# GKZ discriminant and Multiplicities

Jesse Huang and Peng Zhou

###### Abstract.

Let $T=(\mathbb{C}^{*})^{k}$ act on $V=\mathbb{C}^{N}$ faithfully and
preserving the volume form, i.e.
$(\mathbb{C}^{*})^{k}\hookrightarrow\text{SL}(V)$. On the B-side, we have
toric stacks $Z_{W}$ labelled by walls $W$ in the GKZ fan, and toric stacks
$Z_{/F}$ labelled by faces of a polytope corresponding to minimal semi-
orthogonal decomposition (SOD) components. The B-side multiplicity
$n^{B}_{W,F}$ is the number of times ${Coh}(Z_{/F})$ appears in the SOD of
$Coh(Z_{W})$. On the A-side, we have the GKZ discriminant loci components
$\nabla_{F}\subset(\mathbb{C}^{*})^{k}$, and its tropicalization
$\nabla^{trop}_{F}\subset\mathbb{R}^{k}$. The A-side multiplicity
$n^{A}_{W,F}$ is defined as the multiplicity of the tropical complex
$\nabla^{trop}_{F}$ on the wall $W$. We prove that $n^{A}_{W,F}=n^{B}_{W,F}$,
confirming a conjecture in Kite-Segal [KS22] inspired by [APW17]. Our proof is
based on a lemma about the B-side SOD multiplicities, which allows us to
reduce to lower dimensions just as in the A-side [GKZ][Ch 11].

##  1\. Introduction

Homological mirror symmetry (HMS) for toric varieties is a well-studied
subject, yet it can still offer new insights to classical problems. Our paper
concerns a numerical conjecture that is a shadow of a full categorical
conjecture [APW17, KS22, HK22]. We first sketch the categorical conjecture. On
the B-side, we have a toric Calabi-Yau GIT problem and we study the derived
equivalences and semi-orthogonal decompositions that arises from wall-
crossing. On the A-side we have a fiberwise partially wrapped Fukaya category
[AA21] associated to a fibration $\pi:Y\to B$ and a superpotential
$W:Y\to\mathbb{C}$, and we want to study the ’pushforward’ of
$\operatorname{Fuk}(Y,W)$ along $\pi$ to get a Fukaya category on $B$ with
categorical coefficient. For $b\in B$, let $Y_{b}$ be the fiber over $b$ and
$W_{b}$ the restriction of $W$, then there is a discriminant loci
$\nabla\subset B$ where the fiberwise wrapped Fukaya category
$\operatorname{Fuk}(Y_{b},W_{b})$ become ’degenerate’. The full HMS predicts
that, for each wall crossing $W$ (corresponding to certain asymptotic region
of $\nabla$), there is a B-side perverse schober coming from SOD of
$\operatorname{Coh}(Z_{W})$ where $Z_{W}$ is certain toric stack associated to
$W$, and there is an A-side analog coming from a transversal curve (annuli)
intersecting the discriminant loci $\nabla$ in the asymptotic region of the
wall. Our main theorem is a verification that the two schobers has the same
number of singularities of each type.

The full conjecture is explained beautifully in the Kite-Segal paper [KS22],
which is based on the physicists conjecture [APW17]. In the remaining part of
the introduction, we will focus on the numerical conjecture and state our main
results.

###  1.1. Main Result

Our input data is a collection of lattice points $q_{1},\cdots,q_{N}$ in
$\mathbb{Z}^{k}$, where $q_{i}=(q_{i1},\cdots,q_{ik})$, called weights. We
assume that $q_{i}$ generate the linear space $\mathbb{R}^{k}$, and they
satisfy the balanced condition $\sum_{i}q_{i}=0$. Equivalently, the input data
is a full rank linear map

$Q:\mathbb{Z}^{N}\to\mathbb{Z}^{k},\quad e_{i}\mapsto q_{i},$

such that $Q(1,\cdots,1)=0$. From this input data, we can setup two problems
(called A-side and B-side) as follows.

First, we consider the dual of $Q$,
$Q^{\vee}:\mathbb{Z}^{k}\to\mathbb{Z}^{N}$, and let $A$ denote the cokernel
map

$A:\mathbb{Z}^{N}\to\mathsf{N}$

where $\mathsf{N}$ may have torsion if $Q$ is not surjective.

For the sake of the introduction, we assume that $Q$ is surjective and thus
$\mathsf{N}$ is a lattice. We further assume that for any
$i\in[N]=\\{1,\cdots,N\\}$, $a_{i}=A(e_{i})$ are distinct. This is for easy
quotation of results in [GKZ]. We also abuse notation and let $A$ denote the
set $\\{a_{i}\\}$. Both of the assumptions can be easily removed, as described
in the appendix.

####  1.1.1. The A-side setup

The A-side problem concerns with a holomorphic function $W:Y\to\mathbb{C}$ and
a fibration $\pi:Y\to B$. Here
$Y=(\mathbb{C}^{*})^{N},B=(\mathbb{C}^{*})^{k}$, and

$W:(\mathbb{C}^{*})^{N}\to\mathbb{C},\quad W(z)=z_{1}+\cdots+z_{N},$
$\pi=Q_{\mathbb{C}^{*}}:(\mathbb{C}^{*})^{N}\to(\mathbb{C}^{*})^{k},\quad
Q_{\mathbb{C}^{*}}(z)=(\prod_{i=1}^{N}z_{i}^{q_{i1}},\cdots,\prod_{i=1}^{N}z_{i}^{q_{ik}}).$

Gelfand-Kapranov-Zelevinsky [GKZ] defined a polynomial $E_{A}$ of $N$
variables, called principal $A$-determinant, whose vanishing loci is a variety
$\widetilde{\nabla}_{GKZ}\subset(\mathbb{C}^{*})^{N}$. This variety descend
along $\pi$ to a variety $\nabla_{GKZ}\subset B$.

Let $\Pi$ be the convex hull of $a_{i}$. By the Calabi-Yau condition
$\sum_{i}q_{i}=0$, $\Pi$ is a $k-1$ dimensional polytope in an affine
$\mathbb{Z}^{k-1}$. Let $\mathcal{F}_{0}$ denote the faces of $\Pi$ ($\Pi$
itself is a face), and let $\mathcal{F}\subset\mathcal{F}_{0}$ denote the set
of ’minimal faces’, where a $F$ is minimal if one remove any point in $F\cap
A$ then the remaining points still generate the affine span of $F$.

For each minimal face $F$, there is a irreducible variety $\nabla_{F}\subset
B$, and the divisor $\nabla_{GKZ}$ can be decomposed as

$\nabla_{GKZ}=\sum_{F\in\mathcal{F}}m_{F}\cdot\nabla_{F},$

where $m_{F}$ is some explicit integer (not to be confused with our A-side
multiplicity). In [APW17], the multiplicity $m_{F}$ is interpreted as the rank
of $K_{0}$ of certain Higgs problem (Section 2.4). Even better, if $F\neq\Pi$,
then $\nabla_{F}$ is the pullback along certain torus quotient $\pi_{F}:B\to
B_{F}$ of a discrimiant $\nabla^{\prime}_{F}\subset B_{F}$.

The tropicalization $\nabla^{trop}_{GKZ}\subset\mathbb{R}^{k}$ of
$\nabla_{GKZ}\subset(\mathbb{C}^{*})^{k}$ is the codimension-1 part of the
GKZ-fan. Given a subvariety $V\subset(\mathbb{C}^{*})^{k}$, we define its
tropicalization as follows. Consider the map
$\operatorname{Log}:(\mathbb{C}^{*})^{k}\to\mathbb{R}^{k}$ by componentwise
$z\mapsto\log|z|$. Then, we define $V^{trop}=\lim_{t\to
0^{+}}t\cdot\operatorname{Log}(V)$. As shown in [GKZ], the Newton polytope for
the defining polynomial of $\nabla_{GKZ}$ is the secondary polytope, whose
exterior normal fan is the GKZ fan $\Sigma_{GKZ}\subset\mathbb{R}^{k}$.

The tropicalization method [Mik04] output not just a set, but a polyhedral
complex with multiplicities. It allows for easy computation of intersection
multiplicities.

######  Example 1.1.

See Figure 1. The tropicalization of $x+y=1$ in $(\mathbb{C}^{*})^{2}$ is a
tropical curve with weight $1$ on each leg, where the three legs represent the
approximate equations $x=1,y=1,x+y=0$. The tropicalization of $x^{2}+y^{3}=1$
is a tropical curve consist of three legs, representing the three region
$x^{2}\approx 1,y^{3}\approx 1$ and $x^{2}+y^{3}\approx 0$. The weight $2$
over the leg $x^{2}\approx 1$ means there are two branches of complex
solutions $x=\pm 1$ above the tropical ’shadow’.

$1$$1$$1$$x+y=1$$3$$2$$1$$x^{2}+y^{3}=1$ Figure 1. Tropicalization gives
balanced polyhehral complex. The integers labels multiplicities on each cell.

∎

Let $W$ be a wall in the GKZ fan. We can ask for the multiplicity of a
tropical complex $\nabla^{trop}_{GKZ}$ or $\nabla^{trop}_{F}$ along $W$, and
denote them by $n^{A}_{W,GKZ},n^{A}_{W,F}$. We get

$n^{A}_{W,GKZ}=\sum_{F\in\mathcal{F}}m_{F}\cdot n^{A}_{W,F}.$

It is not too hard to compute $n^{A}_{W,GKZ}$ in terms of volumes of some
polytope, but to compute $n^{A}_{W,F}$, some extra work is needed. The key
observation, made in [GKZ][Ch 11], is that $\nabla_{F}$ actually comes from a
lower dimensional problem, and one can solve for $n^{A}_{W,F}$ recursively (or
express the result as an alternating sum, if $m_{F}=1$). We borrow this idea
to the B-side and show such recursive relation also holds for SOD
multiplicity, as predicted by mirror symmetry.

####  1.1.2. The B-side setup

Let $(C^{*})^{k}$ acts on $\mathbb{C}^{N}$ with weight $q_{1},\cdots,q_{N}$.
There is a GKZ fan in $\mathbb{R}^{k}$ that labels all possible GIT quotients.
The GKZ fan is the ’shadow’ of the $\mathbb{R}_{\geq 0}^{N}$ (the moment
polytope of $\mathbb{C}^{N}$ for the $(\mathbb{C}^{*})^{N}$-action) under the
map $Q_{\mathbb{R}}:\mathbb{R}^{N}\to\mathbb{R}^{k}$.

The GIT quotient stacks $X_{C}$ for different chambers
$C\subset\mathbb{R}^{r}$ are derived equivalent. For two adjacent chambers
$C_{1},C_{2}$ separated by a wall $W$, we have $\mathbb{Z}$ many equivalence
functors

$\phi_{i}=\phi_{i,W,C_{1},C_{2}}:\operatorname{Coh}(X_{C_{1}})\to\operatorname{Coh}(X_{C_{2}}),i\in\mathbb{Z}$

where $i$ labels of a choice of window subcategory [HHP08, Seg11, HL15, BFK19]
in the GIT quotient $X_{W}=[V//_{\theta}T]$, $\theta\in W$.

It is interesting to study the autoequivalence induced by window shift
$\mu=\phi_{1}^{-1}\phi_{0}$ (for simplicity, we fix a window shift here). They
comes from a spherical functors

$S:\Phi=\operatorname{Coh}(Z_{W})\to\Psi=\operatorname{Coh}(X_{C_{1}})$

in that $\mu=\operatorname{cone}(1\to SS^{l})[-1]$, where $\Phi$ is called the
vanishing cycle category, and $\Psi$ the nearby cycle category. Hence, this
defines B-model perverse Schober over a disk.

Halpern-Leistner and Shipman [HLS16] showed that if $\Phi$ admits a semi-
orthogonal decomposition $\Phi=\langle T_{1},\cdots,T_{m}\rangle$, then we
have several fractional spherical twists $S_{i}:T_{i}\to\Psi$, and the
monodromy $\mu=\mu_{1}\circ\cdots\circ\mu_{m}$.

Kite and Segal [KS22] identified the possible SOD factors $Z_{/F}$ for
$\operatorname{Coh}(Z_{W})$, where $F$ runs through minimal faces of the
polytope $\Pi$. They have shown that different SOD has Jordan-Hölder property,
namely the multiplicity $n_{W,F}^{B}$ of $\operatorname{Coh}(Z_{/F})$ in
$\operatorname{Coh}(Z_{W})$ is well-defined. They conjectured that
$n^{A}_{W,F}=n^{B}_{W,F}$ and proved it when rank $k$ of the action torus is
$\leq 2$.

####  1.1.3. Statement of Results

Our first result is about toric GIT problem, possibly non-CY. Let
$Q:\mathbb{Z}^{N}\to\mathbb{Z}^{k}$ be a toric GIT problem, $C$ a chamber in
the GKZ fan and let $H\subset\mathbb{R}^{k}$ be a relevant subspace i.e.
$H=\operatorname{cone}(\\{q_{i}:q_{i}\in H\\})$ (Definition 2.12). Let $X_{C}$
be the GIT quotient corresponding to chamber $C$, and let $Z_{H}$ be the
irreducible SOD component for relevant subspace $H$. We are interested in the
SOD multiplicity of $\operatorname{Coh}(Z_{H})$ in
$\operatorname{Coh}(X_{C})$, denoted as $[X_{C}:Z_{H}]$. Let
$[N]_{H}=\\{i\in[N]:q_{i}\in H\\}$ be the index set of the weights lying on
$H$. Then we have a Coulomb GIT problem (Section 2.4)

$Q_{/H}:\mathbb{Z}^{N}/\mathbb{Z}^{[N]_{H}}\to\mathbb{Z}^{k}/(\mathbb{Z}^{k}\cap
H)$

where the chamber $C$ descend to a chamber $C/H$, with corresponding quotient
$X_{C/H}$, the relevant subspace $H$ quotient to a point $H/H$, and
$Z_{H/H}=pt$.

######  Theorem 1.2 (Lemma 2.18).

Let $Q$ be any toric GIT problem, $C$ a chamber, $H$ a relevant subspace. Then
the multiplicity is invariant under passing to the Coulomb problem $Q_{/H}$

$[X_{C}:Z_{H}]=[X_{C/H}:Z_{H/H}].$

######  Theorem 1.3 (Theorem 3.4).

Let $W$ be a codimension one cone in the GKZ fan of a CY problem. For any
minimal face $F$ of the polytope $\Pi$, let $n_{W,F}^{A}$ denote the
intersection multiplicity (A-side multiplicity) defined by the tropical
complex, and $n_{W,F}^{B}$ denote the semiorthogonal decomposition
multiplicity (B-side multiplicity). Then

$n_{W,F}^{A}=n_{W,F}^{B}.$

In addition, we obtain a recursive formula relating multiplicities to the
ranks of K-theory, hence volumes of stacky fans (see Section 2.3.2).

######  Theorem 1.4 (Proposition 2.19).

Let $C$ be a GKZ chamber, and ${\bf\Sigma}$ the corresponding stacky fan. For
each minimal face $F\in\mathcal{F}$, let $n_{F}:=[X_{\bf\Sigma}:Z_{/F}]$
denote the SOD multiplicity of $\operatorname{Coh}(Z_{/F})$ in
$\operatorname{Coh}(X_{\bf\Sigma})$. Then we have a system of linear equations
labelled also by minimal faces $\mathcal{F}$,

$\operatorname{rank}([X_{{\bf\Sigma}\cap F}]):=\sum_{F^{\prime}\leq
F}n_{F^{\prime}}\operatorname{rank}([Z_{F/F^{\prime}}]).$ (1.1)

where equation labelled by $F$ only involves $n_{F^{\prime}}$ with face
$F^{\prime}\leq F$, and the coefficient of ’leading term’ $n_{F}$ is
$\operatorname{rank}([Z_{F/F}])=1$.

###  1.2. Related Work

Homological mirror symmetry for toric variety has been extensively studied,
using Floer theoretic technique [Abo06, Han19, HH20] and using the microlocal
sheaf method [FLTZ11, FLTZ12, Kuw20, Zho19, GPS18], see a recent review using
GIT quotient [She21].

In the context of mirror symmetry to toric GIT problem, it is well-understood
on the B-side [HHP08, BFK19, HL15, Seg11] how to use window subcategory to do
wall-crossing between adjacent chambers, and sometimes in nice cases (e.g.
quasi-symmetric case) how to do wall-crossing among all chambers
simultaneously [HLS20, ŠVdB19].

On the A-side mirror to toric GIT, if we use microlocal sheaf as the A-model,
then thanks to the functoriality of coherent constructible correspondence
[Bon06, FLTZ11, Kuw20, Zho19], we can translate all the B-side VGIT result to
the A-side [Zho20, HZ20], window categories into window skeletons etc. If we
use the more traditional Fukaya category A-model, then the question is more
subtle and harder to solve [DKK16, BDF+15, Ker17]. The program of Ballard-
Diemer-Favero-Katzarkov-Kerr is about matching the A-side and B-side SOD.

In general, the above result on B-model (and on microlocal A-model) only sees
the codimension-1 wall crossing, and did not see the entire GKZ discriminant
locus, the recent work of Kite-Segal [KS22] in some sense remedies the above
deficiency.

The numerical version of of the full conjecture $n^{A}_{W,F}=n^{B}_{W,F}$ has
been partially proven by Kite-Segal when $\dim_{\mathbb{C}}T=1,2$. And Horja-
Katzarkov [HK22] proved an ’integrated’ numerical equality

$\sum_{F}n^{A}_{W,F}\operatorname{rank}(Z_{/F})=\sum_{F}n^{B}_{W,F}\operatorname{rank}(Z_{/F}),$

which does most of the heavy-lifting for us.

To get the full categorical conjecture [APW17, KS22, HK22], one need to
identify the A-model irreducible SOD component mirror to the B-model
counterpart $Z_{/F}$, and find a way to book-keep the relations between
various SOD components. We leave these to future work.

###  1.3. Outline

We will mainly work on the B-side. In the next section we introduce the
necessary notation for toric GIT and review the notion of Coulomb and Higgs
problems for a toric GIT, then we prove our main Lemma. Most of the content
are review or slight generalization of [KS22]. Then, in the last section, we
compare the A-side and B-side multiplicity for toric CY GIT wall-crossing.
There are examples at the end of each section, which might help to counter the
heavy notations.

##  2\. The B-side

In this section, we first setup the general toric GIT problem, then introduce
the GKZ fan to relate various phases of GIT quotients (M-side). These data can
be equally well encoded in certain triangulations problem (N-side). Next, we
introduce the notion of a Coulomb-Higgs GIT problem, which represent a sub-
quotient of the original GIT problem. See [KS22, HK22, GKZ] for more details.
Then we prove our main lemma, namely the SOD multiplicity is invariant under
passing to the Coulomb problem associated to a minimal face. Given the lemma,
we then deduce a recursive formula for SOD multiplicities.

###  2.1. Toric GIT setup

Our starting point is a torus $(\mathbb{C}^{*})^{k}$ acting on
$V=\mathbb{C}^{N}$ with weights $q_{1},\cdots,q_{N}\in\mathbb{Z}^{k}$. More
invariantly, we have a rank $k$ lattice $L\simeq\mathbb{Z}^{k}$, and
$L_{\mathbb{C}^{*}}=L\otimes_{\mathbb{Z}}\mathbb{C}^{*}\simeq(\mathbb{C}^{*})^{k}$
acts on $\mathbb{C}^{N}$ by factoring through
$L_{\mathbb{C}^{*}}\to(\mathbb{C}^{*})^{N}$. This induces a map on the
cocharacter lattices and character lattices

$Q^{\vee}:L\to\mathbb{Z}^{N},\quad Q:(\mathbb{Z}^{N})^{\vee}\to L^{\vee}.$

We assume the $\operatorname{coker}(Q)$ is finite, or equivalently $Q^{\vee}$
is injective.

We have a short exact sequence (SES), called ’N-sequence’

$0\to L\xrightarrow{Q^{\vee}}\mathbb{Z}^{N}\xrightarrow{A}\mathsf{N}\to 0$

where $\mathsf{N}$ might have torsion

$0\to\mathsf{N}_{tors}\to\mathsf{N}\to\mathsf{N}_{free}\to 0.$

Let $\bar{A}:\mathbb{Z}^{N}\to\mathsf{N}_{free}$ denote the obvious
composition. Apply the $\operatorname{Hom}(-,\mathbb{Z})$ to the ’N-sequence’,
and use injective resolution
$\mathbb{Z}\to\mathbb{Q}\to\mathbb{Q}/\mathbb{Z}$, then we have a long exact
sequece, the ’M-sequence’

$0\to\mathsf{M}_{0}\xrightarrow{A^{\vee}}(\mathbb{Z}^{N})^{\vee}\xrightarrow{Q}L^{\vee}\to\mathsf{M}_{1}\to
0,$

where
$\mathsf{M}_{0}=\operatorname{Hom}(\mathsf{N},\mathbb{Z})=\operatorname{Hom}(\mathsf{N}_{free},\mathbb{Z})$
is a lattice, and
$\mathsf{M}_{1}=\operatorname{Ext}^{1}(\mathsf{N},\mathbb{Z})=\operatorname{Hom}(\mathsf{N}_{tors},\mathbb{Q}/\mathbb{Z})$
is the Pontryagin dual to the torsion subgroup $\mathsf{N}_{tors}$.

If $G$ is an abelian group, we denote
$(\cdots)_{G}:=(\cdots)\otimes_{\mathbb{Z}}G$, where $G$ can be
$\mathbb{R},\mathbb{Z},\mathbb{T}=\mathbb{R}/\mathbb{Z},\mathbb{C},\mathbb{C}^{*}$.
For example, if we apply $(\cdots)_{\mathbb{R}}$ to
$Q:(\mathbb{Z}^{N})^{\vee}\to L^{\vee}$, then we will get
$Q_{\mathbb{R}}:(\mathbb{R}^{N})^{\vee}\to L^{\vee}_{\mathbb{R}}$.

Let the co-character lattice
$X_{*}(({\mathbb{C}^{*}})^{N})\simeq\mathbb{Z}^{N}$ be equipped with the
standard basis $e_{1},\cdots,e_{N}$, and let the character lattice
$X^{*}(({\mathbb{C}^{*}})^{N})\simeq(\mathbb{Z}^{N})^{\vee}$ be equipped with
the dual basis $e_{1}^{\vee},\cdots,e_{N}^{\vee}$. Define

$\widetilde{a}_{i}=A(e_{i})\in\mathsf{N},\quad
a_{i}=\bar{A}(e_{i})\in\mathsf{N}_{free},\quad q_{i}=Q(e_{i}^{\vee})\in
L^{\vee}.$

###  2.2. M-side, chambers and walls

By the M-side, we mean the objects living on spaces in the M-sequence, for
example about the map $Q_{\mathbb{R}}:(\mathbb{R}^{N})^{\vee}\to
L^{\vee}_{\mathbb{R}}$. Here for the simplicity of notation, we write
$(\mathbb{R}^{N})^{\vee}$ as $\mathbb{R}^{N}$.

We recall the definition of GKZ fan (rather GKZ stratification)
$\Sigma_{GKZ}(Q)$ in $L^{\vee}_{\mathbb{R}}$ (aka GIT fan or secondary fan)
for a toric GIT problem $Q$. We omit $(Q)$ if there is no danger of confusion.

By the GKZ stratification $\Sigma_{GKZ}(Q)$, we mean the stratification of
$L^{\vee}_{\mathbb{R}}$ generated by $Q_{\mathbb{R}}$ images of various faces
of the positive quadrant $P_{N}=\mathbb{R}_{\geq 0}^{N}$. A top dimensional
strata is called a chamber, and a codimension-1 strata is called a wall. The
support of the GKZ fan is $Q_{\mathbb{R}}(\mathbb{R}_{\geq 0}^{N})$. If the
support is not the full $L^{\vee}_{\mathbb{R}}$, then its complement is still
a GKZ strata, and we call it the empty chamber.

Here $P_{N}$ is the image of the moment map of $(\mathbb{C}^{*})^{N}$ acting
on $\mathbb{C}^{N}$

$\mu_{N}:\mathbb{C}^{N}\to\mathbb{R}^{N},\quad(z_{i})_{i}\mapsto(|z_{i}|^{2})_{i}.$

And the moment map of $L_{\mathbb{C}^{*}}$ acting on $\mathbb{C}^{N}$ is
$\mu=Q_{\mathbb{R}}\circ\mu_{N}$.

For any $c\in L^{\vee}_{\mathbb{R}}$, we define the GIT quotient stack

$X_{c}=[(\mathbb{C}^{N})^{ss}_{c}/L_{\mathbb{C}^{*}}],\quad(\mathbb{C}^{N})^{ss}_{c}=\\{z\in\mathbb{C}^{N}\mid\overline{L_{\mathbb{C}^{*}}\cdot
z}\cap\mu^{-1}(c)\neq\emptyset\\}.$

More concretely, the positive quadrant $P_{N}$ is stratiefied by faces $\tau$,
$P_{N}=\sqcup\tau$ which induces a stratification of $\mathbb{C}^{N}$ into
strata $(\mathbb{C}^{N})_{\tau}=\mu_{N}^{-1}(\tau)$, and
$(\mathbb{C}^{N})^{ss}_{c}=\sqcup\\{(\mathbb{C}^{N})_{\tau}\mid
Q(\overline{\tau})\cap c\neq\emptyset\\}.$ From the latter description, it is
clear that $X_{c}$ is constant when $c$ varies within a GKZ strata $C$, hence
we also write $X_{c}$ as $X_{C}$.

######  Definition 2.1.

For any GKZ strata $C$, we call the GIT quotient $X_{C}$ a ’phase’ of the
toric GIT problem.

Let $\det V=\sum_{i}q_{i}\in L^{\vee}$ denote the weight of
$L_{\mathbb{C}^{*}}$ acting on $\det V$. The toric GIT is called Calabi-Yau
(CY), if $\det V=0$.

Let $W$ be a wall separating two chambers $C_{+},C_{-}$. We choose the $\pm$
sign so that on the wall $\det V$ is pointing towards $C_{+}$. Let
$\lambda_{W}\in L$ be an integral primitive vector conormal to the wall. Let
$d_{W}=|\langle\lambda_{W},\det V\rangle|$. If $d_{W}=0$, i.e, $\det V$ is
parallel to $W$, we say there is a balanced wall-crossing and we have (non-
canonical) derived equivalence
$\operatorname{Coh}(X_{C_{+}})\simeq\operatorname{Coh}(X_{C_{-}})$. If
$d_{W}>0$, then we have Semi-orthogonal decomposition

$\operatorname{Coh}(X_{C_{+}})=\langle\operatorname{Coh}(X_{C_{+}}),\operatorname{Coh}(Z_{W}),\cdots,\operatorname{Coh}(Z_{W})\rangle,$

where $Z_{W}$ is some toric variety defined in […], and the factor
$\operatorname{Coh}(Z_{W})$ repeats $d_{W}$ times.

We will abuse notation and let $X$ also denote $\operatorname{Coh}(X)$.

Therefore, starting from any point $c\in L^{\vee}_{\mathbb{R}}$, we may form
the ray in the direction of $-\det V$. This is called the ’straight-line’ run
in [DKK16]. If $c$ is generic, then the run will only encounter walls, and
ends in a chamber $C$ that contains $-\det V$ in its closure. We call such a
chamber $C$ and the corresponding phase $X_{C}$ minimal. It is possible that
$X_{C}=\emptyset$ or that there are several minimal chambers, but upto derived
equivalence, the minimal phase is unique. Let $X_{\min}(Q)$ denote the minimal
phase (possibly empty) for toric GIT problem.

For a non-CY problem, we can associate a CY problem to by adding a weight
$q_{N+1}=-\det V$. The new problem is

$0\to
L\xrightarrow{\hat{Q}}\mathbb{Z}^{N+1}\xrightarrow{\hat{A}}\mathsf{N}\oplus\mathbb{Z}\to
0$

We have $\hat{A}(e_{i})=(A(e_{i}),1)$ for $i\in[N]$, and
$\hat{A}(e_{N+1})=(0,1)$.

Going the other way around, suppose we have a toric CY GIT problem, then pick
any $q_{i}$ (which by assumption are all nonzero), say $q_{N+1}$ by
relabelling, we can delete it and get a toric non-CY GIT problem.

Assuming $Q$ is a non-CY problem, then the toric CY GKZ stratification
$\Sigma_{GKZ}(\hat{Q})$ refines the non-CY GKZ stratification
$\Sigma_{GKZ}(Q)$. The new walls in $\Sigma_{GKZ}(\hat{Q})$ are all parallel
to $\det V$.

###  2.3. N-side, local triangulations and stacky fan

By the N-side, we mean the objects living on spaces in the N-sequence, for
example about the map $A:\mathbb{Z}^{N}\to\mathsf{N}$. Recall
$a_{i}=\bar{A}(e_{i})\in\mathsf{N}_{free}\subset\mathsf{N}_{\mathbb{R}}$ and
$\widetilde{a}_{i}=A(e_{i})\in\mathsf{N}$.

####  2.3.1. Localized Marked Polytope Subdivision

Let $\Delta_{N}=\operatorname{conv}(0,e_{1},\cdots,e_{N})$ be the standard
$N$-simplex in $\mathbb{R}^{N}$. Let $\Pi=A_{\mathbb{R}}(\Delta_{N})$ be the
image of $\Delta_{N}$ in $\mathsf{N}_{\mathbb{R}}$, which is also the convex
hull $\operatorname{conv}(0,a_{1},\cdots,a_{N})$.

We start with a pair of piecewise linear (PL) functions on $\mathbb{R}^{N}$
and $(\mathbb{R}^{N})^{\vee}$ related by Legendre transformation. Let

$\varphi_{N}(x_{1},\cdots,x_{N}):=\min(0,x_{1},\cdots,x_{N}):(\mathbb{R}^{N})^{\vee}\to\mathbb{R}.$

It is a concave function, with $\varphi_{N}^{-1}(0)=P_{N}=(\mathbb{R}_{\geq
0})^{N}$. By Legendre transformation, we have

$\psi_{N}(y_{1},\cdots,y_{N})=\max_{x\in\mathbb{R}}(\varphi(x)-(x,y)):\mathbb{R}^{N}\to\mathbb{R}.$

$\psi_{N}$ is a convex function, and explicitly

$\psi_{N}(x)=\begin{cases}0&x\in\Delta_{N}\\\
+\infty&x\notin\Delta_{N}\end{cases}$

Given any point $c\in L^{\vee}_{\mathbb{R}}$, choose its lift $b\in
Q^{-1}(c)$. Then, we can identify $\mathsf{M}_{\mathbb{R}}\simeq Q^{-1}(c)$ by

$f_{b}:\mathsf{M}_{\mathbb{R}}\to Q^{-1}(c),\quad\xi\mapsto
b+A_{\mathbb{R}}^{\vee}(\xi)=(b_{1}+(a_{1},\xi),\cdots,b_{N}+(a_{N},\xi)).$

The restriction of $\varphi_{N}$ on $Q^{-1}(b)$ pullback to
$\mathsf{M}_{\mathbb{R}}$ by $f_{b}$ gives

$\varphi_{b}(\xi):=\varphi_{N}(b+A_{\mathbb{R}}^{\vee}(\xi))=\min(0,b_{1}+a_{1}(\xi),b_{2}+a_{2}(\xi),\cdots,b_{N}+a_{N}(\xi)).$

Its Legendre transformation is

$\psi_{b}(\eta)=\max_{\xi\in\mathsf{M}_{\mathbb{R}}}(\varphi_{b}(\xi)-(\eta,\xi)).$

For uniform treatment later, it is useful to introduce another pair
$(a_{N+1},b_{N+1})=(0,0)\in\mathsf{N}_{\mathbb{R}}\times\mathbb{R}$ for the
origin.

The overgraph of $\psi_{b}$

$\Gamma_{\geq\hat{\psi}_{b}}:=\\{(x,y)\in\Pi\times\mathbb{R}\mid
y\geq\psi_{b}(x)\\}$

is the convex hull of the upward rays

$\bigcup_{i=1}^{N+1}\bar{a}_{i}\times\\{y\geq
b_{i}\\}\subset\mathsf{N}_{\mathbb{R}}\times\mathbb{R}.$

We record the subset $S\subset[N+1]$ where the tip of the ray touches the
graph, i.e.

$S=\\{i\in[N+1]\mid\psi_{b}(a_{i})=b_{i}\\}.$

For $x\in\Pi$, if $x=a_{i}$ for some $i\in S$, we abuse notation and say $x$
is marked by $a_{i}$. It is possible $x$ is marked by more than one $a_{i}$.

The maximal linearity domain of $\psi_{b}$ defines a polytope subdivision of
$\Pi$, and the vertices of the polytopes and possibly some interior points are
marked by some $a_{i}$.

The polytope subdivision together with marking points $S$ is called a coherent
(multi-)marked polytope subdivision, and marked subdivision for short.

If we choose a different lift $b\in Q^{-1}(c)$, then $\psi_{b}$ only change by
a linear function and does not affect the coherent subdivision. For a generic
choice of $c$, we have triangulations of $\Pi$ with no interior marked points,
and each vertex of the simplices is marked exactly once.

We say a marked subdivision uses $a_{i}$ if $a_{i}$ (rather $i$) appears in
the marking. We say two marked subdivisions are equivalent at $a_{i}$ if they
have the same collection (possibly empty) marked polytopes that contains
$a_{i}$ in the marking.

######  Definition 2.2.

A marked subdivision localized at $a_{i}$ is an equivalence class of marked
subdivisions modulo equivalence relations at $a_{i}$.

We refer to the index $N+1$, and the point $a_{N+1}$, as the ’base point’.

For any $b\in L^{\vee}_{\mathbb{R}}$, we may use the above procedure to obtain
a marked subdivision $T(b)$ of $\Pi$.

######  Proposition 2.3.

  1. (1)

For any $b\in L^{\vee}_{\mathbb{R}}$, the subdivision $T(b)$ uses the base
point $a_{N+1}$ if and only if $b$ is not in the empty chamber of
$\Sigma_{GKZ}(Q)$.

  2. (2)

For any $b_{1},b_{2}\in L^{\vee}_{\mathbb{R}}$, $b_{1}$ and $b_{2}$ are in the
same GKZ strata of $\Sigma_{GKZ}(Q)$ if and only if $T(b_{1})$ is equivalent
to $T(b_{2})$ at the base point $a_{N+1}$.

###### Proof.

$T(b)$ uses $a_{N+1}$ if and only if $Q_{\mathbb{R}}^{-1}(b)$ intersect
$\mathbb{R}_{\geq 0}^{N}$. $b_{1}$ and $b_{2}$ are in the same GKZ strata, if
and only if as we move $b$ straightly from $b_{1}$ to $b_{2}$, the fiber
$Q_{\mathbb{R}}^{-1}(b)$ intersects each face $\tau$ of $\mathbb{R}_{\geq
0}^{N}$ in the same way. Each non-empty intersection of $\tau\cap
Q_{\mathbb{R}}^{-1}(b)$ corresponds to a polytope (possibly not of top
dimension) in $T(b)$ marked by $a_{N+1}$. ∎

######  Remark 2.4.

If the GIT problem $Q$ is Calabi-Yau, then any marked subdivision of $\Pi$
uses $a_{N+1}$ and the equivalence relation is trivial. In this case, we have
a bijection between GKZ chambers and marked triangulation of $\Pi$.

If the GIT problem $Q$ is not CY, then we may consider the associated CY
problem $\hat{Q}$. Then chambers in $\Sigma_{GKZ}(\hat{Q})$ corresponds to
marked triangulation of of $\Pi$; and chambers in $\Sigma_{GKZ}(Q)$
corresponds to marked triangulations of $\Pi$ localized at $a_{N+1}$.

####  2.3.2. Stacky Fan and its Volume

Let $C$ be a chamber of GKZ fan, and let $T(C)$ be the corresponding marked
triangulation localized at the base point. This defines a stacky fan in
$\mathsf{N}$ as follows.

Recall the definition of simplicial stacky fan following [BCS05]. Let
$\mathsf{N}$ be a finitely generated abelian group, $\Sigma$ be a rational
simplicial polyhedran fan in $\mathsf{N}_{\mathbb{R}}$, and
$\\{v_{i}\\}_{i\in\Sigma^{1}}\subset\mathsf{N}$ such that
$(v_{i})_{\mathbb{R}}$ generate the corresponding ray in $\Sigma$. The triple
${\bf\Sigma}=(\mathsf{N},\Sigma,\\{v_{i}\\})$ is called a stacky fan.

Roughly speaking, modulo torsion, we can think of a stacky fan ${\bf\Sigma}$
as a collection of simplices ${\hat{\sigma}}\subset\mathsf{N}_{\mathbb{R}}$
with shared vertices $(v_{i})_{\mathbb{R}}$.

We write $|\Sigma|$ for the union of the cones $\sigma$, and we write
$|{\bf\Sigma}|$ for the union of the simplices ${\hat{\sigma}}$.

We normalize volume $\operatorname{vol}_{\mathbb{R}}$ on
$\mathsf{N}_{\mathbb{R}}$ such that a minimal simplex with vertices in
$\mathsf{N}_{free}$ has unit volume. Then we define the stacky volume by

$\operatorname{vol}(\mathbf{\Sigma}):=|\mathsf{N}_{tors}|\cdot\operatorname{vol}_{\mathbb{R}}(|{\bf\Sigma}|).$

By design of the volume, we have the following result

######  Proposition 2.5 ([HK22]).

The volume of the stacky fan equals the rank of the $K_{0}$ group of
$X_{\bf\Sigma}$,

$\operatorname{rank}(K_{0}(X_{\bf\Sigma}))=\operatorname{vol}({\bf\Sigma}).$

####  2.3.3. Minimal phases and Minimal Fans

Given a toric GIT problem $A:\mathbb{Z}^{N}\twoheadrightarrow\mathsf{N}$ or
$Q:(\mathbb{Z}^{N})^{\vee}\to L^{\vee}$, on the $M$-side, we have some minimal
chambers $C$, i.e. those containing $-\det(V)$. On the N-side, we have some
minimal stacky fans corresponding to the minimal chambers. They have the same
support $|{\bf\Sigma}^{min}|$ which we describe now.

Let $S=A_{\mathbb{R}}(\mathbb{Z}_{\geq 0}^{N})$ and
$S_{+}=A_{\mathbb{R}}(\mathbb{Z}_{\geq 0}^{N}\backslash\\{0\\})$ and define
(slightly abusing notation)

$|{\bf\Sigma}^{min}|=\operatorname{conv}(S)-\operatorname{conv}(S_{+}).$

We see $|{\bf\Sigma}^{min}|=\emptyset$ if and only if $S=S_{+}$, or
$0\in\Pi=\operatorname{conv}(a_{i})$, or the support of the GKZ fan is not the
whole $L^{\vee}_{\mathbb{R}}$.

######  Proposition 2.6 ([HK22]).

We have the rank-volume relation

$\operatorname{rank}(K_{0}(X_{min}))=\operatorname{vol}(|{\bf\Sigma}^{min}|)=|\mathsf{N}_{tors}|\cdot\operatorname{vol}_{\mathbb{R}}(\operatorname{conv}(S)-\operatorname{conv}(S_{+})).$

###  2.4. Coulomb and Higgs GIT problems

We follow [KS22, APW17] and introduce the a sub and quotient GIT problem,
called Coulomb and Higgs problem respectively.

####  2.4.1. From subsets of $[N]$

For any subset $\Gamma\subset[N]$, we can define a Coulomb problem (as a GIT
problem of a subspace acted on by a subgroup) and a Higgs problem (as a
quotient space acted on by a quotient group).

${0}$${0}$${0}$${0}$${L_{\Gamma}}$${\mathbb{Z}^{\Gamma}}$${\mathsf{N}_{\Gamma}}$${0}$${0}$${L}$${\mathbb{Z}^{N}}$${\mathsf{N}}$${0}$${0}$${L/L_{\Gamma}}$${\mathbb{Z}^{N}/\mathbb{Z}^{\Gamma}}$${\mathsf{N}/\mathsf{N}_{\Gamma}}$${0}$${0}$${0}$${0}$$\scriptstyle{Q_{\Gamma}^{\vee}}$$\scriptstyle{A_{\Gamma}}$$\scriptstyle{Q^{\vee}}$$\scriptstyle{A}$$\scriptstyle{Q_{/\Gamma}^{\vee}}$$\scriptstyle{A_{/\Gamma}}$

Dualize the first two columns, and then apply the snake lemma, we get

${0}$${0}$${0}$${...}$${0}$${(\mathsf{N}/\mathsf{N}_{\Gamma})^{\vee}_{0}}$${(\mathbb{Z}^{N}/\mathbb{Z}^{\Gamma})^{\vee}}$${(L/L_{\Gamma})^{\vee}}$${(\mathsf{N}/\mathsf{N}_{\Gamma})^{\vee}_{1}}$${0}$${0}$${\mathsf{M}_{0}}$${(\mathbb{Z}^{N})^{\vee}}$${L^{\vee}}$${\mathsf{M}_{1}}$${0}$${0}$${(\mathsf{M}_{\Gamma})_{0}}$${(\mathbb{Z}^{\Gamma})^{\vee}}$${L^{\vee}_{\Gamma}}$${(\mathsf{M}_{\Gamma})_{1}}$${0}$${...}$${0}$${0}$${0}$$\scriptstyle{A_{/\Gamma}}$$\scriptstyle{Q_{/\Gamma}}$$\scriptstyle{A^{\vee}}$$\scriptstyle{Q}$$\scriptstyle{A_{\Gamma}^{\vee}}$$\scriptstyle{Q_{\Gamma}}$

######  Definition 2.7.

Let $\Gamma\subset[N]$. We call the GIT problem associated to $Q_{\Gamma}$ (or
$A_{\Gamma}$) as the Coulomb problem for $\Gamma$, and the GIT problem
associated to $Q_{/\Gamma}$ and $A_{/Ga}$ as the Higgs problem for $\Gamma$.

More generally, if we have $\Gamma_{1}\subset\Gamma_{2}\subset[N]$, we may
form the GIT problem

$L_{\Gamma_{2}}/L_{\Gamma_{1}}\xrightarrow{Q^{\vee}_{\Gamma_{2}/\Gamma_{1}}}\mathbb{Z}^{\Gamma_{2}}/\mathbb{Z}^{\Gamma_{1}}\xrightarrow{A_{\Gamma_{2}/\Gamma_{1}}}\mathsf{N}_{\Gamma_{2}}/\mathsf{N}_{\Gamma_{1}}$

as the Coulomb-Higgs problem for the pair $\Gamma_{1}\subset\Gamma_{2}$.

####  2.4.2. From subspaces and faces

Although the Coulomb-Higgs problems can be defined for general subsets
$\Gamma_{1}\subset\Gamma_{2}\subset[N]$, they often arises from subspaces
$H\subset L^{\vee}_{\mathbb{R}}$ and faces $F\subset\Pi$.

######  Definition 2.8.

A subspace $H\subset L^{\vee}_{\mathbb{R}}$ is called weight supported if
$H=\text{span}_{\mathbb{R}}\\{q_{i}\in H\\}$.

A subset $F\subset\Pi$ is called a face if there is a linear function
$l:\mathsf{N}_{\mathbb{R}}\to\mathbb{R}$, such that $l|_{\Pi}\geq 0$ and
$F=l^{-1}(0)\cap\Pi$.

Recall $[N]_{H}=\\{i:q_{i}\in H\\}$ and $[N]_{F}=\\{i:a_{i}\in F\\}$.

######  Definition 2.9.

If $H_{1}\subset H_{2}\subset L^{\vee}_{\mathbb{R}}$ are a pair of weight
supported subspaces, we define the Coulomb-Higgs problem $Q_{H_{2}/H_{1}}$

$Q_{H_{2}/H_{1}}:\mathbb{Z}^{[N]_{H_{2}}-[N]_{H_{1}}}\to(L^{\vee}\cap
H_{2})/H_{1}.$

If $H_{1}=0$, we get a Higgs problem $Q_{H_{2}}$; if
$H_{2}=L^{\vee}_{\mathbb{R}}$, we get a Coulomb problem $Q_{/H_{1}}$.

If $F_{1}\subset F_{2}\subset\Pi$ is a pair of faces, we define the Coulomb-
Higgs problem $A_{F_{2}/F_{1}}$

$A_{F_{2}/F_{1}}:\mathbb{Z}^{[N]_{F_{2}}-[N]_{F_{1}}}\to\mathsf{N}_{F_{2}}/\mathsf{N}_{F_{1}},\quad\mathsf{N}_{F}:=\mathsf{N}_{\Gamma=[N]_{F}}.$

If $F_{1}=0$, we get a Coulomb problem $A_{F_{2}}$; if $F_{2}=\Pi$, we get a
Higgs problem $A_{/F_{1}}$.

####  2.4.3. Minimal faces and relevant subspaces

Here we again follow [KS22, APW17] and introduce certain important Coulomb
Higgs problem, that describes the SOD irreducible components.

######  Definition 2.10.

Let $S$ be a subset of $[N]$. Recall that $a_{i}=A_{\mathbb{R}}(e_{i})$,
$q_{i}=Q_{\mathbb{R}}(e_{i}^{\vee})$.

  1. (1)

We say $S$ is $A_{\mathbb{R}}$-redundant, if there exists $c_{i}\neq 0$ for
each $i\in S$, such that $\sum_{i\in S}c_{i}a_{i}=0$.

  2. (2)

We say $S$ is $A_{\mathbb{R}}$-saturated, if there is a linear function
$l:\mathsf{N}_{\mathbb{R}}\to\mathbb{R}$, such that $S=\\{i:l(a_{i})=0\\}$.

  3. (3)

We say $S$ is $A_{\mathbb{R}}$-extremally-saturated, if there is a linear
function $l:\mathsf{N}_{\mathbb{R}}\to\mathbb{R}$, such that
$S=\\{i:l(a_{i})=0\\}$, and $l(a_{i})\geq 0$ for all $i\in[N]$.

  4. (4)

We say $S$ is $Q_{\mathbb{R}}$-redundant, if there exists $c_{i}\neq 0$ for
each $i\in S$, such that $\sum_{i\in S}c_{i}q_{i}=0$.

  5. (5)

We say $S$ is $Q_{\mathbb{R}}$-saturated, if there is a linear function
$l:L^{\vee}_{\mathbb{R}}\to\mathbb{R}$, such that $S=\\{i:l(q_{i})=0\\}$.

  6. (6)

We say $S$ is $Q_{\mathbb{R}}$-positively-redundant if there exists $c_{i}>0$
for $i\in S$, such that $\sum_{i\in S}c_{i}q_{i}=0$.

######  Proposition 2.11.

Let $S$ be

  1. (1)

$S$ is $A_{\mathbb{R}}$-redundant if and only if $S^{c}$ is
$Q_{\mathbb{R}}$-saturated.

  2. (2)

$S$ is $Q_{\mathbb{R}}$-redundant if and only if $S^{c}$ is
$A_{\mathbb{R}}$-saturated.

  3. (3)

$S$ is $Q_{\mathbb{R}}$-positively redundant if and only if $S^{c}$ is
$A_{\mathbb{R}}$-extremally-saturated.

###### Proof.

We only prove the first statement, as the second follows verbatim by changing
.

For (1), by definition, $S$ is $A_{\mathbb{R}}$-redundant, if there exists
$c_{i}\neq 0$ for each $i\in S$, such that $\sum_{i\in S}c_{i}a_{i}=0$. By
setting $c_{i}=0$ for $i\notin S$, we get an element
$\vec{c}=(c_{i})\in\mathbb{R}^{N}$, such that $A(\vec{c})=0$, i.e, $\vec{c}\in
L$. Hence $\vec{c}$ defines a linear function
$l:L^{\vee}_{\mathbb{R}}\to\mathbb{R}$, such that $l(q_{i})=0$ if and only if
$c_{i}=0$, i.e $i\in S^{c}$. Thus $S^{c}$ is $Q_{\mathbb{R}}$-saturated. The
argument can also be reversed, hence we get the equivalence.

For (2), we only need to change the above argument by swapping
$A_{\mathbb{R}}$ with $Q_{\mathbb{R}}$, and $a_{i}$ with $q_{i}$. And (3) is
the same as (2) where the two positivity conditions matches. ∎

Recall $\Pi=A_{\mathbb{R}}(\mathbb{R}_{+}^{N})\subset\mathsf{N}_{\mathbb{R}}$.
A subset $F\subset\Pi$ is called a face is there is a linear function
$l:\mathsf{N}_{\mathbb{R}}\to\mathbb{R}$, such that $l(\Pi)\geq 0$ and
$F=l^{-1}(0)$. For example, $\Pi$ is always a face of itself. And if $\Pi$
contains a linear subspace $H$, then any face $F\supset H$.

######  Definition 2.12.

Let $F$ is a face of $\Pi$ and define $[N]_{F}=\\{i:a_{i}\in F\\}$. By
construction $[N]_{F}$ is $A_{\mathbb{R}}$ extremally saturated. We say $F$ is
a minimal face of $\Pi$ if $[N]_{F}$ is $A_{\mathbb{R}}$ redundant.

Let $H\subset L^{\vee}_{\mathbb{R}}$ be a weight supported subspace, and
define $[N]_{H}=\\{i:q_{i}\in H\\}$. We say $H$ is a relevant subspace of
$L^{\vee}_{\mathbb{R}}$ if $[N]_{H}$ is $Q_{\mathbb{R}}$-positively-redundant
and $Q_{\mathbb{R}}$ saturated.

######  Remark 2.13.

Our definitions generalize the definitions of minimal faces and relevant
subspaces in [KS22] in the toric CY setting.

We obtain the following slight generalization of Proposition 4.15 in [KS22].

######  Proposition 2.14.

There is a bijection between minimal faces of $\Pi$ and relevant subspaces of
$L^{\vee}_{\mathbb{R}}$, such that if a minimal face $F$ corresponds to a
relevant subspace $H$ then $[N]_{F}=[N]_{H}^{c}$.

###### Proof.

Let $F$ be a minimal face, then $[N]_{F}$ is $A_{\mathbb{R}}$ extremally-
saturated and redundant, hence $[N]_{F}^{c}$ is $Q_{\mathbb{R}}$-positively-
redundant and saturated. Let $H=\text{span}\\{q_{i}:i\in[N]_{F}^{c}\\}$, we
then get a relevant subspace. Going backward is similar. ∎

If $F$ is a minimal face (of $\Pi$), we will abuse notation and also call
$[N]_{F}$ a minimal face. Similarly, if $H$ is a relevant subspace, we will
also call $[N]_{H}$ is a relevant subspace.

####  2.4.4. GIT quotients

Here we define some GIT quotients associated with the Higgs and Coulomb
problems.

######  Definition 2.15.

If $H$ is a relevant subspace, and $F$ is the corresponding minimal face, with
$\Gamma=[N]_{F}$, then we use $Z_{H}=Z_{/F}=Z_{/\Gamma}$ to denote the minimal
phase in the Higgs problem $Q_{/H}$ $A_{F}$.

In general, for any GIT problem $Q_{*}$ or $A_{*}$, where
$*=F,H,F_{1}/F_{2}\cdots$, we use $Z_{*}$ to denote the minimal phase in that
problem.

If $W$ is a wall in $\Sigma_{GKZ}(Q)$, let $H=\text{span}_{\mathbb{R}}(W)$,
then we define $Z_{W}$ as the phase $W$ of the Higgs problem $Q_{H}$.

The following result and proof is essentially also due to [KS22].

######  Proposition 2.16.

Let $F$ be a face of $\Pi$, and $A_{F}$, $Q_{F}$ be the Coulomb problem
associated to $F$. Then we have map of lattices

$\pi_{F}:L^{\vee}\to L^{\vee}_{F}.$

The map $(\pi_{F})_{\mathbb{R}}$ is compatible with the GKZ stratifications
$\Sigma_{GKZ}(Q)$ and $\Sigma_{GKZ}(Q_{F})$, i.e. image of a strata is a
strata.

###### Proof.

Let $\widetilde{\Sigma}_{GKZ}(Q)$ be the pullback stratification of
$\mathbb{R}^{N}$, and $\widetilde{\Sigma}_{GKZ}(Q_{F})$ be that of
$\mathbb{R}^{[N]_{F}}$. Suffice to show that under the quotient map
$\widetilde{\pi}_{F}:\mathbb{R}^{N}\to\mathbb{R}^{[N]_{F}}$, the image of a
strata is a strata.

Let $b\in\mathbb{R}^{N}$, and $b_{F}=\pi_{F}(b)\in\mathbb{R}^{[N]_{F}}$. Then
$b$ defines a PL convex function $\psi_{b}$ on $\Pi$, and $b_{F}$ defines a PL
convex function $\psi_{b_{F}}$ on $F$. It is easy to check
$\psi_{b_{F}}=\psi_{b}|_{F}$. Hence when $b$ varies within a strata, ie,
induced localized marked subdivision remains invariant, the boundary $F$
subdivision also remains invariant. Thus $\pi_{F}$ sends a strata inside a
strata.

On the other hand, consider the lift (a section of $\pi_{F}$)

$\iota_{F}:\mathbb{R}^{[N]_{F}}\to\mathbb{R}^{N},\quad x\mapsto(x-b_{F})+b.$

We claim that $\iota_{F}$ also sends a strata into a strata. Indeed, as we
vary $b_{F}$ such that the localized marked subdivision of $F$ remains
invariant, then we can extend the variation of $b_{F}$ to that of $b$ by
keeping other $b_{i}$s not in the face $F$ fixed. The resulting localized
subdivision of $\Pi$ remains fixed. ∎

Given the above result, if $C$ is a chamber corresponding to the and $F$ is a
face, then we have the a well-defined chamber $C_{F}$ in the Coulomb problem
$Q_{F}$. If $C$ corresponds to a stacky fan ${\bf\Sigma}$, then we use
${\bf\Sigma}_{F}$ for $C_{F}$. Concretely, ${\bf\Sigma}_{F}$ is the stacky fan
$(\mathsf{N}_{\Gamma},\Sigma\cap(\mathsf{N}_{\Gamma})_{\mathbb{R}},S\cap\Gamma)$.
We restrict the simplices ${\hat{\sigma}}$ to the face $F$ and coarsen
$\mathsf{N}\cap\mathbb{R}F$ to sub-lattice $\mathsf{N}_{\Gamma}$.

###  2.5. Main Lemma

First we recall some results from [KS22].

######  Proposition 2.17 ([KS22]).

The irreducible SOD components in a toric GIT problem $Q$ are labelled by the
set of relevant subspaces, or equivalently, the set of minimal faces. Let
$X_{C}$ be any phase and $H\subset L^{\vee}_{\mathbb{R}}$ any relevant
subspaces, then the multiplicity $n^{B}_{C,H}=[X_{C}:Z_{H}]$, as the number of
times that $\operatorname{Coh}(Z_{H})$ appears in any complete SOD of
$\operatorname{Coh}(X_{C})$, is well-defined.

Our main lemma equate the multiplicity $[X_{C}:Z_{H}]$ to its counterpart
$[X_{C/H}:Z_{H/H}]$ in the Coulomb problem $Q_{/H}$.

Recall we have a quotient map $\pi_{H}:L^{\vee}_{\mathbb{R}}\to
L^{\vee}_{\mathbb{R}}/H$. As Kite-Segal observed, the quotient $\pi_{H}$ is
compatible with fans $\Sigma_{GKZ}(Q)$ and $\Sigma_{GKZ}(Q_{/H})$. Indeed, a
strata in $\Sigma_{GKZ}(Q)$ represent a localized marked subdivision, and
passing to $\Sigma_{GKZ}(Q_{/H})$ means we intersect the subdivision with the
face $F$ corresponding to $H$, hence is still a localized marked subdivision.

If $C$ is a chamber in $\Sigma_{GKZ}(Q)$, let $C/H$ denote the corresponding
chamber in $\Sigma_{GKZ}(Q_{/H})$. Different chambers $C$ can result in the
same quotient chamber $C/H$.

Let $X_{C/H}$ denote the phase of $C/H$ in Coulomb problem $Q_{/H}$, and let
$Z_{H/H}$ denote the minimal phase in the Coulomb-Higgs problem $Q_{H/H}$,
which actually is a point.

######  Lemma 2.18.

Let $C$ be a non-empty GKZ chamber and $H$ a relevant subspace, then

$[X_{C}:Z_{H}]=[X_{C/H}:Z_{H/H}].$

###### Proof.

We induct on $r=\operatorname{codim}_{\mathbb{R}}H$. If $r=0$,
$H=L^{\vee}_{\mathbb{R}}$. Since by assumption $H$ is relevant, hence
$Z_{H}=X_{min}\neq\emptyset$. We have $[X_{C}:Z_{H}]=[X_{C}:X_{min}]=1.$ And
we also have $X_{C/H}=pt$, hence $[X_{C/H}:Z_{H/H}]=1$.

Suppose the statement hold when $r<n$, and consider the case $r=n$. For any
chamber $C$, consider a sequence of “monotone decreasing run” $C=C_{0}\leadsto
C_{1}\leadsto\cdots\leadsto C_{m}$, where $C_{m}$ is a minimal phase, and let
$W_{i}$ denote the wall separating $C_{i-1}$ and $C_{i}$. For any wall $W$,
write $W\parallel H$ if $\text{span}_{\mathbb{R}}(W)\supset H$.

For any wall $W$ with $W\not\parallel H$, we have $[Z_{W}:Z_{H}]=0$. Thus

$[X_{C}:Z_{H}]=\sum_{W}d_{W}[Z_{W}:Z_{H}]=\sum_{W\parallel
H}d_{W}[Z_{W}:Z_{H}],$

where summation of $W$ runs over all the wall crossing in the run i.e.
$W=W_{i}$ for some $i$.

For $W$ with $W\parallel H$, $[Z_{W}:Z_{H}]$ is computed in the Higgs problem
$Q_{\text{span}_{\mathbb{R}}W}$. By our induction hypothesis, we have
$[Z_{W}:Z_{H}]=[Z_{W/H},Z_{H/H}]$. Hence

$[X_{C}:Z_{H}]=\sum_{W\parallel H}d_{W}[Z_{W/H}:Z_{H/H}].$

On the other hand, the sequence of “upstair chambers” in
$L_{\mathbb{R}}^{\vee}$ is sent to a sequence of “downstair chambers” in
$L_{\mathbb{R}}^{\vee}/H$ (possibly with repetitions). In particular, if
$W\not\parallel H$, then the two chambers separated by $W$ are sent to the
same chamber in $L^{\vee}_{\mathbb{R}}/H$. Hence the downstairs wall crossing
corresponds to those upstairs wall crossing with $W\parallel H$. We get

$[X_{C/H}:Z_{H/H}]=\sum_{W\parallel H}d_{W/H}[Z_{W/H}:Z_{H/H}].$

Hence we only need to prove $d_{W}=d_{W/H}$, and this follows from
$\pi_{H}^{*}(\lambda_{W/H})=\lambda_{W}$ (up to sign ambiguity of
$\lambda_{W}$). ∎

Since a GKZ chamber $C$ corresponds to a coherent stacky fan ${\bf\Sigma}$,
and a relevant subspace $H$ corresponds to a minimal face $F$, we may also
denote $X_{\bf\Sigma}=X_{C}$ and $Z_{/F}=Z_{H}$, and denote
$n^{B}_{{\bf\Sigma},/F}=n^{B}_{C,H}$. Let $\Gamma=[N]_{F}$ denote set of
$a_{i}$ in the minimal face. Then, the above statement is

$[X_{{\bf\Sigma}}:Z_{/F}]=[X_{{\bf\Sigma}\cap F}:Z_{F/F}]$

###  2.6. Recursive Formula for Multiplicity

Let $\mathcal{F}$ be the set of minimal faces indexing the SOD components, and
let $\mathbb{Z}\mathcal{F}$ be formal $\mathbb{Z}$-linear combination of
$\mathcal{F}$. We denote the basis of $\mathbb{Z}\mathcal{F}$ by $[Z_{/F}]$.
There is a rank map by taking the rank of the $K_{0}$ group of SOD components

$\operatorname{rank}:\mathbb{Z}\mathcal{F}\to\mathbb{Z},\quad\operatorname{rank}([Z_{/F}])=\operatorname{rank}(K_{0}(\operatorname{Coh}(\mathbb{Z}_{/F})))$

Explicitly, we have
$\operatorname{rank}([Z_{/F}])=\operatorname{vol}({\bf\Sigma}^{min}(A_{/F})=i(\Gamma)u(\Gamma)$,
where $i(\Gamma)$ and $u(\Gamma)$ are define in [GKZ] or [HK22]. (See also
Section 2.3.3).

######  Proposition 2.19.

Let $C$ be a GKZ chamber, and ${\bf\Sigma}$ the corresponding stacky fan. For
each minimal face $F\in\mathcal{F}$, let $n_{F}:=[X_{\bf\Sigma}:Z_{/F}]$
denote the SOD multiplicity of $\operatorname{Coh}(Z_{/F})$ in
$\operatorname{Coh}(X_{\bf\Sigma})$. Then we have a system of linear equations
labelled also by minimal faces $\mathcal{F}$,

$\operatorname{rank}([X_{{\bf\Sigma}\cap F}]):=\sum_{F^{\prime}\leq
F}n_{F^{\prime}}\operatorname{rank}([Z_{F/F^{\prime}}]).$ (2.1)

where equation labelled by $F$ only involves $n_{F^{\prime}}$ with face
$F^{\prime}\leq F$, and the coefficient of ’leading term’ $n_{F}$ is
$\operatorname{rank}([Z_{F/F}])=1$.

###### Proof.

We define class $[X_{\bf\Sigma}]\in\mathbb{Z}\mathcal{F}$ as

$[X_{\bf\Sigma}]:=\sum_{F}[X_{{\bf\Sigma}}:Z_{/F}][Z_{/F}].$

For each minimal face $F$, we have

$[X_{{\bf\Sigma}\cap F}]:=\sum_{F^{\prime}\leq F}[X_{{\bf\Sigma}\cap
F}:Z_{F/F^{\prime}}][Z_{F/F^{\prime}}].$ (2.2)

By our main Lemma (2.18), we have that

$[X_{{\bf\Sigma}\cap F}:Z_{F/F^{\prime}}]=[X_{{\bf\Sigma}\cap F\cap
F^{\prime}}:Z_{F^{\prime}/F^{\prime}}]=[X_{{\bf\Sigma}\cap
F^{\prime}}:Z_{F^{\prime}/F^{\prime}}],$

is independent of $F$. Denote
$n_{F^{\prime}}=[X_{{\bf\Sigma}},Z_{/F^{\prime}}]$ as the unknown variables.
Taking rank on both sides in Eq (2.2), we get the desired proposition. ∎

###  2.7. Examples

Here we illustrate how to get the SOD multiplicity of $n^{B}_{{\bf\Sigma},F}$
in two different ways. The first way is by wall crossing, and count how each
wall $Z_{W}$ decompose into various $Z_{/F}$. The second way is to use the
recursive formula we had.

######  Example 2.20.

Consider the following $a_{i}$ points in $\mathsf{N}=\mathbb{Z}^{2}$. Our
initial stacky fan is ${\bf\Sigma}=\operatorname{conv}(\\{0\\}\cup A)$, shown
in yellow in the first figure. We have

$X_{{\bf\Sigma}}=[\mathbb{C}/\mathbb{Z}_{2}]\times[\mathbb{C}/\mathbb{Z}_{3}].$

Then, we show a sequence of shrinking ${\bf\Sigma}$ (shown in yellow), and
show how the lost volume during circuit transition contribute to various SOD
components. The total voluem of ${\bf\Sigma}$ is 6. There are 3 minimal faces

$F_{1}=\Pi,\quad F_{2}=\operatorname{conv}(0,a),\quad
F_{3}=\operatorname{conv}(0,b),\quad F_{4}=0$

where $a=(0,3),b=(2,0)$. They corresponds to

$Z_{/F_{1}}=pt,\quad Z_{/F_{2}}=\mathbb{C},\quad Z_{/F_{3}}=\mathbb{C},\quad
Z_{/F_{4}}=\mathbb{C}^{2}$

They all have $\operatorname{rank}(Z_{/F_{i}})=1$. And we have the
decomposition of total volume as

$6=2(F_{1})+2(F_{2})+1(F_{3})+1(F_{4}).$

Figure 2. Multiplicities from Volume allocation.

Next, we compute the SOD multiplicities again using the recursive relation.
Denote $n_{i}=[X_{\bf\Sigma},Z_{/F_{i}}]$, and denote
$m_{i}=\operatorname{rank}(Z_{/F_{i}})$. First, since $F_{4}$ is smallest
face, we have $n_{4}=1$. Then, consider the equation for face $F_{3}$. $F_{3}$
has a proper subface $F_{4}$, and we have $m_{4}=1,m_{3}=1$. We have

$2=\operatorname{rank}(X_{{\bf\Sigma}\cap F_{3}})=n_{3}+n_{4}=n_{3}+1$

hence $n_{3}=1$. Next, we compute $n_{2}$ using $F_{2}$ equation, we get

$3=n_{2}+n_{4}=n_{2}+1$

hence $n_{2}=2$. Lastly, we compute $n_{1}$, using $F_{1}$, which gives

$6=n_{1}+n_{2}+n_{3}+n_{4}=n_{1}+2+1+1$

Thus $n_{1}=2$. ∎

The next example shows that the irreducible SOD components really depends on
the entire GIT problem, instead of just the initial phase $X_{\bf\Sigma}$ that
we try to decompose.

######  Example 2.21.

Consider the following set of $a_{1},\cdots,a_{5}$ in
$\mathsf{N}=\mathbb{Z}^{2}$.

$a_{1}$$a_{2}$$a_{3}$$a_{4}$$a_{5}$ Figure 3. SOD components depend on the
entire GIT problem

We have 4 minimal faces

$F_{1}=(0,0),\quad F_{2}=\operatorname{conv}(0,a_{2}),\quad
F_{3}=\operatorname{conv}(0,a_{5}),\quad
F_{4}=\operatorname{conv}(0,a_{1},\cdots,a_{5}),$

with corresponding irreducible components $Z_{i}=Z_{/F_{i}}$

$Z_{1}=\mathbb{C}\times[\mathbb{C}/\mathbb{Z}_{2}],\quad
Z_{2}=\mathbb{C},\quad Z_{3}=\mathbb{C},\quad Z_{4}=pt.$

The initial stacky fan ${\bf\Sigma}=\operatorname{conv}(0,a_{2},a_{4})$, with
$X_{{\bf\Sigma}}=[\mathbb{C}/\mathbb{Z}_{2}]\times[\mathbb{C}/\mathbb{Z}_{2}]$.
And we have SOD and rank (volume) equation

$X_{{\bf\Sigma}}=Z_{1}+Z_{2}+Z_{4},\quad 4=2+1+1.$

Note that it is important that we have the point $a_{5}$ in the GIT problem.
This makes $Z_{2}=\mathbb{C}$.

If we change the problem by deleting $a_{5}$. Then, $a_{i}$ still generate
$\mathsf{N}=\mathbb{Z}^{2}$, and we still have the above 4 minimal faces.
However, we have different SOD components, in that
$Z_{2}=[\mathbb{C}/\mathbb{Z}_{2}]$. This results in a different SOD for
$X_{\bf\Sigma}$

$X_{{\bf\Sigma}}=Z_{1}+Z_{2},\quad 4=2+2.$

∎

##  3\. Multiplicity Conjecture for toric CY GIT

Consider a toric GIT problem $Q:{({\mathbb{Z}^{\vee}})^{N}}\to L^{\vee}$ that
satisfies the Calabi-Yau condition, i.e. $\sum_{i=1}^{N}Q(e_{i}^{\vee})=0$.

In this case, GKZ defined a principal $A$-determinant $E_{A}$, an integer
polynomial with $N$ variables. $E_{A}$ has many nice properties, its Newton
polytope equals the secondary polytope, and its zero-loci descent to a
hypersurface $\nabla_{GKZ}\subset L^{\vee}_{\mathbb{C}^{*}}$, the GKZ
discriminant loci.

It is known that divisor $\nabla_{GKZ}$ has decomposition into irreducible
components $\nabla_{\Gamma}$,

$\nabla_{GKZ}=\sum_{\Gamma\in\mathcal{F}}m_{\Gamma}\nabla_{\Gamma},$

where $\mathcal{F}$ denote the set of minimal faces, where
$m_{\Gamma}=\operatorname{rank}(K_{0}(Z_{\Gamma}))$ are certain
multiplicities.

It is also known that the tropicalization of $\nabla_{GKZ}$ is the union of
walls in $\Sigma_{GKZ}$. We will now decorate the walls by multiplicities in
free abelian group generated by minimal faces.

Let $\mathbb{Z}\mathcal{F}$ denote the free abelian group generated by
$\mathcal{F}$, with basis denoted by $[\Gamma]$. Then for each wall $W$, we
define

$[W]:=\sum_{\Gamma}[Z_{W}:Z_{\Gamma}]\cdot[\Gamma]=\sum_{\Gamma}n^{B}_{W,\Gamma}\cdot[\Gamma].$

Let $\mathcal{W}$ denote the set of walls in $\Sigma_{GKZ}(Q)$. Let
$\mathbb{Z}\mathcal{W}$ be the free abelian group generated by $\mathcal{W}$,
with basis denoted as $\langle W\rangle$. Then any tropical complex supported
on $\Sigma_{GKZ}$ is equivalent to a non-negative linear combination of the
walls. In particular, we have

$\nabla_{\Gamma}^{trop}:=\sum_{W\in\mathcal{W}}n^{A}_{W,\Gamma}\langle
W\rangle$

######  Definition 3.1.

Let $Q$ be a toric CY GIT problem. A tropical complex is an element in
$\mathbb{Z}\mathcal{W}$, and a decorated tropical complex is an element in
$\mathbb{Z}(\mathcal{W}\times\mathcal{F})$. We define the A-model tropical
complexes as

$\nabla^{A}_{GKZ}=\sum_{\Gamma}\nabla_{\Gamma}^{trop}\cdot[\Gamma]=\sum_{\Gamma,W}n^{A}_{W,\Gamma}\langle
W\rangle[\Gamma],\quad\nabla_{\Gamma}^{A}=\sum_{W}n^{A}_{W,\Gamma}\langle
W\rangle$

We define the B-model tropical complexes as

$\nabla^{B}_{GKZ}=\sum_{W}\langle
W\rangle[W]=\sum_{W,\Gamma}n^{B}_{W,\Gamma}\langle
W\rangle[\Gamma],\quad\nabla_{\Gamma}^{B}=\sum_{W}n^{B}_{W,\Gamma}\langle
W\rangle$

We write
$\nabla^{A}_{GKZ}(Q),\nabla^{A}_{\Gamma}(Q),\nabla^{B}_{GKZ}(Q),\cdots$ to
emphasize the dependence on $Q$.

Our goal is to show that $\nabla^{A}(Q)=\nabla^{B}(Q)$, or more concretely for
each $\Gamma$, $\nabla^{A}_{\Gamma}(Q)=\nabla^{B}_{\Gamma}(Q)$. We first
reduce the task to showing just the top dimensional minimal face.

Let $Q_{\Gamma}:({\mathbb{Z}^{\vee}})^{\Gamma}\to L^{\vee}_{\Gamma}$ denote
the Coulomb problem for minimal face $\Gamma$. Let
$\pi_{\Gamma}:L^{\vee}_{\mathbb{R}}\to(L^{\vee}_{\Gamma})_{\mathbb{R}}$.
Define
$\pi_{\Gamma}^{*}:\mathbb{Z}(\mathcal{W}_{\Gamma}\times\mathcal{F}_{\Gamma})\to\mathbb{Z}(\mathcal{W}\times\mathcal{F})$.
For a wall $W_{\Gamma}\in\mathcal{W}_{\Gamma}$, we define
$\pi_{\Gamma}^{*}(\langle
W_{\Gamma}\rangle)=\langle\pi^{-1}W_{\Gamma}\rangle=\sum_{W\in\mathcal{W},\pi_{\Gamma}(W)=W_{\Gamma}}\langle
W\rangle$. For any $\Gamma^{\prime}\in\mathcal{F}_{\Gamma}$, since
$\mathcal{F}_{\Gamma}\subset\mathcal{F}$, we have
$\pi_{\Gamma}^{*}([\Gamma^{\prime}])=[\Gamma^{\prime}]$ .

######  Proposition 3.2.

$\nabla^{A}_{\Gamma}(Q)=\pi_{\Gamma}^{*}(\nabla^{A}_{\Gamma}(Q_{\Gamma}))$ and
$\nabla^{B}_{\Gamma}(Q)=\pi_{\Gamma}^{*}(\nabla^{B}_{\Gamma}(Q_{\Gamma}))$.

###### Proof.

On the A-side, this follows from relation from the complex hypersurfaces
$\nabla_{\Gamma}(Q)=\pi_{\Gamma}^{-1}(\nabla_{\Gamma}(Q_{\Gamma}))$.

On the B-side, we know from Lemma 2.18 that
$[Z_{W}:Z_{\Gamma}]=[Z_{W_{\Gamma}}:Z_{\Gamma/\Gamma}]$. Thus

$\nabla_{\Gamma}^{B}(Q)=\sum_{W}[Z_{W}:Z_{\Gamma}]\langle
W\rangle=\sum_{W:W\parallel\Gamma}[Z_{W_{\Gamma}}:Z_{\Gamma/\Gamma}]\langle
W\rangle=\sum_{W_{\Gamma}\in\mathcal{W}_{\Gamma}}[Z_{W_{\Gamma}}:Z_{\Gamma/\Gamma}]\pi_{\Gamma}^{*}(\langle
W_{\Gamma}\rangle)=\pi_{\Gamma}^{*}(\nabla^{B}_{\Gamma}(Q_{\Gamma})).$

∎

Recall we have a rank map
$\operatorname{rank}:\mathbb{Z}\mathcal{F}\to\mathbb{Z}$, where

$\operatorname{rank}([\Gamma]):=\operatorname{rank}(K_{0}(Z_{\Gamma})).$

And we extends by linearity to
$\operatorname{rank}:\mathbb{Z}(\mathcal{W}\times\mathcal{F})\to\mathbb{Z}(\mathcal{W}).$

Horja-Katzarkov proved a numerical version of the desired theorem

######  Proposition 3.3 ([HK22], Theorem 3.5).

$\operatorname{rank}(\nabla^{A}_{GKZ})=\operatorname{rank}(\nabla^{B}_{GKZ})$

###### Proof.

In loc.cit, we have
$\operatorname{rank}([W])=\sum_{\Gamma}n^{A}_{W,\Gamma}\operatorname{rank}([\Gamma])$.
Hence

$\operatorname{rank}(\nabla^{B}_{GKZ})=\sum_{W}\langle
W\rangle\operatorname{rank}([W])=\sum_{W,\Gamma}n^{A}_{W,\Gamma}\operatorname{rank}([\Gamma])=\operatorname{rank}(\nabla^{A}_{GKZ}).$

∎

Now we are ready to prove our main theorem.

######  Theorem 3.4.

For any toric CY GIT problem $Q$, we have
$\nabla^{A}_{GKZ}(Q)=\nabla^{B}_{GKZ}(Q)$, i.e.
$n^{A}_{W,\Gamma}=n^{B}_{W,\Gamma}$ for any wall $W$ and minimal faces
$\Gamma$.

###### Proof.

We induct on the rank of $L$. In the case $\operatorname{rank}L=0$, there is
no wall, hence nothing to prove. Assume the case for $\operatorname{rank}L<n$
is proven, and we have $\operatorname{rank}L=n$.

Let $\Gamma_{0}$ be largest minimal face, concretely
$\Gamma_{0}=\\{i:q_{i}\neq 0\\}$. For any minimal face $\Gamma\neq\Gamma_{0}$,
we have $\operatorname{rank}L_{\Gamma}<n$, hence by induction
$\nabla_{\Gamma}^{A}(Q_{\Gamma})=\nabla_{\Gamma}^{B}(Q_{\Gamma})$. By
Proposition 3.2, we have $\nabla_{\Gamma}^{A}(Q)=\nabla_{\Gamma}^{B}(Q)$ for
all minimal faces $\Gamma\neq\Gamma_{0}$. Thus, we have

$\nabla^{A}_{GKZ}(Q)-\nabla^{B}_{GKZ}(Q)=\sum_{\Gamma}(\nabla^{A}_{\Gamma}(Q)-\nabla^{B}_{\Gamma}(Q))[\Gamma]=(\nabla^{A}_{\Gamma_{0}}(Q)-\nabla^{B}_{\Gamma_{0}}(Q))[\Gamma_{0}]$

Now, take $\operatorname{rank}$ on both sides, using
$\operatorname{rank}([\Gamma_{0}])=\operatorname{rank}(K_{0}\operatorname{Coh}(Z_{\Gamma_{0}}))=\operatorname{rank}(K_{0}\operatorname{Coh}(\mathbb{A}^{\Gamma_{0}^{c}}))=1\neq
0$, and Horja-Katzarkov result Proposition 3.3, we get

$0=\nabla^{A}_{\Gamma_{0}}(Q)-\nabla^{B}_{\Gamma_{0}}(Q).$

Thus $\nabla^{A}_{GKZ}(Q)=\nabla^{B}_{GKZ}(Q)$. ∎

###  3.1. Examples

######  Example 3.5.

Consider the example in [KS22, Example 3.7] with an additional weight $-\det
V=(-1,-2)$ (marked in blue) to get a toric CY GIT problem. There are three
relevant subspaces:

$H_{0}=L^{\vee}_{\mathbb{R}},\quad
H_{1}=\text{span}_{\mathbb{R}}(q_{3},q_{4},q_{6}),\quad H_{2}=\\{0\\}$

We compute the tropical complexes in this CY problem.

$q_{1},q_{2}$$q_{3},q_{4}$$q_{5}$$q_{6}$$q_{7}$$\nabla^{B}_{GKZ}$$\nabla^{B}_{H_{1}}$$2$$\nabla^{B}_{H_{2}}$
Figure 4. Decomposition of $\nabla^{B}_{GKZ}=\sum_{H}\nabla^{B}_{H}$, where
$H$ labels the relevant subspaces.

($H_{0}$) Since the problem is CY, every chamber is a minimal chamber. The
Coulomb problem is the trivial problem
$\mathbb{Z}^{0}\xrightarrow{0}\mathbb{Z}^{0}$, and
$\nabla_{H_{0}}^{B}=\emptyset$.

($H_{1}$) From the Higgs problem $\Gamma_{1}^{c}=\\{1,3,4\\}$ we know
$Z_{H_{1}}=\mathbb{A}^{1}$. The Coulomb problem for $\Gamma_{1}^{c}$ is the
Atiyah flop $Q=(1,1,-1,-1)$. We claim $\nabla_{H_{1}}^{B}$ has multiplicity 1
on the contributing walls, which is a result proven in [KS22] when
$\operatorname{rank}L=1$. (Indeed, any irreducible hypersurface in
$\mathbb{C}^{*}$ is a point with multiplicity 1).

($H_{2}$) In this case, the Coulomb problem is the entire GIT problem. The
picture on the right is the tropical complex $\nabla_{H_{2}}^{B}$ where one of
the walls has multiplicity 2. These multiplicities can be computed by looking
at the Higgs problem. For the wall with multiplicity 2, $Z_{W}=\mathbb{P}^{1}$
and $Z_{\Gamma}=pt$, and
$\operatorname{Coh}(\mathbb{P}^{1})=\langle\operatorname{Coh}(pt),\operatorname{Coh}(pt)\rangle$.
For other walls, $Z_{W}=pt$ and $Z_{\Gamma}=pt$. ∎

######  Example 3.6.

This example shows that the conjectural formula [HLS16, Remark 4.7] is false.

Consider toric CY GIT problem $Q:\mathbb{Z}^{5}\to\mathbb{Z}^{3}$, with weight
matrix

$Q=\begin{pmatrix}1&1&0&0&-n&n-2\\\ 0&0&1&0&1&-2\\\ 0&0&0&1&0&-1\end{pmatrix}$

and let $q_{i}$ be the column vectors. Assume $n\geq 2$. Consider the $xy$
plane, which contains weights $q_{1}=q_{2}=(1,0),q_{3}=(0,1),q_{5}=(-n,1)$
(omitting the 3rd coordinate). The following is a picture of the $xy$ plane
for $n=2$.

$q_{1},q_{2}$$q_{3}$$q_{5}$$\mathbb{F}_{2}$$\mathbb{P}^{2}_{1,1,2}$$W_{1}$$W_{2}$

There are two phases on this plane, corresponding to weighted projective space
$Z_{W_{1}}=\mathbb{P}^{2}_{1,1,n}$ ($W_{1}=\operatorname{cone}(q_{3},q_{5})$)
and Hirzebruch surface $Z_{W_{2}}=\mathbb{F}_{n}$ (for
$W_{2}=\operatorname{cone}(q_{1},q_{3})$). We have SOD multiplicities (see
also the example in the introduction of [BDF+15])

$[\mathbb{P}^{2}_{1,1,n}]=(n+2)[pt],\quad[\mathbb{F}_{n}]=4[pt].$

This means a local curve transverse to wall $W_{1}$ will intersect
$\nabla_{GKZ}$ at $n+2$ points, and a local curve transverse to wall $W_{2}$
will intersect $\nabla_{GKZ}$ at $4$ points. On the other hand, the formula
given by [HLS16] would always predict $4$ (since the 4 weights $q_{i}$ on the
xy plane are all primitive vectors), no matter which wall the curve
intersects.

One can also verify that $\nabla^{B}_{GKZ}$ here is a balanced tropical
complex. Consider a local tropical curve $C$ with tangent direction $(0,0,1)$
intersects $\nabla^{B}_{GKZ}$ near point $q_{3}$. If $C$ intersects at
$W_{1}$, then we get intersection multiplicity

$C\cdot\nabla^{B}_{GKZ}=(C\cdot\langle
W_{1}\rangle)\cdot\operatorname{rank}(Z_{W_{1}})=1\cdot(n+2)=n+2,$

where tropical intersection number $C\cdot W$ is defined as the unsigned
volume $|\det(v_{1}\wedge v_{2}\wedge v_{3})|$, where $v_{i}\in\mathbb{Z}^{3}$
and $v_{1}$ is primitive generator of $C$ and $v_{2},v_{3}$ are generators of
the sublattice $\text{span}_{\mathbb{R}}(W)\cap\mathbb{Z}^{3}$. If we move the
curve to the right, $C$ intersects with $W_{2}$ and an additional wall
$W_{3}=\operatorname{cone}(q_{3},q_{6})$, and we get

$C\cdot\nabla_{GKZ}^{B}=(C\cdot W_{2})\operatorname{rank}(Z_{W_{2}})+(C\cdot
W_{3})\operatorname{rank}(Z_{W_{3}})=1\cdot 4+(n-2)\cdot 1=n+2,$

where $Z_{W_{3}}=pt$.

∎

##  Appendix A Adaptation of GKZ’s setup

In the paper, we quoted results from [GKZ] whereas the setup do not quite
match. In the appendix, we show that with minor modification, the generalized
case reduces to the original GKZ setup, hence the GKZ conclusion still
applies. In GKZ, one start with a finite set $A\subset\mathbb{Z}^{k-1}$, where
as in our setup, we can have a map $A:\mathbb{Z}^{N}\to\mathsf{N}$, where
$\mathsf{N}$ can have torsion and $a_{i}=A(e_{i})$ can coincide with each
other. Here in the appendix we show that the difference in the setup is minor.

First, we generalize $A$ to be a ’multi-set’ while keeping $\mathsf{N}$ a
lattice. Suppose we have
$A:\mathbb{Z}^{N}\to\mathbb{Z}^{k-1}\times\mathbb{Z}$, and let $A^{\prime}$
denote the image set $\\{A(e_{i})\\}$, then we have a map
$s:[N]\twoheadrightarrow A^{\prime}$. We claim that the GKZ discriminant loci
for $A$ is a pull-back of that for $A^{\prime}$, where the fiber is a product
of pair-of-pants. Indeed, consider the example,

$A:\mathbb{Z}^{3}\to\mathbb{Z},\quad e_{i}\mapsto 1$

then the corresponding $W=(c_{1}+c_{2}+c_{3})z$, and the discrimiants happens
exactly at $c_{1}+c_{2}+c_{3}=0$, which after quotient by $\mathbb{C}^{*}$
become the pair-of-pants in $L^{\vee}_{\mathbb{C}^{*}}=(\mathbb{C}^{*})^{2}$.

Next, we generalize the case where $\mathsf{N}$ have torsion. Suppose
$Q:(\mathbb{Z}^{N})^{\vee}\to L^{\vee}$ has finite cokernel. Define
$\hat{L}^{\vee}=im(Q)\subset L^{\vee}$, then we have

$0\to\hat{L}^{\vee}\to L^{\vee}\to\mathsf{M}_{1}\to 0.$ (A.1)

Dualize, we get

$0\to L\to\hat{L}\to\mathsf{N}_{tors}\to 0$ (A.2)

where we used that
$\mathsf{N}_{tors}=\operatorname{Ext}^{1}(\mathsf{M}_{1},\mathbb{Z})$. In
other words, $\hat{L}$ is the saturation of $L$ in $\mathbb{Z}^{N}$.

$0\to(\mathsf{M}_{0})_{\mathbb{C}^{*}}\to(C^{*})^{N}\to\hat{L}_{\mathbb{C}^{*}}^{\vee}\to
1.$ (A.3)

Apply $\operatorname{Hom}(-,\mathbb{C}^{*})$ to Eq (A.1),

$0\to\mathsf{M}_{1}\to\hat{L}^{\vee}_{\mathbb{C}^{*}}\to
L^{\vee}_{\mathbb{C}^{*}}\to 1$ (A.4)

Thus, although the fiber of $(C^{*})^{N}\to L_{\mathbb{C}^{*}}^{\vee}$ has
$|\mathsf{M}_{1}|$ disconnected component of
$(\mathsf{M}_{0})_{\mathbb{C}^{*}}$, and the point $b\in
L_{\mathbb{C}^{*}}^{\vee}$ is ’bad’ if the function $W$ restrict any component
is singular (at finite place or at infinity), we can first work with
$\hat{Q}:(\mathbb{Z}^{N})^{\vee}\to\hat{L}^{\vee}$, and get
$\nabla_{GKZ}(\hat{Q})\subset\hat{L}_{\mathbb{C}^{*}}^{\vee}$, then
pushforward it along $\hat{L}_{\mathbb{C}^{*}}^{\vee}\to
L^{\vee}_{\mathbb{C}^{*}}$ to get $\nabla_{GKZ}(\hat{Q})$.

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