uniformly on compacts in the region
(3.2) for an appropriate choice of $\hat{\psi}$.

###### Proof.

The idea of the proof are the same as that of Proposition 3.8 and we leave the
details to the reader. ∎

Our next goal is to understand the asymptotic behavior of

$\Gamma_{c}(t^{-\psi(v_{1})}x_{1},\ldots,t^{-\psi(v_{n})}x_{n})$

for real $t\to+\infty$. We can assume $x_{i}$ to be generic nonzero complex
numbers, so that for large enough $t$ we fall within the range of convergence
of $\Gamma$.

For each $c$ we consider the minimum cone $\sigma(c)$ of $\Sigma$ that
contains $c$. We have $c=\sum_{j\in\sigma(c)}c_{j}v_{j}$. It defines a twisted
sector $\gamma(c)=\sum_{j\in\sigma(c)}\\{c_{j}\\}v_{j}$. We also consider the
dual twisted sector
$\gamma^{\vee}(c)=\sum_{j\in\sigma(c),c_{j}\not\in{\mathbb{Z}}}(1-\\{c_{j}\\})v_{j}$.
There is a special element

$\displaystyle-c=\sum_{i\in I}(-c_{i})v_{i}$ (3.6)

in $L_{c,\gamma^{\vee}(c)}$.

######  Lemma 3.10.

As $t\to+\infty$, we have for $c\in C\cap N$ and $\gamma\neq\gamma^{\vee}(c)$
the $\gamma$ summand of
$\Gamma_{c}(t^{-\psi(v_{1})}x_{1},\ldots,t^{-\psi(v_{n})}x_{n})$ is
$o(t^{\psi(c)})$. For $\gamma=\gamma^{\vee}(c)$ we have

$\displaystyle\Gamma_{c}(t^{-\psi(v_{1})}x_{1},\ldots)=t^{\psi(c)}\prod_{i=1}^{n}{\rm
e}^{\frac{D_{i}}{2\pi{\rm i}}(\log x_{i}-\psi(v_{i})\log
t)}\prod_{i=1}^{n}\frac{x_{i}^{-c_{i}}}{\Gamma(1-c_{i}+\frac{D_{i}}{2\pi{\rm
i}})}(1+o(1)).$

###### Proof.

Let $\gamma=\sum_{j\in\sigma(\gamma)}\gamma_{j}v_{j}$. Let $(l_{i})$ be an
element of $L_{c,\gamma}$. The contribution to
$\Gamma_{c}(t^{-\psi(v_{1})}x_{1},\ldots)$ is only nonzero if the set of $i$
for which $l_{i}\in{\mathbb{Z}}_{<0}$ together with $\sigma(\gamma)$ is a cone
in $\Sigma$. Consequently, $i$ for which $l_{i}$ are negative lie in a cone of
$\Sigma$. Therefore,

$\displaystyle\sum_{l_{i}<0}(-l_{i})\psi(v_{i})=\psi(\sum_{l_{i}<0}(-l_{i})v_{i})=\psi(c+\sum_{l_{i}>0}l_{i}v_{i})\leq\sum_{l_{i}>0}l_{i}\psi(v_{i})+\psi(c),$
(3.7)

which implies

$\displaystyle-\sum_{i=1}^{n}l_{i}\psi(v_{i})\leq\psi(c).$ (3.8)

Now notice that the equality in (3.8) holds if and only if the minimal cone of
$\sum_{l_{i}<0}(-l_{i})v_{i}$ is a cone in $\Sigma$ which contains $c$ and all
$v_{i}$ with $l_{i}>0$. This cone would then contain $c$ and all $v_{i}$ for
which $l_{i}\neq 0$. This means that $l_{i}=-c_{i}$, which implies that
$\gamma=\gamma^{\vee}(c)$. This gives the claimed asymptotic contribution.

It is not enough to bound the asymptotic behavior of each individual term as
$t\to\infty$, one also needs to ensure that the rest of the terms _together_
do not contribute to anything larger than $o(t^{\psi(c)})$. This follows
either from the estimates of Proposition 3.8 or simply from the fact that we
have absolute convergence at $\bf x$ and then all other terms decay faster.
Indeed, if we have an absolutely convergent series $\sum_{i\geq 0}a_{i}$ and
then consider $\sum_{i\geq 0}a_{i}t^{\alpha_{i}}$ with $\alpha_{0}-\alpha_{i}$
larger than some positive $\varepsilon$, then as $t\to\infty$ we have

$\sum_{i\geq 0}a_{i}t^{\alpha_{i}}=a_{0}t^{\alpha_{0}}(1+o(1))$

because

$\Big{|}\sum_{i>0}a_{i}t^{\alpha_{i}-\alpha_{0}}\Big{|}\leq
t^{-\varepsilon}\sum_{i>0}|a_{i}|.$

We can apply it to our situation since $(l_{i})$ are in a countable set and
there exists $\varepsilon>0$ so that for all other terms the inequality (3.8)
is strict by at least $\varepsilon$. The logarithmic terms
$\prod_{i}(t^{-\psi(v_{1})}x_{i})^{\frac{D_{i}}{2\pi{\rm i}}}$ can be absorbed
by a slight change of $\varepsilon$. ∎

We can state a similar result for $\Gamma^{\circ}$. For $d\in C^{\circ}$ we
consider the element of $L_{d,\gamma^{\vee}(d)}$

$-d=\sum_{i\in\sigma(d)}(-d_{i})v_{i}.$

######  Lemma 3.11.

As $t\to+\infty$, we have for $c\in C\cap N$ and $\gamma\neq\gamma^{\vee}(d)$
the $\gamma$ summand of
$\Gamma^{\circ}_{d}(t^{-\psi(v_{1})}x_{1},\ldots,t^{-\psi(v_{n})}x_{n})$ is
$o(t^{\psi(d)})$. For $\gamma=\gamma^{\vee}(d)$ we have

$\displaystyle\Gamma_{d}(t^{-\psi(v_{1})}x_{1},\ldots)=t^{\psi(d)}\prod_{i=1}^{n}{\rm
e}^{\frac{D_{i}}{2\pi{\rm i}}(\log x_{i}-\psi(v_{i})\log
t)}\prod_{i=1}^{n}\frac{x_{i}^{-d_{i}}}{\Gamma(1-d_{i}+\frac{D_{i}}{2\pi{\rm
i}})}$
$\displaystyle\left(\prod_{i\in\sigma(d)}D_{i}^{-1}\right)F_{\sigma(d)}(1+o(1)).$

###### Proof.

The proof is analogous to that of Lemma 3.10 and is left to the reader. ∎

Now we use this information about the asymptotic behavior of $\Gamma$ and
$\Gamma^{\circ}$ to compute the constant
$\langle\Gamma,\Gamma^{\circ}\rangle=\sum_{c,d,I}\xi_{c,d,I}\operatorname{Vol}_{I}\left(\prod_{i\in
I}x_{i}\right)\Gamma_{c}\otimes\Gamma^{\circ}_{d}$ where $\xi$ are defined in
Theorem 2.4.

As in Section 2, let $I$ be a subset of $\\{1,\ldots,n\\}$ of size ${\rm
rk}N$, which may or may not be a cone in $\Sigma$. Let $c$ and $d$ be such
that $c+d=\sum_{i\in I}v_{i}$ and $c+\varepsilon v,d-\varepsilon
v\in\sum_{i\in I}{\mathbb{R}}_{\geq 0}v_{i}$ for small $\varepsilon>0$. The
following observation is key.

######  Proposition 3.12.

Under the above assumptions on $c,d,I$ we have

$\lim_{t\to+\infty}\prod_{i=1}^{n}(t^{-\psi(v_{i})}x_{i})\Gamma_{c}(t^{-\psi(v_{1})}x_{1},\ldots)\Gamma^{\circ}_{d}(t^{-\psi(v_{1})}x_{1},\ldots)=0$

unless $\gamma(d)=\gamma^{\vee}(c)$ and $I$ contains $\sigma(\gamma(c))$.

###### Proof.

Since $c$ and $d$ are contained in $\sum_{i\in I}{\mathbb{R}}_{\geq 0}v_{i}$
and $c+d=v_{I}$, we have

$c=\sum_{i\in I}\alpha_{i}v_{i},~{}~{}d=\sum_{i\in I}(1-\alpha_{i})v_{i}$

with $\alpha_{i}\in[0,1]$. Convexity of $\psi$ implies that

$\displaystyle\psi(c)\leq\sum_{i}\alpha_{i}\psi(v_{i}),~{}~{}\psi(d)\leq\sum_{i}(1-\alpha_{i})\psi(v_{i})$
(3.9)

which leads to $\psi(c)+\psi(d)-\sum_{i}\psi(v_{i})\leq 0$, so we can use
Propositions 3.10 and 3.11 to see that the leading power of $t$ is
nonpositive. In fact, it is negative, unless the inequalities in (3.9) are
equalities, which means that the subset of $I$ for which $\alpha_{i}>0$ is a
cone in $\Sigma$, and similarly for the subset of $\alpha_{i}<1$. This implies
the claim. ∎

######  Proposition 3.13.

If $\gamma(c)=\gamma^{\vee},~{}\gamma(d)=\gamma=\sum_{i\in I}\gamma_{i}v_{i}$,
then we define $I_{c}$ to be the subset of $I$ such that the coefficients
$c_{i}$ of $c$ are equal to $1$ and similarly for $I_{d}$. The asymptotic
behavior as $t\to\infty$ is

$\displaystyle\prod_{i=1}^{n}(t^{-\psi(v_{i})}x_{i})\Gamma_{c}(t^{-\psi(v_{1})}x_{1},\ldots)\Gamma^{\circ}_{d}(t^{-\psi(v_{1})}x_{1},\ldots)=o(1)+\frac{1}{(2\pi{\rm
i})^{\operatorname{rk}N-|\sigma(\gamma)|}}$
$\displaystyle\cdot\frac{D_{I_{c}}}{\prod_{i\in\sigma(\gamma)}\Gamma(\gamma_{i}+\frac{D_{i}}{2\pi{\rm
i}})\prod_{i\in\operatorname{Star}(\sigma(\gamma))\backslash\sigma(\gamma)}\Gamma(1+\frac{D_{i}}{2\pi{\rm
i}})}\prod_{i=1}^{n}{\rm e}^{\frac{D_{i}}{2\pi{\rm i}}(\log
x_{i}-\psi(v_{i})\log t)}$
$\displaystyle\bigotimes\frac{F_{I_{d}}}{\prod_{i\in\sigma(\gamma)}\Gamma(1-\gamma_{i}+\frac{D_{i}}{2\pi{\rm
i}})\prod_{i\in\operatorname{Star}(\sigma(\gamma))\backslash\sigma(\gamma)}\Gamma(1+\frac{D_{i}}{2\pi{\rm
i}})}\prod_{i=1}^{n}{\rm e}^{\frac{D_{i}}{2\pi{\rm i}}(\log
x_{i}-\psi(v_{i})\log t)}$

in $H_{\gamma}\otimes H_{\gamma^{\vee}}^{c}$.

###### Proof.

The proof of Proposition 3.12 shows that the only contribution other than
$o(1)$ can come from the terms that give better than $o(t^{\psi(c)})$ and
$o(t^{\psi(d)})$ contributions to the asymptotic behavior of $\Phi_{c}$ and
$\Phi_{d}$. So by Propositions 3.10 and 3.11 the only contributions come from
elements of $L_{c,\gamma^{\vee}}$ and $L_{d,\gamma}$ given by

$-c=\sum_{i\in I}(-c_{i})v_{i},~{}~{}-d=\sum_{i\in I}(c_{i}-1)v_{i}.$

For $i\in\sigma(\gamma)$ we note that $\gamma_{i}=1-c_{i}$ if $c_{i}\in(0,1)$.
For $i\in I_{c}$ we use

$\frac{1}{\Gamma(1-c_{i}+\frac{D_{i}}{2\pi{\rm
i}})}=\frac{1}{\Gamma(\frac{D_{i}}{2\pi{\rm i}})}=\frac{\frac{D_{i}}{2\pi{\rm
i}}}{\Gamma(1+\frac{D_{i}}{2\pi{\rm i}})}$

and similarly for $i\in I_{d}$, and the result follows. ∎

Now we recall that $\langle\Gamma,\Gamma^{\circ}\rangle$ is constant.

######  Corollary 3.14.

The constant pairing $\langle\Gamma,\Gamma^{\circ}\rangle$ lies in
$\bigoplus_{\gamma}H_{\gamma}\otimes H_{\gamma^{\vee}}^{c}$ and is given by

$\displaystyle\frac{1}{(2\pi{\rm
i})^{\operatorname{rk}N}}\bigoplus_{\gamma}\sum_{\begin{subarray}{c}c\in
C,d\in C^{\circ}\\\
|I|=\operatorname{rk}N\end{subarray}}\xi_{c,d,I}\operatorname{Vol}_{I}(2\pi{\rm
i})^{|\sigma(\gamma)|}\frac{D_{I_{c}}}{\widehat{\Gamma}_{\gamma}}\otimes\frac{F_{I_{d}}}{\widehat{\Gamma}_{\gamma^{\vee}}}$

where
$\widehat{\Gamma}_{\gamma}=\prod_{i\in\sigma(\gamma)}\Gamma(\gamma_{i}+\frac{D_{i}}{2\pi{\rm
i}})\prod_{i\in\operatorname{Star}(\sigma(\gamma))\backslash\sigma(\gamma)}\Gamma(1+\frac{D_{i}}{2\pi{\rm
i}})$ and similarly for $\widehat{\Gamma}_{\gamma^{\vee}}$. There also holds
for each $k$

$\displaystyle 0=\bigoplus_{\gamma}\sum_{\begin{subarray}{c}c\in C,d\in
C^{\circ}\\\
|I|=\operatorname{rk}N\end{subarray}}\xi_{c,d,I}\operatorname{Vol}_{I}(2\pi{\rm
i})^{|\sigma(\gamma)|}\Big{(}D_{k}\frac{D_{I_{c}}}{\widehat{\Gamma}_{\gamma}}\Big{)}\otimes\frac{F_{I_{d}}}{\widehat{\Gamma}_{\gamma^{\vee}}}$
$\displaystyle+\bigoplus_{\gamma}\sum_{\begin{subarray}{c}c\in C,d\in
C^{\circ}\\\
|I|=\operatorname{rk}N\end{subarray}}\xi_{c,d,I}\operatorname{Vol}_{I}(2\pi{\rm
i})^{|\sigma(\gamma)|}\frac{D_{I_{c}}}{\widehat{\Gamma}_{\gamma}}\otimes\Big{(}D_{k}\frac{F_{I_{d}}}{\widehat{\Gamma}_{\gamma^{\vee}}}\Big{)}.$

###### Proof.

Proposition 3.13 gives the asymptotic behavior of
$\langle\Gamma,\Gamma^{\circ}\rangle$ as a polynomial in $\log x_{i}$.
However, we also know it is a constant by Theorem 2.4. The first statement of
the proposition is reading off the constant term of the polynomial and the
second statement is reading off the coefficient by $\log x_{k}$. ∎

##  4\. Euler characteristic pairing

Now we are ready to prove that the pairing of Gamma series
$\langle\Gamma,\Gamma^{\circ}\rangle$ is inverse to the Euler characteristic
pairing on $\mathbb{P}_{\Sigma}$. Before we state the main theorem of this
section, we have the following useful observation, which is an orbifold analog
of the relationship between the $\Gamma$-class and the Todd class of a smooth
manifold. Recall that $*$ is the duality map on $H$ defined in Proposition
3.7.

######  Lemma 4.1.

$(\widehat{\Gamma}_{\gamma})^{*}\widehat{\Gamma}_{\gamma^{\vee}}=(2\pi{\rm
i})^{|\sigma(\gamma)|}(-1)^{\deg\gamma^{\vee}}\operatorname{Td}(\gamma^{\vee})$.

###### Proof.

We can expand
$(\widehat{\Gamma}_{\gamma})^{*}\widehat{\Gamma}_{\gamma^{\vee}}$ as

$\displaystyle\prod_{i\in\sigma(\gamma)}\Gamma(\gamma_{i}+\frac{D_{i}}{2\pi{\rm
i}})^{*}\Gamma(1-\gamma_{i}+\frac{D_{i}}{2\pi{\rm i}})$
$\displaystyle\cdot\prod_{i\in\operatorname{Star}(\sigma(\gamma))\backslash\sigma(\gamma)}\Gamma(1+\frac{D_{i}}{2\pi{\rm
i}})^{*}\Gamma(1+\frac{D_{i}}{2\pi{\rm i}})$ $\displaystyle=$
$\displaystyle\prod_{i\in\sigma(\gamma)}\Gamma(\gamma_{i}-\frac{D_{i}}{2\pi{\rm
i}})\Gamma(1-\gamma_{i}+\frac{D_{i}}{2\pi{\rm i}})$
$\displaystyle\cdot\prod_{i\in\operatorname{Star}(\sigma(\gamma))\backslash\sigma(\gamma)}\Gamma(1-\frac{D_{i}}{2\pi{\rm
i}})\Gamma(1+\frac{D_{i}}{2\pi{\rm i}}).$

We use the identity $\Gamma(z)\Gamma(1-z)=-\frac{2\pi{\rm i}\,{\rm e}^{\pi{\rm
i}z}}{1-{\rm e}^{2\pi{\rm i}z}}$ to rewrite the first product as

$\displaystyle(-2\pi{\rm i})^{|\sigma(\gamma)|}{\rm
e}^{\sum_{i\in\sigma(\gamma)}\pi{\rm i}\gamma_{i}}{\rm
e}^{-\frac{1}{2}\sum_{i\in\sigma(\gamma)}D_{i}}\prod_{{}_{i\in\sigma(\gamma)}}\frac{1}{1-{\rm
e}^{2\pi{\rm i}\gamma_{i}-D_{i}}}.$

For the second product, we use $\Gamma(1-\frac{z}{2\pi{\rm
i}})\Gamma(1+\frac{z}{2\pi{\rm i}})=\frac{z{\rm e}^{-\frac{z}{2}}}{1-{\rm
e}^{-z}}$ to rewrite it as

$\displaystyle{\rm
e}^{-\frac{1}{2}\sum_{i\in\operatorname{Star}(\sigma(\gamma))\backslash\sigma(\gamma)}D_{i}}\prod_{i\in\operatorname{Star}(\sigma(\gamma))\backslash\sigma(\gamma)}\frac{D_{i}}{1-{\rm
e}^{-D_{i}}}.$

Putting the two formulas together, we get

$\displaystyle(\widehat{\Gamma}_{\gamma})^{*}\widehat{\Gamma}_{\gamma^{\vee}}=(2\pi{\rm
i})^{|\sigma(\gamma)|}(-1)^{\deg\gamma^{\vee}}{\rm
e}^{-\frac{1}{2}\sum_{i\in\operatorname{Star}(\sigma(\gamma))}D_{i}}\frac{\prod_{i\in\operatorname{Star}(\sigma(\gamma))\backslash\sigma(\gamma)}D_{i}}{\prod_{i\in\operatorname{Star}(\sigma(\gamma))}{1-{\rm
e}^{-D_{i}}}}$ $\displaystyle=(2\pi{\rm
i})^{|\sigma(\gamma)|}(-1)^{\deg\gamma^{\vee}}\operatorname{Td}(\gamma^{\vee})$

where we used
$\sum_{i\in\operatorname{Star}(\sigma(\gamma))}D_{i}=\sum_{i=1}^{n}D_{i}=0$. ∎

Now we can state and prove the main theorem of this section. Recall that we
defined the pairing $\langle\cdot,\cdot\rangle$ on solutions of the better-
behaved GKZ systems. When we apply it to $\Gamma$ and $\Gamma^{\circ}$, we get
a constant element of $H\otimes H^{c}$.

######  Theorem 4.2.

The constant pairing $\langle\Gamma,\Gamma^{\circ}\rangle$ is equal up to a
constant factor to the inverse of the Euler characteristic pairing
$\chi(-,-):H\otimes H^{c}\rightarrow{\mathbb{C}}$.

###### Proof.

It’s clear that we can consider each twisted sector individually. For a fixed
$\gamma$, the statement is equivalent to the assertion that

$\displaystyle\sum_{\begin{subarray}{c}c\in C,d\in C^{\circ}\\\
|I|=\operatorname{rk}N\end{subarray}}\xi_{c,d,I}\operatorname{Vol}_{I}(2\pi{\rm
i})^{|\sigma(\gamma)|}\chi\left(P,\frac{F_{I_{d}}}{\widehat{\Gamma}_{\gamma^{\vee}}}\right)\frac{D_{I_{c}}}{\widehat{\Gamma}_{\gamma}}=P$

holds for all classes $P\in H_{\gamma}$. Since the class
$\widehat{\Gamma}_{\gamma}$ is invertible in $H_{\gamma}$, dividing by it
induces an automorphism on the cohomology, hence it suffices to prove

$\displaystyle\sum_{\begin{subarray}{c}c\in C,d\in C^{\circ}\\\
|I|=\operatorname{rk}N\end{subarray}}\xi_{c,d,I}\operatorname{Vol}_{I}(2\pi{\rm
i})^{|\sigma(\gamma)|}\chi\left(\frac{P}{\widehat{\Gamma}_{\gamma}},\frac{F_{I_{d}}}{\widehat{\Gamma}_{\gamma^{\vee}}}\right)\frac{D_{I_{c}}}{\widehat{\Gamma}_{\gamma}}=\frac{P}{\widehat{\Gamma}_{\gamma}}$
(4.1)

for all $P$. We prove this by induction on the degree of $P$.

The base case $\deg{P}=0$ corresponds to $P=1_{\gamma}$. Since

$\chi\left(\frac{1_{\gamma}}{\widehat{\Gamma}_{\gamma}},\frac{F_{I_{d}}}{\widehat{\Gamma}_{\gamma^{\vee}}}\right)=0$

unless $|I_{d}|={\operatorname{rk}}N-|\sigma(\gamma)|$, the equation becomes

$\displaystyle\sum_{|I_{d}|=\operatorname{rk}N-|\sigma(\gamma)|}\xi_{\gamma^{\vee},\gamma+v_{I_{d}},I_{d}\sqcup\sigma(\gamma)}\operatorname{Vol}_{I_{d}\sqcup\sigma(\gamma)}(2\pi{\rm
i})^{|\sigma(\gamma)|}\chi\left(\frac{1_{\gamma}}{\widehat{\Gamma}_{\gamma}},\frac{F_{I_{d}}}{\widehat{\Gamma}_{\gamma^{\vee}}}\right)\frac{1_{\gamma}}{\widehat{\Gamma}_{\gamma}}=\frac{1_{\gamma}}{\widehat{\Gamma}_{\gamma}}.$
(4.2)

Then by definition of $\chi$ and Lemma 4.1, we have

$\displaystyle\chi\left(\frac{1_{\gamma}}{\widehat{\Gamma}_{\gamma}},\frac{F_{I_{d}}}{\widehat{\Gamma}_{\gamma^{\vee}}}\right)$
$\displaystyle=\frac{1}{|\operatorname{Box}(\sigma(\gamma))|}\int_{\gamma^{\vee}}\operatorname{Td}(\gamma^{\vee})\left(\frac{1}{\widehat{\Gamma}_{\gamma}}\right)^{*}\frac{F_{I_{d}}}{\widehat{\Gamma}_{\gamma^{\vee}}}$
$\displaystyle=\frac{1}{|\operatorname{Box}(\sigma(\gamma))|}\int_{\gamma^{\vee}}\frac{F_{I_{d}}}{(\widehat{\Gamma}_{\gamma})^{*}\widehat{\Gamma}_{\gamma^{\vee}}}\operatorname{Td}(\gamma^{\vee})$
$\displaystyle=\frac{1}{|\operatorname{Box}(\sigma(\gamma))|}\int_{\gamma^{\vee}}\frac{F_{I_{d}}}{(2\pi{\rm
i})^{|\sigma(\gamma)|}(-1)^{\deg\gamma^{\vee}}}$
$\displaystyle=\frac{(-1)^{\deg\gamma^{\vee}}}{(2\pi{\rm
i})^{|\sigma(\gamma)|}\operatorname{Vol}_{\overline{I_{d}}}|\operatorname{Box}(\sigma(\gamma))|}$

here $\operatorname{Vol}_{\overline{I_{d}}}$ denotes the volume of the cone
$\overline{\sigma_{I_{d}}}$ in the quotient fan $\Sigma/\sigma(\gamma)$. Note
that we have

$\displaystyle\operatorname{Vol}_{I_{d}\sqcup\sigma(\gamma)}=\operatorname{Vol}_{\overline{I_{d}}}|\operatorname{Box}(\sigma(\gamma))|$

hence (4.2) becomes

$\displaystyle\sum_{|I_{d}|=\operatorname{rk}N-|\sigma(\gamma)|}(-1)^{\deg\gamma^{\vee}}\xi_{\gamma^{\vee},\gamma+v_{I_{d}},I_{d}\sqcup\sigma(\gamma)}=1.$

If we perturb $\gamma^{\vee}$ by $\varepsilon v$, then it will fall in the
interior of exactly one maximal cone in $\Sigma$, and the corresponding
coefficient $\xi$ is the only nonzero term in the sum above (recall the
definition of $\xi_{c,d,I}$ in Theorem 2.4), which is equal to

$\displaystyle(-1)^{\deg\gamma^{\vee}}(-1)^{\deg{\gamma^{\vee}}}=1$

So the base case is proved.

Now we assume the equality (4.1) holds for all classes of degree less than
$m$. Since the cohomology $H_{\gamma}$ is generated as an algebra by classes
$D_{k}$, it suffices to prove the identity

$\displaystyle\sum_{\begin{subarray}{c}c\in C,d\in C^{\circ}\\\
|I|=\operatorname{rk}N\end{subarray}}\xi_{c,d,I}\operatorname{Vol}_{I}(2\pi{\rm
i})^{|\sigma(\gamma)|}\chi\left(\frac{D_{k}P}{\widehat{\Gamma}_{\gamma}},\frac{F_{I_{d}}}{\widehat{\Gamma}_{\gamma^{\vee}}}\right)D_{I_{c}}=D_{k}P$

for each $D_{k}P$ where $P\in H_{\gamma}$ is of degree $m-1$. Since $D_{k}$ is
skew-symmetric with respect to the $\chi$ pairing, the above statement can be
rewritten as

$\displaystyle D_{k}P=-\sum_{\begin{subarray}{c}c\in C,d\in C^{\circ}\\\
|I|=\operatorname{rk}N\end{subarray}}\xi_{c,d,I}\operatorname{Vol}_{I}(2\pi{\rm
i})^{|\sigma(\gamma)|}\chi\left(\frac{P}{\widehat{\Gamma}_{\gamma}},\frac{D_{k}F_{I_{d}}}{\widehat{\Gamma}_{\gamma^{\vee}}}\right)D_{I_{c}}.$

On the other hand, we can multiply the induction assumption for $P$ by $D_{k}$
to get

$\displaystyle\sum_{\begin{subarray}{c}c\in C,d\in C^{\circ}\\\
|I|=\operatorname{rk}N\end{subarray}}\xi_{c,d,I}\operatorname{Vol}_{I}(2\pi{\rm
i})^{|\sigma(\gamma)|}\chi\left(\frac{P}{\widehat{\Gamma}_{\gamma}},\frac{F_{I_{d}}}{\widehat{\Gamma}_{\gamma^{\vee}}}\right)D_{k}\,D_{I_{c}}=D_{k}P.$

Compare these two identities. It suffices to show

$\begin{split}0=\sum_{\begin{subarray}{c}c\in C,d\in C^{\circ}\\\
|I|=\operatorname{rk}N\end{subarray}}\xi_{c,d,I}&\operatorname{Vol}_{I}\left(D_{k}\cdot\frac{D_{I_{c}}}{\widehat{\Gamma}_{\gamma}}\right)\otimes\frac{F_{I_{d}}}{\widehat{\Gamma}_{\gamma^{\vee}}}\\\
&+\sum_{\begin{subarray}{c}c\in C,d\in C^{\circ}\\\
|I|=\operatorname{rk}N\end{subarray}}\xi_{c,d,I}\operatorname{Vol}_{I}\frac{D_{I_{c}}}{\widehat{\Gamma}_{\gamma}}\otimes\left(D_{k}\cdot\frac{F_{I_{d}}}{\widehat{\Gamma}_{\gamma^{\vee}}}\right)\end{split}$
(4.3)

which follows from Corollary 3.14. ∎

######  Remark 4.3.

Theorem 4.2 implies, in particular, that the pairing of Theorem 2.4 is
nondegenerate and is independent of $v$. We are not aware of a direct proof of
this fact.

We conclude this section by an explanation of our motivation behind the
definition of the coefficients $\xi_{c,d,I}$ in Theorem 2.4. This definition
is inspired by the following fan displacement resolution of diagonal formula
of Fulton-Sturmfels [6].

######  Proposition 4.4.

Let $X$ be the toric variety corresponds to a complete fan $\Sigma$ in a
lattice $N$, denote the diagonal embedding $X\hookrightarrow X\times X$ by
$\delta$. Let $\sigma\in\Sigma$ be any cone and $v$ a generic point in $N$,
then the diagonal class decomposes as

$\displaystyle[\delta(V(\sigma))]=\sum_{\sigma_{1},\sigma_{2}}m_{\sigma_{1},\sigma_{2}}^{\sigma}\cdot[V(\tau_{1})\times
V(\tau_{2})]$

where $m_{\sigma_{1},\sigma_{2}}^{\sigma}=[N:N_{\sigma_{1}}+N_{\sigma_{2}}]$
and the sum is over all cones $\sigma_{1},\sigma_{2}\in\Sigma$ with
$\operatorname{codim}\sigma_{1}+\operatorname{codim}\sigma_{2}=\operatorname{codim}\sigma$
and $\sigma\subseteq\sigma_{1},\sigma_{2}$ such that
$(v+\sigma_{1})\cap\sigma_{2}\not=\emptyset$.

Note that the coefficient $m_{\sigma_{1},\sigma_{2}}^{\sigma}$ is exactly the
volume $\operatorname{Vol}_{\sigma_{1}\cup\sigma_{2}}$ of the cone spanned by
$\sigma_{1}$ and $\sigma_{2}$. This formula cannot be applied to our case
directly, since the toric varieties they worked with are complete while ours
are not. Nevertheless we have the following relationship between the
definition of $\xi_{c,d,I}$ and the conditions occurred in Fulton-Sturmfels
formula.

######  Proposition 4.5.

Let $c,d\in\sigma_{I}$ and $v$ be a generic point in $C^{\circ}$. Then both
$c+\varepsilon v$ and $d-\varepsilon v$ lies in $\sigma_{I}^{\circ}$ for all
sufficiently small $\varepsilon>0$ if and only if

$\displaystyle(v+\sigma(c))\cap\sigma(d)\not=\emptyset$

where $\sigma(c)$ denotes the minimal cone of $\Sigma$ that contains $c$.

###### Proof.

Assume both $c+\varepsilon v$ and $d-\varepsilon v$ lies in
$\sigma_{I}^{\circ}$. Then we can write $c+\varepsilon v=\sum_{i\in
I}s_{i}v_{i}$ where all $s_{i}\in(0,1)$. Recall that
$I=\sigma(c)\cup\sigma(d)=I_{c}\sqcup I_{d}\sqcup\sigma(\gamma(c))$, this
equation can be rewritten into the form $v=v_{1}-v_{2}$, where
$v_{1}\in\sigma(c)$ and $v_{2}\in\sigma(d)$, which is equivalent to the second
statement. The other direction can be proved similarly. ∎

######  Remark 4.6.

We believe our methods should allow one to give a new proof of the Fulton-
Sturmfels formula, which could be done by restricting our results to the
twisted sectors that are compact. We do not go into details further in this
paper.

##  5\. Extensions and open questions

There is a more general version of the better-behaved GKZ systems which
includes a parameter $\beta\in N_{\mathbb{C}}$, with $\beta=0$ case being the
one we considered so far. Namely, the torus homogeneity equations of
Definition 1.1 read

$\sum_{i=1}^{n}\langle\mu,v_{i}\rangle
x_{i}\partial_{i}\Phi_{c}+\langle\mu,c-\beta\rangle\Phi_{c}=0$

and similarly for $\Psi_{d}$. Much of what we did in this paper is applicable
to the pair of better behaved GKZ systems with parameters $\pm\beta$. For
instance, we readily observe that our argument in Section 2 goes through for
arbitrary parameter $\beta$ to give a pairing between spaces of solutions to
$\operatorname{bbGKZ}(C,\beta)$ and $\operatorname{bbGKZ}(C^{\circ},-\beta)$.

We would like to see what happens in the limit given by a regular subdivision
$\Sigma$ for a generic $\beta$. While there are certain versions of $H$ and
$H^{c}$ considered in [8] it will be easier for our purposes to simply write
${\rm Vol}(\Delta)$ linearly independent solutions given by $\Gamma$-series,
essentially along the lines of the solutions of the original GKZ paper [7].

Let $\Sigma$ be a regular subdivision of $C$. For each maximum-dimensional
cone $\sigma$ we consider ${\rm Vol}(\sigma)$ linearly independent solutions
in the large Kähler limit of ${\mathbb{P}}_{\Sigma}$, in bijection with the
elements $\gamma$ of $N/\sum_{i\in\sigma}{\mathbb{Z}}v_{i}$. Namely, we define
the set $L_{c,\gamma,\sigma;\beta}\subset{\mathbb{C}}^{n}$ by

$\sum_{i=1}^{n}l_{i}v_{i}=\beta-c$

and the properties $l_{i}\in{\mathbb{Z}}$ for all $i\not\in\sigma$ and
$c+\sum_{i\not\in\sigma}l_{i}v_{i}=-\gamma\hskip-3.0pt\mod\sum_{i\in\sigma}{\mathbb{Z}}v_{i}.$
Then for each $\gamma$ we define a solution $\Phi^{\gamma,\sigma}$ of
$\operatorname{bbGKZ}(C,\beta)$ by

$\Phi^{\gamma,\sigma}_{c}(x_{1},\ldots,x_{n})=\sum_{l\in
L_{c,\gamma,\sigma;\beta}}\prod_{i=1}^{n}\frac{x_{i}^{l_{i}}}{\Gamma(1+l_{i})}.$

We define $\Gamma$-series solutions $\Psi^{\gamma,\sigma}$ to
$\operatorname{bbGKZ}(C^{\circ},-\beta)$ in the same way by

$\Psi^{\gamma,\sigma}_{d}(x_{1},\ldots,x_{n})=\sum_{l\in
L_{d,\gamma,\sigma;-\beta}}\prod_{i=1}^{n}\frac{x_{i}^{l_{i}}}{\Gamma(1+l_{i})}.$

Note that in the case of generic $\beta$ every solution of
$\operatorname{bbGKZ}(C^{\circ},-\beta)$ can be uniquely extended to solutions
of $\operatorname{bbGKZ}(C,-\beta)$. It is not hard to show that these
$\Phi_{c}$ and $\Psi_{d}$ converge uniformly on compacts in the region (3.2)
for an appropriate choice of $\hat{\psi}$. Moreover, as $\sigma$ and $\gamma$
vary, we get bases of the space of solutions, with linear independence assured
by them lying in different eigenspaces of the monodromy operators for small
loops around $x_{i}=0$.

Monodromy considerations imply that for the pairing
$\langle\cdot,\cdot\rangle$ of Section 2 we have
$\langle\Phi^{\gamma,\sigma},\Psi^{\gamma^{\prime},\sigma^{\prime}}\rangle=0$
unless $\sigma=\sigma^{\prime}$ and
$\gamma=-\gamma^{\prime}\hskip-3.0pt\mod\sum_{i\in\sigma}{\mathbb{Z}}v_{i}$.
In the latter case, the constant contribution will happen for
$l_{i}+l_{i}^{\prime}=0$ for $i\not\in I$ and $l_{i}+l_{i}^{\prime}=-1$ for
$i\in I$. If any of $l_{i},l_{i}^{\prime}$ is a negative integer, then the
corresponding term vanishes, due to a pole of $\Gamma$, so we may assume that
they are nonnegative for $i\not\in\sigma$, which then implies that

$I=\sigma;~{}l_{i}+l_{i}^{\prime}=-1,{\rm~{}for~{}}i\in\sigma;~{}l_{i}=l_{i}^{\prime}=0{\rm~{}for~{}}i\not\in\sigma.$

This implies that
$c=-\gamma\hskip-3.0pt\mod\sum_{i\in\sigma}{\mathbb{Z}}v_{i}$ and
$d=\gamma\hskip-3.0pt\mod\sum_{i\in\sigma}{\mathbb{Z}}v_{i}$.

We claim that for any $\gamma$ there exists exactly one pair $(c,d)$ in
$\sigma$ satisfying this constraint and $\xi_{c,d,\sigma}\not=0$. The
definition of the coefficients $\xi$ of the pairing implies that we must also
have $c+d=\sum_{i\in\sigma}v_{i}$ with $c+\varepsilon v$ and $d-\varepsilon v$
in the corresponding cone $\sum_{i\in\sigma}{\mathbb{R}}_{\geq 0}v_{i}$ for
all small $\varepsilon>0$. We can write $\beta$, $v$ and $\gamma$ uniquely as

$\beta=\sum_{i\in\sigma}\beta_{i}v_{i},~{}v=\sum_{i\in\sigma}s_{i}v_{i},~{}\gamma=\sum_{i}\gamma_{i}v_{i}$

with $\gamma_{i}\in[0,1)$. It is then easy to see that $\xi_{c,d,\sigma}$ is
nonzero if and only if

$\displaystyle c$ $\displaystyle=\sum_{\\{i:\gamma_{i}\neq
0\\}}(1-\gamma_{i})v_{i}+\sum_{\\{i:\gamma_{i}=0,s_{i}<0\\}}v_{i},$
$\displaystyle d$ $\displaystyle=\sum_{\\{i:\gamma_{i}\neq
0\\}}\gamma_{i}v_{i}+\sum_{\\{i:\gamma_{i}=0,s_{i}>0\\}}v_{i}.$

Thus for $\gamma_{i}\neq 0$ we have
$l_{i}=\beta_{i}-1+\gamma_{i},~{}l_{i}^{\prime}=-\beta_{i}-\gamma_{i}.$ For
$\gamma_{i}=0$ and $s_{i}>0$ we have
$l_{i}=\beta_{i},~{}l_{i}^{\prime}=-1-\beta_{i}$ and for $\gamma_{i}=0$ and
$s_{i}<0$ we have $l_{i}=-1+\beta_{i},~{}l_{i}^{\prime}=-\beta_{i}.$ In
particular,

$\deg(c)=-\deg(\gamma)+{\operatorname{rk}}N-\\#\\{i:\gamma_{i}=0,s_{i}>0\\}.$

Therefore the pairing is given by

$\displaystyle\langle\Phi^{\gamma,\sigma},\Psi^{-\gamma,\sigma}\rangle$
$\displaystyle=(-1)^{\deg(c)}\,{\rm
Vol(\sigma)}\prod_{\gamma_{i}\not=0}\frac{1}{\Gamma(\beta_{i}+\gamma_{i})\Gamma(1-\beta_{i}-\gamma_{i})}$
$\displaystyle\prod_{\gamma_{i}=0,s_{i}>0}\frac{1}{\Gamma(1+\beta_{i})\Gamma(-\beta_{i})}\prod_{\gamma_{i}=0,s_{i}<0}\frac{1}{\Gamma(\beta_{i})\Gamma(1-\beta_{i})}$
$\displaystyle=(-1)^{\deg(c)}\,{\rm
Vol(\sigma)}\prod_{\gamma_{i}\not=0}\frac{{\rm e}^{2\pi{\rm
i}(\beta_{i}+\gamma_{i})}-1}{2\pi{\rm i}\,{\rm e}^{\pi{\rm
i}(\beta_{i}+\gamma_{i})}}$
$\displaystyle\prod_{\gamma_{i}=0,s_{i}>0}\frac{{\rm e}^{2\pi{\rm
i}(\beta_{i}+1)}-1}{2\pi{\rm i}\,{\rm e}^{\pi{\rm
i}(\beta_{i}+1)}}\prod_{\gamma_{i}=0,s_{i}<0}\frac{{\rm e}^{2\pi{\rm
i}\beta_{i}}-1}{2\pi{\rm i}\,{\rm e}^{\pi{\rm i}\beta_{i}}}$
$\displaystyle=\frac{(-1)^{\deg(c)}\,{\rm Vol(\sigma)}}{(2\pi{\rm
i})^{{\operatorname{rk}}N}}{\rm e}^{-\pi{\rm
i}\sum_{i\in\sigma}(\beta_{i}+\gamma_{i})}\prod_{\gamma_{i}=0,s_{i}>0}{\rm
e}^{\pi{\rm i}}\prod_{i\in\sigma}({\rm e}^{2\pi{\rm
i}(\beta_{i}+\gamma_{i})}-1)$ $\displaystyle=\frac{{\rm
Vol(\sigma)}}{(2\pi{\rm i})^{{\operatorname{rk}}N}}{\rm e}^{-\pi{\rm
i}\deg(\beta)-2\pi{\rm i}\deg(\gamma)}\prod_{i\in\sigma}(1-{\rm e}^{2\pi{\rm
i}(\beta_{i}+\gamma_{i})})$ $\displaystyle=\frac{e^{-\pi{\rm
i}\deg(\beta)}{\rm Vol(\sigma)}}{(2\pi{\rm
i})^{{\operatorname{rk}}N}}\prod_{i\in\sigma}(1-e^{2\pi{\rm
i}(\beta_{i}+\gamma_{i})}).$

######  Remark 5.1.

An immediate consequence of the above calculation is that the pairing
$\langle\cdot,\cdot\rangle$ is non-degenerate for a generic $\beta$.

Further directions. We conclude this section by stating some open problems
related to our construction, in no particular order.

  * •

Is the pairing of this paper nondegenerate for all $\beta$? We know this to be
the case for $\beta=0$ and $\beta$ generic, and it seems likely to be always
true.

  * •

We would like to settle the analytic continuation conjecture of [1] to extend
the main result of [4] to the better-behaved GKZ systems. One consequence of
Theorem 4.2 is that it should be enough to just work with the usual $K$-theory
and the compactly supported version should follow from duality.

  * •

What is the HMS counterpart of our pairing from the point of view of Fukaya-
Seidel categories for the mirror potential? Our formula for the pairing is
quite simple, so presumably so should be the mirror version of it. We refer to
[5], [11] for background.

  * •

Solutions to bbGKZ systems come with a lattice structure inherited from the
$K$-theory of ${\mathbb{P}}_{\Sigma}$ (it is independent of $\Sigma$). Can
this structure be locally defined outside of the region of convergence of any
$\Gamma$-series?

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