# Mirror symmetry for truncated cluster varieties

Benjamin Gammage and Ian Le  Department of Mathematics, Harvard University
Mathematical Sciences Institute, Australian National University
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In the algebraic setting, cluster varieties were reformulated by Gross-
Hacking-Keel as log Calabi-Yau varieties admitting a toric model. Building on
work of Shende-Treumann-Williams-Zaslow in dimension 2, we describe the mirror
to the GHK construction in arbitrary dimension: given a truncated cluster
variety, we construct a symplectic manifold and prove homological mirror
symmetry for the resulting pair. We also describe how our construction can be
obtained from toric geometry, and we relate our construction to various
aspects of cluster theory which are known to symplectic geometers.

###### Contents

  1. 1 Introduction
  2. 2 Cluster varieties from non-toric blowups
  3. 3 Symplectic geometry background
  4. 4 The mirror construction
  5. 5 Skeleta from nodal trades

##  1\. Introduction

Cluster varieties are algebraic varieties which are glued together from torus
charts using birational maps that are called _cluster transformations_. They
are described by starting with a seed, $\mathfrak{s}$, which we will define
below. Each seed has an associated seed torus, which is one of the torus
charts for the cluster variety. It also contains some combinatorial
information which explains how to _mutate_ seeds. By mutating, one contains a
(often infinite) set of seed tori. Mutation also gives a prescription for
gluing these seed tori together. The result of gluing these tori together is
called a _cluster variety_. Often, one prefers to consider the affinization of
this space, in which case it is this affinization which is referred to as the
cluster variety.

Cluster varieties come in two flavors, $\mathcal{X}$ and $\mathcal{A}$, which
have the same seed data, but use somewhat different birational transformations
to glue seed tori. For each seed $\mathfrak{s}$, there is a natural dual seed
$\mathfrak{s}^{\vee}$. Fock and Goncharov conjectured that the cluster
varieties $\mathcal{X}(\mathfrak{s})$ and $\mathcal{A}(\mathfrak{s}^{\vee})$
satisfied an intricate set of conjectures that they called the _duality
conjectures_. Moreover, they suspected that these duality conjectures were a
consequence of homological mirror symmetry between $\mathcal{X}(\mathfrak{s})$
and $\mathcal{A}(\mathfrak{s}^{\vee})$

In [GHK11, GHK13], Gross, Hacking, and Keel successfully placed the duality
conjectures within the framework of mirror symmetry for log Calabi-Yau
varieties. To start with, they reinterpreted the cluster mutation formula as
the birational map given by an elementary transformation of
$\mathbb{P}^{1}$-bundles. This allowed them to give an algebro-geometric
construction of cluster varieties: begin with a toric variety $\overline{X}$
whose boundary is purely of codimension 1 (hence a disjoint union of tori);
blow up a disjoint union of subtori (determined by the seed data) which are
codimension 1 in the boundary tori; delete the strict transform of the
original toric boundary.

Above dimension 2, the naïve result of performing this blow-up-and-delete
operation described in [GHK13] may produce a variety which is “almost” a
cluster variety, in the sense that the former may differ in codimension 2 from
the traditional Fock-Goncharov definition of cluster variety. (In particular,
they have the same affinization.) In order to distinguish them, we refer to
the former as a truncated cluster variety. These are the varieties whose
mirrors we produce in this paper.

We describe here a mirror to the GHK construction: beginning with the
Liouville sector (symplectic Landau-Ginzburg model) mirror to $\overline{X},$
perform Weinstein handle attachments as indicated by the seed data. Using
recent advances [GPS19, GPSb, GPSa, NS] in the theory of Weinstein Fukaya
categories, we are able to compute the Fukaya categories of the resulting
spaces and match them with categories of coherent sheaves on the truncated
cluster varieties. We can thus express many features of cluster algebra quite
naturally in the language of symplectic geometry. Many of these features have
already been observed in the 2-dimensional case in [STWZ19, STW], subsequently
made explicit in [HK]; we offer a new perspective on the mirror-symmetry
result proved there, as well as a generalization to arbitrary dimensions.

The basic Weinstein handle attachment we perform is a familiar part of two-
dimensional symplectic geometry, in the form of Symington’s “nodal trade”
[Sym03]. The local model for this move, its relation to cluster theory, and
its higher-dimensional analogues have been previously studied [Via16, PT20,
Nad]. However, our work is the first to describe the mirror to a general
truncated cluster variety in terms of a series of these nodal trade
operations.

###  1.1. Mirror construction and plan of the paper

Our construction is most indebted to [STW], which studied cluster structures
inside the Fukaya category of a Weinstein 4-manifold $W$ obtained from a
cotangent bundle $T^{*}\Sigma$ by Weinstein handle attachments. In the case
when $\Sigma=T$ is a 2-torus, we propose to reintepret their construction in
the framework of homological mirror symmetry, treating $T$ as the fiber of an
SYZ fibration: the wrapped Fukaya category $\mathcal{W}(W)$ can be understood
as the category of coherent sheaves $\textup{Coh}(W^{\vee})$ on a mirror
cluster variety.

The best framework for understanding this mirror cluster variety turns out to
be the Gross-Hacking-Keel construction, which we review in Section 2. Their
construction, beginning with a toric variety, performs a certain blow-up-and-
delete operation in the toric divisors. Our observation is that if $W^{\vee}$
is mirror to a Weinstein manifold obtained from the handle attachments
described in [STW], the blow-up-and-delete operations of $W^{\vee}$ are in
bijection with the handle attachments of $W$.

We prove that in arbitrary dimensions, the basic blow-up-and-delete operation
of [GHK13] is mirror to a basic Weinstein disk attachment. A seed
$\mathfrak{s},$ which in [GHK13] is treated as a recipe for producing a
cluster variety $U$, we treat as a recipe for producing a Weinstein manifold
$U^{\vee}.$ After some review of microlocal sheaf theory in Section 3, in
Section 4 we describe this Weinstein manifold and prove our main theorem, a
homological mirror symmetry result for this pair:

######  Theorem.

There is an equivalence of dg categories
$\textup{Coh}(U)\cong\mathcal{W}(U^{\vee})$ between the category of coherent
sheaves on the truncated cluster variety $U$ and the wrapped Fukaya category
of the Weinstein manifold $U^{\vee}.$

We also mention that cluster mutations can be understood in our framework,
referring to [PT20, STW, Nad] for details. Finally, we explain in Section 5
how the Weinstein manifolds we discuss may appear naturally in Kähler geometry
as complements of certain anticanonical divisors in toric varieties, obtained
as particular smoothings of the toric boundary divisor.

###  1.2. Future questions

As mentioned above, we restrict ourselves in this paper to varieties obtained
from the construction of [GHK13], which may differ in codimension 2 from the
cluster varieties which have traditionally arisen in representation theory and
geometry. The mirrors to these latter cluster varieties will be Weinstein
manifolds with skeleta more complicated than those described here. In the
context of our construction in Section 5, such a Weinstein manifold may appear
as the complement, in a toric variety, of a divisor obtained from the toric
boundary divisor by a potentially complicated smoothing, outside the scope of
the smoothings mentioned in this paper. In future work, we will describe these
more complicated Lagrangian skeleta in the special cases of positroid cells
and Richardson varieties, and explain their relation to mirror symmetry.

Those more complicated skeleta are also desirable because they admit
deformations to more singular skeleta which are actually holomorphic
Lagrangian in a certain hyperkähler structure. (In the simplest example of a
skeleton which is a torus with disk glued in, this deformation collapses the
disk to produce a nodal curve.) Such skeleta have close connections with
representation theory and holomorphic symplectic geometry; in the toric
hyperkähler setting, these skeleta have been used to prove mirror-symmetry
results in [MW, GMW].

However, already in the cases we consider, it seems likely that the Lagrangian
skeleta we describe can be used to produce interesting results in symplectic
geometry. For instance, the skeleton of one of our Weinstein manifolds
$U^{\vee}$ encodes the Reeb dynamics on the contact boundary of $U^{\vee}.$ In
[Pas19], an analysis of Reeb dynamics for $U^{\vee}$ 2-dimensional was used in
order to study its quantum cohomology ring and hence the coordinate ring of
the mirror cluster variety; we hope that our symplectic construction will help
uncover new features of the algebraic geometry of mirror cluster varieties.

Finally, we note that in the 2-dimensional case, the recent paper [HK]
considers not only mirror symmetry for the cluster variety $U$, as we do, but
also for its compactification $X=\bar{U},$ which we do not consider in this
paper. However, this case should be within easy reach of our methods, since
the mirror to $X$ (for a particular complex structure on $X$) is a Weinstein
sector, which should be obtainable from our Weinstein manifolds $U^{\vee}$ by
adding a stop at infinity. We expect that translating the 2-dimensional [HK]
result into this language should, when combined with the constructions of this
paper, suggest a mirror symmetry construction for the compactification $X$ in
arbitrary dimensions.

### Notation

The following notation will be used throughout the paper.

  * •

We pick dual lattices $N,M$, and we write $N_{\mathbb{R}},N_{\mathbb{C}}$ for
$N\otimes_{\mathbb{Z}}\mathbb{R},N\otimes_{\mathbb{Z}}\mathbb{C},$ and
similarly for $M$.

  * •

We write $N_{S^{1}},N_{\mathbb{C}^{\times}}$ for the real and complex tori
$N_{\mathbb{R}}/N,N_{\mathbb{C}}/N,$ and similarly for $M.$ Note that
$N_{\mathbb{C}^{\times}}=N_{\mathbb{R}}/N\times N_{\mathbb{R}}.$ In sections
of this paper where we are only concerned with one of $M$ or $N$ and no
ambiguity can result, we will sometimes denote this real or complex torus by
$T$ or $T_{\mathbb{C}},$ respectively.

  * •

We write $\Sigma$ for a stacky fan of rays in $N_{\mathbb{R}}.$ “Stacky” means
that we keep track not only of the rays
$\langle\psi_{1}\rangle,\ldots\langle\psi_{r}\rangle,$ but of not necessarily
primitive generators $\\{d_{i}\psi_{i}\\}_{i=1}^{r},$ where $\psi_{i}$ is
primitive and $d_{i}\in\mathbb{Z}_{>0}.$

  * •

We denote the toric stack with fan $\Sigma$ by $\overline{X}_{\Sigma}.$ It has
toric divisor $\overline{D}$ and dense open torus
$\overline{T}_{\mathbb{C}}:=N_{\mathbb{C}^{\times}},$ which has cocharacter
lattice $N$ and character lattice $M.$

### Categorical conventions

Throughout this paper, we work with stable $\infty$-categories over a field
$k$ of characteristic 0, which we model as dg categories (so that, for
instance, $\textup{Coh}(X)$ always refers to the dg category of coherent
sheaves on $X$). All functors are derived, and the word “limits/colimit” in a
diagram of categories is always used to refer to a homotopy limit/colimit.

### Acknowledgements

The authors are grateful to Denis Auroux, for explaining the construction of
the Kähler potential in Lemma 5.10; Roger Casals and Dmitry Tonkonog, for
explanations about nodal trades; Andrew Hanlon, for discussions about the GHK
construction; Vivek Shende and Harold Williams, for discussions about the
papers [STWZ19, STW]; Harold again, for several helpful comments on a draft of
this paper; and Daniel Pomerleano, for a correction to Section 5 and for the
content of Remark 5.20. BG is supported by an NSF postdoctoral fellowship,
DMS-2001897.

##  2\. Cluster varieties from non-toric blowups

In this section, we review the Gross-Hacking-Keel construction of cluster
varieties, following [HK18] and then, with an eye toward our future symplectic
constructions, we discuss their SYZ geometry.

###  2.1. Review of the GHK construction

The constructions of the Gross-Hacking-Keel program take place within the
framework of log Calabi-Yau geometry.

######  Definition 2.1.

Let $(X,D)$ be a smooth projective variety equipped with a reduced normal
crossings divisor $D$ satisfying $K_{X}+D=0$. In this case we say that $(X,D)$
is a log Calabi-Yau pair and $U=X\setminus D$ is a log Calabi-Yau variety. We
write $\Omega$ for the unique (up to scalar multiples) holomorphic volume form
on $U$ with simple poles on each component of $D.$

######  Remark 2.2.

In this paper, we will actually be interested only in the pair
$(X^{\prime},D^{\prime})$ obtained from $(X,D)$ as above by deleting all
strata in $D$ of codimension at least 2 in $X.$ The resulting pair does not
strictly fall under the above definition, since $X^{\prime}$ is now only
quasiprojective, but we will abuse notation slightly by continuing to refer to
$(X,D)$ as a log Calabi-Yau pair.

######  Example 2.3.

Let $(\overline{X},\overline{D})$ be an $n$-dimensional toric variety with its
toric boundary divisor. Then $(\overline{X},\overline{D})$ is a log Calabi-Yau
pair with $U=X\setminus
D\cong(\mathbb{C}^{\times})^{n},\Omega=\frac{dx_{1}}{x_{1}}\wedge\cdots\wedge\frac{dx_{n}}{x_{n}}.$

As suggested in Remark 2.2, we will only be interested here in considering log
Calabi-Yau pairs where $D$ has no higher-codimension strata. Such a pair in
the toric case can be obtained from a fan with no higher-dimensional cones.

######  Example 2.4.

Let $\Sigma\subset N_{\mathbb{R}}$ be a fan of cones of dimension at most 1.
In other words, $\Sigma$ contains the zero cone and the rays generated by
primitive vectors $\psi_{1},\ldots,\psi_{r}\in N.$ Then
$(\overline{X}_{\Sigma},\overline{D})$ is a log Calabi-Yau pair where
$\overline{D}=\bigsqcup_{i=1}^{r}\overline{D}_{i}$ is a disjoint union of
copies of $(\mathbb{C}^{\times})^{n-1}.$ More precisely, the component
$\overline{D}_{i}$ corresponding to ray $\langle\psi_{i}\rangle$ is isomorphic
to the quotient torus $(N/\psi_{i})_{\mathbb{C}^{\times}}.$ Note that the
character lattice of the torus $\overline{D}_{i}$ is therefore the orthogonal
$\psi_{i}^{\perp}\subset M.$

######  Remark 2.5.

In fact, we will see that in order to produce cluster varieties whose seed
matrix is not skew-symmetric but only skew-symmetrizable, we will have to
allow $\Sigma$ to be a toric stack. For information on toric stacks, we refer
to [BCS05]. In the setting of Example 2.4, this means only that we equip
$\Sigma$ with a choice of not necessarily primitive generator of each ray, or
equivalently that to each primitive generator $\psi_{i}$, we associate a
positive integer $d_{i}.$ The resulting toric stack $\overline{X}_{\Sigma}$
has coarse moduli space the usual toric variety associated to $\Sigma,$ but
$\overline{X}_{\Sigma}$ has a $\mathbb{Z}/d_{i}$ isotropy group along the
divisor $\overline{D}_{i}.$

The discovery of [GHK11, GHK13] is that log Calabi-Yau varieties related to
the previous example in a simple way possess cluster structure.

######  Construction 2.6.

Let $(\overline{X}_{\Sigma},\overline{D})$ be an $n$-dimensional toric pair as
in Example 2.4, and choose a character $\chi_{i}\in\psi_{i}^{\perp}$ for each
component $\overline{D}_{i}$ of $\overline{D}.$ Let $H\subset\overline{D}$ be
the subvariety whose intersection with the component $\overline{D}_{i}$ is
equal to the linear subspace $\\{\chi_{i}=-1\\}$. We write $(X,D)$ for the
blowup of $\overline{X}$ along $H$ together with the strict transform of
$\overline{D}$.

######  Remark 2.7.

The construction of [GHK13] more generally allows components of $H$ to be
defined by an equation of the form $\chi_{i}=\lambda$ for any
$\lambda\in\mathbb{C}^{\times}.$ We prefer to restrict to the case of
$\lambda=-1,$ since this is a paper about exact symplectic geometry, and a
mirror to the variety $X$ obtained for more general $\lambda$ is expected to
be non-exact. (See [HK, Section 2.2] for a more detailed discussion of this
phenomenon in dimension 2.) However, note that in this paper we do not discuss
mirror symmetry for the variety $X$ but only for $U$, which is not generally
sensitive to the value of $\lambda,$ so this issue will not arise explicitly
in this paper.

Observe that if the divisor $\overline{D}$ has a single component, then we can
choose coordinates so that
$\overline{X}=(\mathbb{C}\times\mathbb{C}^{\times})\times(\mathbb{C}^{\times})^{n-2}$
and $H=\\{(0,-1)\\}\times(\mathbb{C}^{\times})^{n-2}$, so that the space $X$
resulting from the GHK construction is

$X=\operatorname{Bl}_{\\{(0,1)\\}}(\mathbb{C}\times\mathbb{C}^{\times})\times(\mathbb{C}^{\times})^{n-2}.$

A recurring theme in this paper is that Construction 2.6 can be reduced to
repeated applications of this basic move, making this the most important (and
also simplest) case to understand. We begin to study it in the following
example.

######  Example 2.8.

Let $\overline{X}=\mathbb{C}\times\mathbb{C}^{\times}$ and $H=\\{(0,-1)\\},$
as above. Then we can describe the blowup
$X=\operatorname{Bl}_{H}\overline{X}$ as embedded in
$\mathbb{C}\times\mathbb{C}^{\times}\times\mathbb{P}^{1}$ as

$X\cong\\{(x,y),[z:w]\mid
xz-(y+1)w=0\\}\subset\mathbb{C}\times\mathbb{C}^{\times}\times\mathbb{P}^{1}.$

The strict transform $D$ of the toric boundary divisor
$\overline{D}=\\{0\\}\times\mathbb{C}^{\times}$ contains the point $[1:0]$ in
the exceptional $\mathbb{P}^{1},$ and hence the variety $U=X\setminus D$ is
completely contained in the affine chart
$\mathbb{C}\times\mathbb{C}^{\times}\times\mathbb{C}_{z}:$

$U\cong\\{(x,y,z)\mid
xz=y+1\\}\subset\mathbb{C}\times\mathbb{C}^{\times}\times\mathbb{C}.$

If desired, we can also eliminate the variable $y$ to describe $U$ as the
complement

$U=\mathbb{C}^{2}\setminus\\{xz=1\\}.$

######  Definition 2.9.

Let $U=X\setminus D$ for a log Calabi-Yau pair $(X,D)$ obtained via
Construction 2.6. In this case we call $U$ a truncated cluster variety, and we
say that the blowup $(X,D)\to(\overline{X},\overline{D})$ is a toric model for
$U$.

######  Remark 2.10.

Note that in case the dimension of $X$ is equal to 2, the data of the choice
of character $\chi_{i}\in\psi_{i}^{\perp}$ in Construction 2.6 is not
necessary; knowledge of the divisors in which the blow-up is to take place is
sufficient data in this case.

Composition of the inclusion $U\to X$ and the blow-up map $X\to\overline{X}$
defines a birational morphism $U\to\overline{X}$. The inverse birational map
is well-defined on the dense torus $\overline{T}\subset\overline{X},$ giving
an open inclusion $\overline{T}\hookrightarrow U$, defining the toric chart on
$U$ corresponding to toric model $(X,D)\to(\overline{X},\overline{D}).$ We
will see that, as expected, other toric charts can be reached by mutation;
first, we recall the traditional defining data for a cluster variety.

######  Definition 2.11.

A seed $\mathfrak{s}$ is the data of

  * •

The lattice $N$;

  * •

A skew bilinear form $\overline{\sigma}\in\wedge^{2}M$;

  * •

A basis $\\{\psi_{1},\ldots,\psi_{r},\psi_{r+1},\ldots,\psi_{n}\\}$ of $N$,
along with a designated subset $\\{\psi_{1},\ldots,\psi_{r}\\}$ of unfrozen
basis vectors;

  * •

Positive integers $d_{i},$ which can be used to modify $\overline{\sigma}$ to
a non-skew-symmetric form $\epsilon,$ defined by
$\epsilon(\psi_{i},\psi_{j})=\overline{\sigma}(\psi_{i},\psi_{j})d_{j}=:\epsilon_{ij}.$
We say that $\epsilon$ is _skew-symmetrizable_.

One of the main insights of [GHK13] is that the seed data of Definition 2.11
can be expressed in terms of the log Calabi-Yau geometry discussed above.

######  Construction 2.12.

Let $\mathfrak{s}$ be a seed as in Definition 2.11.

  * •

The unfrozen vectors $\psi_{1},\ldots,\psi_{r}$ determine a fan of rays
$\Sigma$ in $N,$ which can be upgraded to a stacky fan by specifying
$d_{i}\psi_{i}$ as generators of the rays in $\Sigma.$ This information
determines a toric stack $\overline{X}_{\Sigma}$ with dense torus
$\overline{T}_{\mathbb{C}}=N_{\mathbb{C}^{\times}}$ and boundary divisors
$D_{i}=(N/\psi_{i})_{\mathbb{C}^{\times}}\times B\mathbb{Z}/d_{i}.$

  * •

For $i\leq r,$ The pairing $\chi_{i}:=\overline{\sigma}(\psi_{i},-)$
determines an element $\chi_{i}\in M,$ which by skew-symmetry of
$\overline{\sigma}$ is contained in $\psi_{i}^{\perp}$ and hence can be
treated as a character of the $(n-1)$-torus $D_{i}.$ We write
$H_{i}=\\{\chi_{i}+1=0\\}.$

  * •

The skew bilinear form $\overline{\sigma}\in\wedge^{2}M\simeq
H^{0}(\Omega_{\overline{X}}^{2}(\log\overline{D}))$ determines a holomorphic
2-form on $\overline{X}$ with simple poles along $\overline{D}$. As explained
in [HK18, Lemma 3.4], the definition of $\chi_{i}$ ensures that
$\overline{\sigma}$ extends to a form in $H^{0}(\Omega_{X}^{2}(\log D)).$

Hence we get a log Calabi-Yau pair $(X,D)$ as in Construction 2.6. We will
call the variety $U=X-D$ the truncated cluster variety associated to the seed
$\mathfrak{s}.$

######  Remark 2.13.

We refer to $U$ as a “truncated cluster variety” to distinguish it from the
more traditional notions of cluster variety. Fock and Goncharov originally
defined cluster varieties as unions of cluster charts; these cluster varieties
come in two types, $\mathcal{X}$ and $\mathcal{A}$. Cluster
$\mathcal{A}$-varieties are quasi-affine; from our point of view, the correct
variety to consider is the affine closure of the cluster
$\mathcal{A}$-variety.

The variety $U$ constructed above is, up to codimension 2, a cluster
$\mathcal{A}$-variety, and it has the same affine closure. It is for this
reason that we call a “truncated cluster variety.” Throughout this paper, we
will only ever work with the variety $U$ as constructed above, so we will
often drop the adjective “truncated.”

The simplest case of this phenomenon is treated in the following example.

######  Example 2.14.

Let $\mathfrak{s}$ be the seed data corresponding to the variety
$U=\mathbb{C}^{2}\setminus\\{xz=1\\}$ from Example 2.8, and write
$\widetilde{\mathfrak{s}}$ for the product $\mathfrak{s}\times\mathfrak{s}$ of
two copies of this seed data. The cluster variety associated to
$\widetilde{\mathfrak{s}}$ ought to be

$\widetilde{U}:=U\times
U=\mathbb{C}^{4}\setminus\\{(x_{1}z_{1}-1)(x_{2}z_{2}-1)=0\\},$

but the blow-up-and-delete construction described above produces instead the
space $\widetilde{U}\setminus\\{(0,0,0,0)\\}.$

Finally, there is also a geometric interpretation of cluster mutation. Recall
how cluster mutation alters seed data:

######  Definition 2.15.

Starting from a seed $\mathfrak{s}$ as in Definition 2.11 and a choice of
unfrozen basis vector $\psi_{k}$, the mutation at $\psi_{i}$ is a new seed
$\mu_{i}\mathfrak{s},$ produced by changing the basis $\\{\psi_{i}\\}$ to the
basis $\\{\psi_{i}^{\prime}\\}$ determined by the formula

$\psi_{i}^{\prime}=\begin{cases}\psi_{i}+[\epsilon_{ik}]_{+}\psi_{k}&\text{for
$i\neq k$}\\\ -\psi_{k}&\text{for $i=k$},\end{cases}$

where we define $[r]_{+}=\max(0,r).$

The geometry underlying mutations is described in [GHK13, Section 3.1], which
we can summarize as follows:

######  Proposition 2.16 ([GHK13]).

Let $(X,D)\to(\overline{X},\overline{D})$ be a log Calabi-Yau variety with
toric model induced from a seed $\mathfrak{s}$ as in Construction 2.12. Given
a choice of an unfrozen vector $\psi_{i}$ in the seed, an elementary
transformation of $\mathbb{P}^{1}$-bundles induces a new toric model
$(X^{\prime},D^{\prime})\to(\overline{X}^{\prime},\overline{D}^{\prime}),$
which corresponds under Construction 2.12 to the mutated seed
$\mu_{i}\mathfrak{s}.$ The induced map of tori
$\overline{T}_{\mathbb{C}}\to\overline{T}_{\mathbb{C}}^{\prime}$ is the
corresponding cluster transformation.

The local model for this elementary transformation of $\mathbb{P}^{1}$-bundles
is described in the following example.

######  Example 2.17.

Let

$\overline{X}=\mathbb{P}^{1}_{[z_{0}:z_{1}]}\times\mathbb{C}^{\times}_{w}\supset\overline{D}=\\{z_{0}z_{1}=0\\}=(\\{0\\}\cup\\{\infty\\})\times\mathbb{C}^{\times},$

and let $H=\\{z_{0}=0,w+1=0\\},$ so that $(\overline{X},\overline{D})$ is a
toric model for the log Calabi-Yau pair
$(X=\operatorname{Bl}_{H}\overline{X},D=\widetilde{\overline{D}})$ discussed
in Example 2.8. Let $\widetilde{H}$ be the strict transform of the line
$\\{w+1=0\\}\subset\overline{X}$. Then the blowdown of $\widetilde{H}$ can be
used to define a new toric model

$f:(X,D)\to(\overline{X}^{\prime},\overline{D}^{\prime})\cong\mathbb{P}^{1}\times\mathbb{C}^{\times},$

where now the new blowup locus $H^{\prime}\subset\overline{D}^{\prime}$ lives
in $\\{\infty\\}\times\mathbb{C}^{\times}$ rather than
$\\{0\\}\times\mathbb{C}^{\times}.$ An SYZ image of this mutatation is
illustrated below in Figure 3.

###  2.2. SYZ bases

In many cases, the symplectic mirror to $U$ which we will construct has
traditionally been described (at least in dimension 2) by means of the toric
base diagrams from [Sym03], which can be used to express Lagrangian torus
fibrations with singularities. In fact, the resulting torus fibrations will be
SYZ fibrations for our mirror construction. We will not use this fact in an
essential way, but for completeness of the exposition we briefly indicate here
the corresponding SYZ fibration on the truncated cluster varieties.

Our construction on both sides of mirror symmetry begins with the usual
Lagrangian torus fibration on $(\mathbb{C}^{\times})^{n}$ and then asks how
the fibration is affected by a modification of this space. In the present
case, this means understanding how the torus fibration on the dense torus
$\overline{T}_{\mathbb{C}}=\overline{X}\setminus\overline{D}$ is affected by
the GHK blow-up-and-delete construction. As this construction is local to each
divisor, we only need to understand it in a single local model, treated in the
following example.

######  Example 2.18 ([HK18, Example 4.4]).

Consider again the basic local model described in Example 2.8, where the log
Calabi-Yau pair $(X,D)$ is obtained from toric model
$(\overline{X},\overline{D})=(\mathbb{C}\times\mathbb{C}^{\times},\\{0\\}\times\mathbb{C}^{\times})$
by blowing up at center $H=\\{(0,-1)\\}.$

From the perspective of the moment-map base $\mathbb{R}_{\geq
0}\times\mathbb{R}$ of $\overline{X},$ the blowup at $H$ has the effect of
deleting a triangle with base on the $y$-axis centered at $(0,0)$ and
identifying two of its edges, as illustrated in Figure 1. The manifold
$U=X\setminus D$ then admits a fibration over the interior of this base, with
generic fiber a torus and nodal (focus-focus) singularity above the vertex of
the triangle. Away from the singular fiber, the character lattices of the
torus fibers equip the base with the structure of an integral affine manifold,
with nontrivial monodromy around the singular fiber given by a Dehn twist in
the vanishing cycle. After taking a branch cut (necessary because of the
monodromy of the integral affine structure), we can embed this integral affine
manifold in $\mathbb{R}^{2}.$

Figure 1. The process of producing the base for a torus fibration with
singularity on the blowup
$\operatorname{Bl}_{(0,1)}(\mathbb{C}\times\mathbb{C}^{\times}),$ by starting
from the moment base of the original toric variety, then deleting a triangle
and then identifying its edges. The dashed line indicates a branch cut in the
integral affine structure.

This example makes clear the relation between the SYZ base of a cluster
surface and the fan $\Sigma$ in a toric model, since each blowup in
Construction 2.6 has a local model as in Example 1, and these models only
affect the SYZ base locally near the components of $H$, which do not interact.
We learned the following construction, and especially the diagram in Figure 2,
from Andrew Hanlon:

######  Construction 2.19.

Let $\Sigma$ be a fan of rays in $\mathbb{R}^{2}.$ The SYZ base $B$ associated
to $\Sigma$ is the integral affine manifold which topologically is
$\mathbb{R}^{2},$ but with singularities in the integral affine structure as
follows: for each ray $v=\begin{pmatrix}\psi_{1}\\\ \psi_{2}\end{pmatrix}$ in
$\Sigma,$ there is a singularity at a point $p_{v}$ on $v$, with monodromy
matrix $\begin{pmatrix}1+\psi_{1}\psi_{2}&-\psi_{1}^{2}\\\
\psi_{2}^{2}&1-\psi_{1}\psi_{2}\end{pmatrix}.$ After taking a branch cut at
each $p_{v}$ pointing in direction $v$, $B$ can be embedded in
$\mathbb{R}^{2}$ in the obvious way, as illustrated in Figure 2.

Figure 2. The fan $\Sigma$ for $\mathbb{C}^{2}\setminus\\{(0,0)\\}$
superimposed on the SYZ base for the $A_{2}$ cluster variety, where the dashed
lines emanating from singularities represent the branch cuts.

######  Proposition 2.20.

Let $U$ be the cluster surface with toric model
$(\overline{X}_{\Sigma},\overline{D}).$ Then the integral affine structure on
the base $B$ constructed in 2 is induced by a Lagrangian torus fibration $U\to
B$ with focus-focus singularities above the singularities of $B.$

###### Proof.

Let $\overline{\Delta}$ be the moment polytope of $\overline{X}_{\Sigma},$ so
that we can begin with the usual moment fibration
$\overline{X}_{\Sigma}\to\overline{\Delta}.$ We would like to modify this
fibration as we perform the GHK blow-up-and-delete construction. But from the
above discussion, we see that such modifications can be performed locally on
the base $\overline{\Delta},$ and that each such modification is given by the
local model of Example 1, modified by an integral affine-linear change of
coordinates $A$ in $SL(2,\mathbb{Z}).$ If $\begin{pmatrix}1\\\ 0\end{pmatrix}$
is an eigenvector for the monodromy matrix of a singularity, and the
transformation $A$ takes it to $\begin{pmatrix}\psi_{1}\\\
\psi_{2}\end{pmatrix},$ then $A$ must conjugate the monodromy matrix
$\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$ to
$\begin{pmatrix}1+\psi_{1}\psi_{2}&-\psi_{1}^{2}\\\
\psi_{2}^{2}&1-\psi_{1}\psi_{2}\end{pmatrix}.$ ∎

######  Remark 2.21.

An analogue of the above proposition holds in higher dimensions, where now we
replace the local model of Example 1 by its product with
$(\mathbb{C}^{\times})^{n-2}.$ As a result, the corresponding SYZ base will no
longer have isolated singularities, but rather will have affine-linear
subspaces of singularities (corresponding to the product of $\mathbb{R}^{n-2}$
with the singularities from the 2-dimensional case). We do not go into detail
about this case, since it is less useful for drawing pictures than the
2-dimensional case is, but we revisit it on the mirror side in Section 5. See
in particular Figure 11 for an illustration of the local model in dimension
$n=3.$

One simple instance when the integral affine geometry described above can be
helpful is in picturing the elementary transformation underlying the mutation
construction of Proposition 2.16.

######  Example 2.22.

Consider again the local model for an elementary transformation of
$\mathbb{P}^{1}$-bundles described in Example 2.17, where $X$ is obtained from
$\overline{X},\overline{X}^{\prime}=\mathbb{P}^{1}\times\mathbb{C}^{\times}$
by non-toric blowup at a single point in the toric boundary divisor
$\\{0\\}\times\mathbb{C}^{\times}$ (respectively
$\\{\infty\\}\times\mathbb{C}^{\times})$.

Figure 3. SYZ bases for
$\operatorname{Bl}_{(0,-1)}\mathbb{P}^{1}\times\mathbb{C}^{\times}$ and
$\operatorname{Bl}_{(\infty,-1)}\mathbb{P}^{1}\times\mathbb{C}^{\times},$
respectively.

Two different SYZ bases for $X$ are depicted in Figure 3, depending on a
choice of these two toric models. After deleting the divisor $D$ from $X$, the
resulting SYZ bases for $U$ will be the same, differing only in a choice of
branch cut used to embed them in $\mathbb{R}^{2}.$

##  3\. Symplectic geometry background

###  3.1. Liouville sectors

We begin by recalling some basic facts about Weinstein manifolds and Liouville
sectors from [GPS19, Eli].

######  Definition 3.1.

A Liouville domain is a compact symplectic manifold $(X,\omega)$ with boundary
together with a choice of primitive $\omega=d\lambda$ such that
$\lambda|_{\partial X}$ is a contact form, or equivalently such that the
Liouville vector field $V=\omega^{-1}(\lambda)$ is outwardly transversal to
the boundary. A Liouville domain $X$ can be completed to a Liouville manifold
$\hat{X}$ by attaching a cylindrical end:

$\hat{X}=X\cup(\partial X\times[0,\infty)),$

where we extend $\lambda$ to $\hat{X}$ as $e^{s}(\lambda|_{\partial X})$ at
the end.

As Liouville domains and Liouville manifolds can be obtained from each other,
we will not be careful about distinguishing them. Features of the boundary
$\partial X$ (for instance, Weinstein hypersurfaces) can be equivalently
described in terms of the ideal contact boundary of $\hat{X}.$

######  Definition 3.2.

The Liouville domain $(X,\omega,\lambda)$ is in addition Weinstein if it can
be equipped with an exhausting Morse-Bott function $f:X\to\mathbb{R}$ for
which the Liouville vector field $V$ is gradient-like.

The Weinstein condition exists in order to rule out any pathological behavior
of the Liouville vector field. All of the Liouville manifolds and sectors we
consider in this paper will be Weinstein, and we will usually suppress the
function $f$ in the notation. However, note that if $X$ is a Stein manifold
with Morse-Bott Kähler potential $f$, then $f$ underlies a Weinstein structure
with $\lambda=d^{c}f$.

Weinstein manifolds are so named in honor of the discovery, in [Wei91], that
they are a class of symplectic manifolds which can be glued together from
simpler pieces. Today, we understand this gluing data, for a Weinstein
manifold $X,$ as specifying a cover of $X$ by Liouville sectors. These are
Liouville manifolds with boundary which can be obtained from Liouville domains
by completing along only a subset of the boundary.

######  Example 3.3.

The manifold $T^{*}S^{1}=S^{1}\times\mathbb{R}$ is a Liouville manifold
obtained as a completion of the Liouville domain $S^{1}\times[-1,1].$
Specifying a finite number of points on the boundary of this Liouville domain
determines a Liouville sector, by completing the boundary away from
neighborhoods of those points.

We will refer to [GPS19, Definition 1.1] for the technical definition of a
Liouville sector, and we will content ourselves here with the simpler but
ultimately equivalent notion from [Eli] of a Weinstein pair.

######  Definition 3.4.

A Weinstein pair is the data of a Weinstein domain $(X,\omega,\lambda)$
together with a real hypersurface $\Sigma\subset\partial X$ such that
$(\Sigma,d\lambda|_{\Sigma},\lambda|_{\Sigma})$ is itself a Weinstein
manifold.

A Weinstein pair $(X,\Sigma)$ gives rise to a Liouville sector in the sense of
[GPS19] by completing $X$ away from a standard neighborhood of $\Sigma.$ Thus
$\Sigma$ becomes the boundary of the resulting Liouville sector, along which
we can perform gluings:

######  Construction 3.5.

Let $(X_{1},\Sigma_{1})$ and $(X_{2},\Sigma_{2})$ be two Weinstein pairs,
together with an isomorphism of Weinstein manifolds
$\Sigma_{1}\cong\Sigma_{2}.$ Then the Weinstein manifolds $X_{1}$ and $X_{2}$
can be glued together along a neighborhood of $\Sigma_{i}$ to produce a new
Weinstein manifold $X_{1}\cup_{\Sigma}X_{2}.$ If the boundary $\Sigma_{i}$ has
several components, we can glue along one of them, and the resuling gluing
will still be a Weinstein pair / Liouville sector.

######  Remark 3.6.

Higher-codimension gluings are also possible, using the theory of Liouville
sectors with (sectorial) corners from [GPSb], but will not be necessary in
this paper.

All of the gluing data describing a Weinstein manifold $X$ can be encapsulated
in a singular Lagrangian $\mathbb{L}\subset X,$ the skeleton of $X.$

######  Definition 3.7.

For $X$ a Weinstein manifold, the skeleton of $X$ is the stable set

$\mathbb{L}:=\\{x\in X\mid\varprojlim_{t\to\infty}\phi_{t}(x)\in X\\}$

consisting of all points $x\in X$ whose image under the flow $\phi_{t}$ of the
Liouville vector field eventually converges.

For $X$ a Liouville sector, we allow $\mathbb{L}$ to contain all those points
which flow to the boundary of $X$. In particular, for $(X,\Sigma)$ a Weinstein
pair, the skeleton $\mathbb{L}_{(X,\Sigma)}$ of the associated Liouville
sector is the union

$\mathbb{L}_{(X,\Sigma)}=\mathbb{L}_{X}\cup\operatorname{Cone}(\Lambda),$

of the skeleton of $X$ with the cone, under the Liouville flow, of the
skeleton $\Lambda$ of $\Sigma.$

Hence the skeleton of a Weinstein gluing $X_{1}\cup_{\Sigma}X_{2}$ is the
gluing of the skeleta of the corresponding sectors along their glued boundary:

$\mathbb{L}_{X_{1}\cup_{\Sigma}X_{2}}=\mathbb{L}_{X_{1}}\cup_{\mathbb{L}_{\Sigma}}\mathbb{L}_{X_{2}}.$

The main source for Liouville sectors in mirror symmetry is via Landau-
Ginzburg models.

######  Example 3.8.

Let $X$ be a complex affine variety, and $\pi:X\to\mathbb{C}$ a global
holomorphic function. We can treat a general fiber $\pi^{-1}(1)$ as a
Weinstein hypersurface inside the contact boundary $\partial^{\infty}X$ and
hence $(X,\pi^{-1}(1))$ as a Weinstein pair. We will often denote the
associated Liouville sector by $(X,\pi).$

###  3.2. Fukaya categories and descent

In [GPS19], a wrapped Fukaya category $\mathcal{W}(X)$ is associated to any
Liouville sector $X$. In the case where $X$ is a Liouville manifold, this is
the usual wrapped Fukaya category of $X$, and in the case where $X$ is a
Landau-Ginzburg sector $(X,\pi),$ this is the Fukaya-Seidel-type category
associated to the fibration $\pi.$ We refer to [GPS19, GPSb, GPSa] for details
of this construction, and we summarize here its most important properties.

######  Proposition 3.9 ([GPS19]).

The wrapped Fukaya category is covariantly functorial for inclusions of
Liouville sectors: a Liouville subsector $i:X\subset Y$ induces a functor
$i:\mathcal{W}(X)\to\mathcal{W}(Y).$

The central result of the series [GPS19, GPSb, GPSa] is that not only do
Liouville sectors glue along their skeleta at the boundary, but their Fukaya
categories glue in precisely the same way:

######  Theorem 3.10 ([GPSb]).

The wrapped Fukaya category satisfies descent for Weinstein sectorial covers:
Let $U_{1},\ldots,U_{n}\subset X$ be Liouville subsectors of $X$ which cover
$X$ whose intersections $U_{I}:=\bigcap_{i\in I}U_{i},$ for $I\subset[n],$ are
also Liouville sectors, and let $\mathcal{P}$ be the poset of inclusions of
the subsectors $U_{I}.$ Then the natural map

$\varinjlim_{I\in\mathcal{P}}\mathcal{W}(U_{I})\to\mathcal{W}(X)$

from the homotopy colimit of the wrapped Fukaya categories of the $U_{I}$ to
the wrapped Fukaya category of $X$ is an equivalence of dg categories.

In other words, if we can present a Weinstein manifold $X$ as a gluing of
various Liouville sectors $U_{i}$ whose Fukaya categories (and corestriction
functors) we understand, then we can compute the Fukaya category of $X$
itself. One trick which is occasionally helpful in such computations is the
following (described for instance at [Gai, Lemma 1.3.3]): if the functors in a
colimit diagram are all continuous, and the categories are all cocomplete,
then the colimit is equivalent to the limit of the opposite diagram obtained
by passing from all functors to their right adjoints.

######  Corollary 3.11.

Let $U_{i}$ and $X$ be as in Theorem 3.10, and write $\mathcal{W}^{\infty}$
for the ind-completion of the wrapped Fukaya category. Then the natural map

$\mathcal{W}^{\infty}(X)\to\varprojlim_{I\in\mathcal{P}^{op}}\mathcal{W}^{\infty}(U_{I})$

is an equivalence. The wrapped Fukaya category $\mathcal{W}(X)$ can therefore
be recovered as the category of compact objects inside the homotopy limit
$\varprojlim_{I\in\mathcal{P}^{op}}\mathcal{W}^{\infty}(U_{I}).$

This is particularly useful in case each Fukaya category $\mathcal{W}(U_{I})$
is equivalent to the category $\textup{Coh}(U_{I}^{\vee})$ of coherent sheaves
on a smooth algebraic variety $U_{I}^{\vee},$ in which case the ind-completed
category
$\mathcal{W}^{\infty}(U_{I})\cong\textup{Ind}\textup{Coh}(U_{I}^{\vee})$ will
be equivalent to the category $\textup{QCoh}(U_{I})$ of quasicoherent sheaves
on $U_{I}.$

The computations of the Fukaya categories $\mathcal{W}(U_{I})$ of individual
subsectors can be accomplished in the case when the subsectors $U_{I}$ are
cotangent bundles (with sectorial structure); in this case, the Fukaya
category can be computed as a category of constructible sheaves on the base,
or, more invariantly, microlocal sheaves along the Lagrangian skeleton:

######  Theorem 3.12 ([GPSa, Theorem 1.1]).

Let $(T^{*}M,\Sigma)$ be a Liouville sector, and $\Lambda\subset S^{*}M$ the
skeleton of $\Sigma$ inside the contact boundary of $T^{*}M$. Write
$\mathit{Sh}^{\infty}_{-\Lambda}(M)$ for the cocomplete dg category of
(possibly infinite-dimensional) sheaves on $M$ whose singular support away
from the zero section is contained in $-\Lambda$. Then there is an equivalence
of categories

$\mathit{Sh}^{\infty}_{-\Lambda}(M)\cong\mathcal{W}^{\infty}(T^{*}M,\Sigma)$

between this category of sheaves and the Fukaya category of the Liouville
sector associated to $(T^{*}M,\Sigma).$

If we write $\mathit{Sh}^{w}_{-\Lambda}(M)$ (for “wrapped constructible
sheaves”) for the category of compact objects in
$\mathit{Sh}^{\infty}_{-\Lambda}(M),$ then we obtain an equivalence

$\mathit{Sh}^{w}_{-\Lambda}(M)\cong\mathcal{W}(T^{*}M,\Sigma).$

######  Remark 3.13.

Note that the category $\mathit{Sh}^{w}_{-\Lambda}(M)$ is not in general
equivalent to the usual category of constructible sheaves on $M$ with singular
support in $-\Lambda$, since it includes some objects whose stalks are not
finite-dimensional. For instance, if $M=S^{1}$ and $\Lambda$ is empty, then
$\mathit{Sh}^{w}_{-\Lambda}(M)$ includes the universal local system
$k[x^{\pm}]$ on $S^{1}.$

######  Remark 3.14.

The minus sign in the theorem, denoting the anti-symplectic involution
$(-):T^{*}M\to T^{*}M$ which acts on the fibers as scaling by $(-1),$ appears
by convention in order to fix the fact that
$\mathit{Sh}_{\Lambda}^{\infty}(M)$ is actually isomorphic not to the Fukaya
category $\mathcal{W}^{\infty}(T^{*}M,\Sigma)$ but to its opposite category;
as the map $(-)$ is anti-symplectic, it acts at the level of categories as an
anti-involution, cancelling out this opposition.

We will not need to perform any serious constructible-sheaf calculations in
this paper: we rely on the computation in [Kuw20], summarized in Theorem 3.23
below. The difficulties for us will consist only in gluing these calculations
together.

######  Remark 3.15.

In case a Weinstein manifold $X$ has a cover by sectors of the form
$(T^{*}M,\Sigma),$ the above theorems show that there exists a cosheaf of
categories on the skeleton $\mathbb{L}_{X}$ of $X$, locally isomorphic to
$\mathit{Sh}^{w}_{-\Lambda}(M),$ whose homotopy colimit recovers the wrapped
Fukaya category of $X$. We denote this cosheaf by $\mu\mathit{Sh}^{w}$, and
call it the cosheaf of wrapped microlocal sheaves along $\mathbb{L}_{X}.$ In
fact, this cosheaf has now beeen constructed in [NS] on the skeleton of a
general Weinstein manifold.

For our discussion of mutations in Section 4.3 below, we will find it helpful
to note one other functoriality of the wrapped Fukaya category. In addition to
its covariant functoriality under embeddings of Liouville sectors, the wrapped
Fukaya category admits a contravariant functoriality under embeddings of
Liouville domains, which is expected to agree with the Viterbo functoriality
of [AS10].

######  Proposition 3.16 ([GPSb, Section 8.3]).

Let $Y$ be a Liouville sector, and $X\subset Y$ a Liouville subdomain. Then
there is a restriction functor $\mathcal{W}(Y)\to\mathcal{W}(X).$

###  3.3. Toric mirror symmetry

We now specialize to the class of Liouville sectors most relevant to our
discussion: those arising from Landau-Ginzburg mirrors to toric varieties.

######  Definition 3.17.

Let $\overline{X}_{\Sigma}$ be a toric variety with fan $\Sigma$ of cones in
$N_{\mathbb{R}}$. Let $\Delta\subset N_{\mathbb{R}}$ be the convex hull of the
generators of the rays in $\Sigma,$ and let
$W_{\Sigma}:M_{\mathbb{C}^{\times}}\to\mathbb{C}$ be a Laurent polynomial with
Newton polytope $\Delta.$ The Landau-Ginzburg model
$(M_{\mathbb{C}^{\times}},W)$ is a Hori-Vafa mirror to
$\overline{X}_{\Sigma}.$

The skeleton of the Liouville sector $(M_{\mathbb{C}^{\times}},W)$ (though not
originally recognized as such) appeared implicitly in work of Bondal [Bon06]
and was studied extensively by Fang-Liu-Treumann-Zaslow [FLTZ12a, FLTZ11,
FLTZ12b].

######  Definition 3.18.

Let $\Sigma$ be a fan of cones in $N_{\mathbb{R}},$ and note that we can treat
$N_{\mathbb{R}}$ as the cotangent fiber of the cotangent bundle
$T^{*}M_{S^{1}}=M_{\mathbb{R}}/M\times N_{\mathbb{R}}.$ The (negative) Bondal
Lagrangian is a conic Lagrangian $\mathbb{L}_{\Sigma}\subset T^{*}M_{S^{1}}$
defined by

(3.19)
$\mathbb{L}_{\Sigma}=\bigcup_{\sigma\in\Sigma}\overline{\sigma^{\perp}}\times\sigma\subset
M_{\mathbb{R}}/M\times N_{\mathbb{R}},$

where we write $\overline{\sigma^{\perp}}$ for the image of the linear
subspace $\sigma^{\perp}\subset M_{\mathbb{R}}$ in the quotient
$M_{\mathbb{R}}/M.$

######  Example 3.20.

Let $\dim n=2$ and $\Sigma$ be the fan whose only nonzero cone is the ray
$\mathbb{R}\langle e_{1}\rangle.$ Then the Lagrangian $\mathbb{L}$ is a torus
with a cylinder attached, as in Figure 4.

Figure 4. The skeleton $\mathbb{L}_{\Sigma}$ for $\Sigma$ the fan of
$\mathbb{C}\times\mathbb{C}^{\times},$ given as the union of a torus (in
black) with a cylinder (in red).

The torus corresponds in the decomposition of Equation (3.19) to the zero cone
in $\Sigma,$ while the cylinder corresponds to the cone $\mathbb{R}\langle
e_{1}\rangle.$

As mentioned in Remark 2.5, in the case of a cluster seed whose inner-product
$\overline{\sigma}$ is skew-symmetric but whose associated form $\epsilon$ is
only skew-symmetrizable (i.e., when not all the integers $d_{i}$ are equal to
1), we need to scale some of the generators of rays in $\Sigma,$ making
$\overline{X}_{\Sigma}$ into a toric stack. In this case, we need to make a
slight modification of the above definition. For $\sigma$ a cone in $\Sigma$,
write $\sigma_{\mathbb{Z}}:=\sigma\cap N$ for the sublattice defined by
$\sigma$, and note that the tori $\overline{\sigma}^{\perp}$ are the
Pontrjagin duals of the quotient lattices $N/\sigma_{\mathbb{Z}}$:

(3.21)
$\overline{\sigma}^{\perp}=\textup{Hom}(N/\sigma_{\mathbb{Z}},\mathbb{R}/\mathbb{Z}).$

Suppose now that $\Sigma$ is a stacky fan, so that the rays in $\Sigma$ are
equipped with not necessarily primitive generators. If $\sigma$ is a cone on
rays equipped with not necessarily primitive generators
$\beta_{1},\ldots,\beta_{k},$ write $\sigma_{\mathbb{Z}}$ for the sublattice
$\langle\beta_{1},\ldots,\beta_{k}\rangle\subset N.$ In analogy with Equation
(3.21), we define a not necessarily connected torus $G_{\sigma}$ as the
Pontrjagin dual

$G_{\sigma}=\textup{Hom}(N/\sigma_{\mathbb{Z}},\mathbb{R}/\mathbb{Z}).$

The group $\pi_{0}(G_{\sigma})$ of connected components of $G_{\sigma}$ takes
into account the isotropy along the toric stratum corresponding to cone
$\sigma.$ The definition of the Lagrangian in the stacky case proceeds
precisely in analogy to Definition 3.18:

######  Definition 3.22.

Let $\Sigma$ be a stacky fan of cones $\sigma,$ and define the groups
$G_{\sigma}$ as above. Then the stacky Bondal Lagrangian is defined as

$\mathbb{L}_{\Sigma}=\bigcup_{\sigma\in\Sigma}G_{\sigma}\times\sigma\subset
M_{\mathbb{R}}/M\times N_{\mathbb{R}}.$

The motivation for studying the Lagrangian $\mathbb{L}_{\Sigma}$ was the
following result, studied by Bondal and FLTZ and eventually proved in full
generality by Kuwagaki (later reproved in many cases in [Zho19]):

######  Theorem 3.23 ([Kuw20]).

There is an equivalence of dg categories
$\mathit{Sh}^{w}_{-\mathbb{L}_{\Sigma}}(M_{S^{1}})\cong\textup{Coh}(\overline{X}_{\Sigma})$
between the category of wrapped constructible sheaves on the torus $M_{S^{1}}$
with singular support in $-\mathbb{L}_{\Sigma}$ and the category of coherent
sheaves on the toric stack $\overline{X}_{\Sigma}.$

The theory of Liouville sectors has made it possible to relate the Lagrangian
$\mathbb{L}_{\Sigma}$ to the Hori-Vafa Landau-Ginzburg model and hence to
reframe the above in more traditional mirror-symmetric language.

######  Theorem 3.24 ([GS, Zho20]).

Let $\Delta\subset N_{\mathbb{R}}$ be a polytope containing 0, and let
$\Sigma$ be the fan of cones on the faces of $\Delta.$ As above, let $W$ be a
Laurent polynomial $M_{\mathbb{C}^{\times}}\to\mathbb{C}$ with Newton polytope
$\Delta.$ Then $\mathbb{L}_{\Sigma}$ is the Lagrangian skeleton of the
Liouville sector associated to $(M_{\mathbb{C}^{\times}},W).$

######  Corollary 3.25.

The dg category $\mathit{Sh}^{w}_{-\mathbb{L}_{\Sigma}}(M_{S^{1}})$ is
equivalent to the wrapped Fukaya category
$\mathcal{W}(M_{\mathbb{C}^{\times}},W)$ of the Liouville sector associated to
the Hori-Vafa Landau-Ginzburg model $(M_{\mathbb{C}^{\times}},W).$

However, Theorem 3.24 as stated does not apply to our situation, since the
hypothesis of the theorem requires that $\Sigma$ contain higher-dimensional
cones, corresponding to the positive-dimensional faces of $\Delta,$ whereas we
are interested in fans which contain only one-dimensional cones.

Nevertheless, in the case of a purely 1-dimensional fan $\Sigma,$ the
Lagrangian $\mathbb{L}_{\Sigma}$ is still the skeleton of a perfectly good
Liouville sector – indeed, its boundary at infinity is a disjoint union of
$(n-1)$-tori, corresponding to the structure of a Weinstein pair of the form
$((\mathbb{C}^{\times})^{n},\bigsqcup_{i=1}^{r}(\mathbb{C}^{\times})^{n-1})$ –
and Theorem 3.23 remains true in this case, so that we must continue to
understand this sector as the mirror to the (no longer complete) toric variety
$\overline{X}_{\Sigma}.$ We are therefore justified in the following notation:

######  Definition 3.26.

We write $\overline{X}_{\Sigma}^{\vee}$ for the Liouville sector with
Lagrangian skeleton $\mathbb{L}_{\Sigma}.$

##  4\. The mirror construction

###  4.1. Weinstein handle attachment

The manifolds in our mirror construction will be obtained from the cotangent
bundle of a torus by a simple Weinstein handle attachment. We begin with the
two-dimensional case, studied extensively in [STW].

######  Construction 4.1.

Let $T=T^{2}$ be a 2-torus and $\psi_{1},\ldots,\psi_{r}:T\to S^{1}$
characters of $T,$ defining codimension-1 subtori $S_{1},\ldots,S_{r}.$ A
choice of coorientations of the subtori $S_{i}$ determines lifts of the
$S_{i}$ to Legendrian circles $\mathcal{L}_{i}$ in the contact boundary
$S^{*}T$ of $T^{*}T.$ We write $W$ for the Weinstein manifold obtained by
attaching one Weinstein handle along each Legendrian $\mathcal{L}_{i}.$

######  Example 4.2.

Let $r=1$ and let $\psi=\psi_{1}$ be the character given in coordinates by
$(\theta_{1},\theta_{2})\mapsto\theta_{1}.$ Then the Weinstein manifold $W$
has skeleton given by the union of the torus $T$ and a Lagrangian disk glued
to $T$ along $S=\\{\psi=1\\},$ as depicted two different ways in Figure 5.

Figure 5. Two views of the skeleton obtained from a single Weinstein disk
attachment, either as a torus with cylinder (the cone on Legendrian
$\mathcal{L}$) attached and then capped off, or as a torus with a disk glued
in directly.

The Weinstein manifold $W$ constructed above admits a simple Liouville-
sectorial cover. Let $M=\mathbb{Z}^{2},T=M_{S^{1}},$ and note that each
character $\psi_{i}$ is a vector in the dual vector space $N_{\mathbb{R}}$;
hence we can think of the collection of characters $\psi_{i}$ as a purely one-
dimensional fan $\Sigma$ in $N_{\mathbb{R}}$. Now the following is an
immediate consequence of Definition 3.26:

######  Lemma 4.3.

The Liouville sector determined by the pair
$(T^{*}M,\bigsqcup\mathcal{L}_{i})$ is the Liouville sector
$\overline{X}_{\Sigma}^{\vee}$ described in Definition 3.26, the mirror to the
(incomplete) toric variety $\overline{X}_{\Sigma}.$

The other piece in the Liouville-sectorial cover we consider will be a
disjoint union of copies of $T^{*}D,$ for $D$ a disk; this is the Weinstein
manifold whose skeleton is the disjoint union of copies of the disk $D.$ Note
that for $A\subset D$ an annulus, the Liouville sector $T^{*}D$ admits
$T^{*}A$ as a subsector.

######  Proposition 4.4.

Let $W$ and $\Sigma$ as above. Then $W$ admits a cover by the Liouville
sectors $\overline{X}_{\Sigma}^{\vee}$ and $\bigsqcup_{i=1}^{r}T^{*}D,$ which
intersect in $\bigsqcup_{i=1}^{r}T^{*}A.$

###### Proof.

This sectorial cover is nothing more than the expression of $W$ as a Weinstein
disk attachment. See Figure 6 for an illustration in the case $r=2.$ ∎

Figure 6. An image of the Liouville-sectorial cover of the Weinstein manifold
obtained by gluing in disks along circles in homology classes $(1,0)$ and
$(0,1)$. The three pieces in the cover are the torus with cylinders attached;
a pair of disks; and their intersection, a pair of cylinders.

So far, we have only recalled the constructions already appearing in [STW]. We
now generalize to higher dimensions by taking a product of the local model for
the above disk gluing with $T^{*}T^{n-2}=(\mathbb{C}^{\times})^{n-2}.$

######  Example 4.5.

Let $W$ be the product of $(\mathbb{C}^{\times})^{n-2}$ with the Weinstein
manifold studied in Example 4.2. Then $W$ can be obtained as follows: starting
with an $n$-torus $T=T^{n},$ define a codimension-1 subtorus $S$ by the
character $\psi:T\to S^{1}$ given by
$(\theta_{1},\ldots,\theta_{n})\mapsto\theta_{1},$ and use a choice of
coorientation to lift $S$ to a Legendrian $\mathcal{L}\subset S^{*}T,$ where
now $\mathcal{L}$ is an $(n-1)$-torus. Then glue $T^{*}T$ to $T^{*}(D\times
T^{n-2})$ along the Legendrian $\mathcal{L}.$

Note that unlike in the 2-dimensional case, our construction in higher
dimensions will not be determined by the choice of Legendrian $\mathcal{L}$:
in addition, we need to know which of the remaining $n-1$ directions contains
the disk capping off $\mathcal{L}$. This will be a direction $\chi$ in the
torus $S.$

######  Definition 4.6.

Let $\chi:S^{1}\to T^{n-1}$ be a cocharacter of an $(n-1)$-torus $T^{n-1},$ so
that $T^{n-1}$ can be expressed as a product $(S^{1})_{\chi}\times T^{n-2}.$
We write $\mathbb{D}_{\chi}\times T^{n-2}$ for the result of replacing the
first factor in this product with a disk.

We are thus ready to combine several copies of the move described in Example
4.5.

######  Construction 4.7.

Let $T$ be the $n$-torus $M_{S^{1}}$, and let $\Sigma$ be a purely
1-dimensional fan in $N_{\mathbb{R}},$ defining a collection of cooriented
codimension-1 subtori $S_{i}\subset T,$ lifting to Legendrians
$\mathcal{L}_{i}\subset S^{*}T.$ Moreover, for each $S_{i}$ choose a
cocharacter $\chi_{i}:S^{1}\to S_{i}$ of the torus $S_{i}.$ Choose a
decomposition $S_{i}\simeq S^{1}_{\chi_{i}}\times T_{i}$, where $T_{i}$ is an
$(n-2)$-torus. Finally, glue a copy of $T^{*}(D_{\chi_{i}}\times T_{i})$ along
each Legendrian $\mathcal{L}_{i}$ as in Example 4.5. We denote the resulting
Weinstein manifold by $U^{\vee}$.

Note that the resulting Weinstein manifold does not depend on the
decomposition $S_{i}=S^{1}_{\chi_{i}}\times T_{i}$. The key point here is that
given two decompositions $S_{i}\simeq S^{1}_{\chi_{i}}\times T_{i}\simeq
S^{1}_{\chi_{i}}\times T_{i}^{\prime}$, there is a canonical isomorphism
$D_{\chi_{i}}\times T_{i}\simeq D_{\chi_{i}}\times T_{i}^{\prime}$.

As in Proposition 4.4, this gluing presentation of the manifold $U^{\vee}$
immediately gives a Liouville-sectorial cover.

######  Proposition 4.8.

The Weinstein manifold $U^{\vee}$ defined in Construction 4.7 is a union of
the Liouville sectors $\overline{X}_{\Sigma}^{\vee}$ and
$\bigsqcup_{i=1}^{r}T^{*}(D_{\chi_{i}}\times T_{i}),$ glued along
$\bigsqcup_{i=1}^{r}T^{*}(S^{1}_{\chi_{i}}\times T_{i}).$

###### Proof.

As in Proposition 4.4, this follows immediately from the construction of
$U^{\vee}$: one can obtain $U^{\vee}$ by beginning with sector
$\overline{X}_{\Sigma}^{\vee}$ and performing gluings along the $r$ disjoint
Legendrians $\mathcal{L}_{i}$ as indicated. ∎

Combined with the codescent prescription of Theorem 3.10, this gives a
presentation of the wrapped Fukaya category of $U^{\vee}$.

######  Corollary 4.9.

The wrapped Fukaya category $\mathcal{W}(U^{\vee})$ of the Weinstein manifold
defined by Construction 4.7 can be presented as a homotopy pushout of the
diagram

(4.10)
---
$\textstyle{\mathcal{W}\left(\bigsqcup_{i=1}^{r}T^{*}(S^{1}_{\chi_{i}}\times
T_{i})\right)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{W}\left(\bigsqcup_{i=1}^{r}T^{*}D_{\chi_{i}}\times
T_{i}\right).}$$\textstyle{\mathcal{W}(\overline{X}_{\Sigma}^{\vee})}$

Since the categories in the pushout diagram (4.10) all have well-understood
interpretations in mirror symmetry, we can rephrase Corollary 4.9 entirely in
terms of the mirror algebraic geometry used by Gross-Hacking-Keel in
Construction 2.6 .

Recall that that construction also began with the construction of a fan
$\Sigma\subset N_{\mathbb{R}}$ of rays $\langle d_{i}\psi_{i}\rangle$,
defining a toric variety $\overline{X}_{\Sigma}$ with boundary divisor

$\overline{D}=\bigsqcup_{i=1}^{r}\overline{D}_{i}\cong\bigsqcup_{i=1}^{r}(N/\psi_{i})_{\mathbb{C}^{\times}}\times
B\mathbb{Z}/d_{i},$

along with characters $\chi_{i}$ of these quotient tori, defining a subvariety

$H\cong\bigsqcup_{i=1}^{r}(\mathbb{C}^{\times})^{n-2}\times
B\mathbb{Z}/d_{i}\subset\overline{D}.$

We write $i:\overline{D}\to\overline{X}_{\Sigma}$ for the inclusion of the
toric boundary divisor and $j:H\to\overline{D}$ for the inclusion of $H$ into
$\overline{D}$. Now we are ready to restate Corollary 4.9:

######  Corollary 4.11.

The wrapped Fukaya category $\mathcal{W}(U^{\vee})$ of $U^{\vee}$ can be
presented as a homotopy pushout of the diagram

(4.12)
---
$\textstyle{\textup{Coh}(\overline{D})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j^{*}}$$\scriptstyle{i_{*}}$$\textstyle{\textup{Coh}(H)}$$\textstyle{\textup{Coh}(\overline{X})}$

###### Proof.

The equivalences between the categories in diagram (4.12) and those in diagram
(4.10) have already been discussed. (For the stacky case, where not all
$d_{i}$ are equal to 1, note that there is an equivalence of dg categories
$\textup{Coh}(B\mathbb{Z}/d_{i})\cong\textup{Coh}\left(\bigsqcup_{j=1}^{d_{i}}\textup{Spec}(k)\right).$)
That the vertical arrows agree is proved in [GS, Section 7.2] (technically the
result there is for the pullback $i^{*}$; the result for the pushforward can
be recovered by passing to right adjoints), and the agreement of the
horizontal arrows is clear. ∎

The above presentation is very pleasant to have, though it may at first seem
perplexing and indeed to an algebraic geometer it will likely not look
“geometric,” due to the opposing functorialities present. Nevertheless, we
will see in our main result, Theorem 4.15 below, that this pushout does indeed
present the category of coherent sheaves on the truncated cluster variety
$U=X\setminus D.$

Before proceeding to that proof, we consider the most basic example.

######  Example 4.13.

Consider again the situation of Example 4.2, where $n=2,$ and $r=1,$ so that
$\chi$ is the character of $T=\mathbb{R}^{2}/\mathbb{Z}^{2}$ given by
$(\theta_{1},\theta_{2})\mapsto\theta_{1}.$ In this case, the diagram (4.12)
becomes

---
$\textstyle{\textup{Coh}(\\{0\\}\times\mathbb{C}^{\times})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\textup{Coh}\left(\\{(0,-1)\\}\right)}$$\textstyle{\textup{Coh}(\mathbb{C}\times\mathbb{C}^{\times}),}$

where the vertical and horizontal maps are the evident pushforward and
pullback functors, respectively.

Let us rewrite this diagram by first passing to ind-completions and right
adjoints as in Corollary 3.11. At the level of categories, this amounts to
replacing coherent by quasicoherent sheaves (using the fact that all varieties
involved are smooth – otherwise, we would have to use ind-coherent sheaves),
so that we obtain the pullback diagram

(4.14)
---
$\textstyle{k[x,y^{\pm}]/(x)\operatorname{-mod}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces
k[x,y^{\pm}]/(x,y+1)\operatorname{-mod}}$$\textstyle{k[x,y^{\pm}]\operatorname{-mod}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

of module categories. The horizontal functor, right adjoint to pullback, is
pushforward, (which on module categories is restriction of scalars); the
vertical functor, as the right adjoint to pushforward, is the shriek pullback
functor, which on a $k[x,y^{\pm}]$-module $M$ is the coextension of scalars

$M\mapsto\textup{Hom}_{k[x,y^{\pm}]}(k[x,y^{\pm}]/(x),M).$

By resolving $k[x,y^{\pm}]/(x)$ as a $k[x,y^{\pm}]$-module, we can also
understand this as the functor taking a module $M$ to the cone of the map
$x:M\to M.$

Let us denote by $\mathcal{C}$ the homotopy limit of diagram (4.14). An object
of $\mathcal{C}$ is a triple $(M,N,\phi)$ consisting of a
$k[x,y^{\pm}]$-module $M$, a $k[x,y^{\pm}]/(x,y+1)$-module $N$, and an
isomorphism

$\textstyle{\phi:\operatorname{Cone}(M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x}$$\textstyle{M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\sim}$$\textstyle{N}$

of $k[x,y^{\pm}]/(x)$-modules (regarding $N$ as one by restriction). The
isomorphism $\phi$ can be reformulated as the imposition on
$\operatorname{Cone}(x:M\to M)$ of the equation that $y+1$ acts as zero, or in
other words, it is the data of a degree-$(-1)$ endomorphism $x^{\prime}$ of
this complex with $d(x^{\prime})=y+1.$

As explained in detail in [STWZ19, Proposition 4.10], if $M$ was concentrated
in degree 0, then $x^{\prime}$ is equivalent to the datum of a (degree-0)
endomorphism of $M$ satisfying $xx^{\prime}=y+1$: pictorially, we have

$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x^{\prime}}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{x}$$\textstyle{M,}$

where the horizontal map $x$ is the differential on $\operatorname{Cone}(x),$
so that $xx^{\prime}=y+1$ expresses the triviality of $y+1$ acting on
$\operatorname{Cone}(x).$

As expected, this agrees with the category
$\textup{QCoh}(U)=k[x,x^{\prime},y^{\pm}]/(xx^{\prime}=y+1)\operatorname{-mod}$
of quasicoherent sheaves on the mirror cluster variety
$U=\mathbb{C}^{2}\setminus\\{xx^{\prime}=1\\}.$

###  4.2. Homological mirror symmetry

Let $(X,D)$ be an $n$-dimensional log Calabi-Yau pair with toric model
$(\overline{X},\overline{D}),$ modeling the log Calabi-Yau variety
$U=X\setminus D$ as in Definition 2.9. Recall that such a pair is determined
by the following two pieces of data:

  * •

A purely one-dimensional fan $\Sigma$ in $N_{\mathbb{R}},$ the union of rays
$\langle d_{1}\psi_{1}\rangle,\ldots\langle d_{r}\psi_{r}\rangle,$ where
$\psi_{i}\in N$ are primitive lattice vectors, and $d_{i}\in\mathbb{Z}_{>0}.$
As is usual in toric geometry, the $\psi_{i}$ can be understood as
cocharacters of the torus $N_{\mathbb{C}^{\times}}.$

  * •

For each $1\leq i\leq r,$ an element $\chi_{i}\in\psi_{i}^{\perp}=M/\psi_{i},$
which we think of as a character on the divisor $D_{i}$ (thought of as a
complex $(n-1)$-torus), defining a hypersurface
$H_{i}=\\{\chi_{i}=-1\\}\subset D_{i}.$

This is the same data used in Construction 4.7 to define the Weinstein
manifold $W^{\vee},$ where now we think of $\psi_{i}\in N$ not as a
cocharacter of $N_{\mathbb{C}^{\times}},$ but rather as a character of the
real $n$-torus $M_{S^{1}},$ and $\chi_{i}$ as as a cocharacter on the subtorus
$S_{i}\subset M_{S^{1}}$ defined by $\psi_{i}.$

The main theorem of this paper is a statement of homological mirror symmetry
for the varieties $U$ and $U^{\vee}.$

######  Theorem 4.15.

There is an equivalence of dg categories

$\textup{Coh}(U)\cong\mathcal{W}(U^{\vee})$

between the category of coherent sheaves on $U$ and the wrapped Fukaya
category of $U^{\vee}.$

The slogan is, “The blow-up-and-delete construction of Gross-Hacking-Keel is
mirror to the Weinstein handle attachment of Example 4.5.” We will deduce this
theorem from the following more general result, which we can understand as a
generalization of the calculation described in Example 4.13:

######  Theorem 4.16.

Let $X$ be a smooth variety or DM stack, $D\subset X$ a divisor, and $H\subset
D$ a smooth subvariety of codimension 2 in $X.$ Let
$\mathfrak{X}=\operatorname{Bl}_{H}X\setminus\widetilde{D}$ and let
$\mathcal{C}$ denote the homotopy pushout of dg categories

(4.17)
$\varinjlim\left({\textup{Coh}(H)\leftarrow\textup{Coh}(D)\to\textup{Coh}(X)}\right).$

Then there is an equivalence

$\mathcal{C}\cong\textup{Coh}(\mathfrak{X})$

between $\mathcal{C}$ and the category of coherent sheaves on $\mathfrak{X}$.

###### Proof.

We begin by writing down the functor
$\mathcal{C}\to\textup{Coh}(\mathfrak{X}).$ From the pushout presentation of
$\mathcal{C},$ such a functor may be specified by the data of functors
$F_{H}:\textup{Coh}(H)\to\textup{Coh}(\mathfrak{X})$ and
$F_{X}:\textup{Coh}(X)\to\textup{Coh}(\mathfrak{X}),$ together with an
equivalence between the pullbacks of $F_{H}$ and $F_{X}$ to $\textup{Coh}(D)$
along the maps in the pushout (4.17).

Consider the following diagram of all the spaces involved in our story, where
$E$ is the exceptional divisor of the blowup, the vertical maps are the
projections, and the horizontal maps are the inclusions:

$\textstyle{\mathfrak{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{j}$$\textstyle{\operatorname{Bl}_{H}X\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p}$$\textstyle{E\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\overline{i}}$$\scriptstyle{q}$$\textstyle{X}$$\textstyle{D\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{k}$$\textstyle{H.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i}$

The functors $F_{H}$ and $F_{X}$ will be defined as the compositions

$\displaystyle F_{H}=j^{*}\overline{i}_{*}q^{*},$ $\displaystyle
F_{X}=j^{*}p^{*}.$

The restriction to $D$ of $F_{X}$ is evidently given by $j^{*}p^{*}k_{*},$ and
we must produce an isomorphism between this functor and the other restriction
$j^{*}\overline{i}_{*}q^{*}i^{*}.$ Note that before applying final composition
with $j^{*},$ these two functors differ, since $p^{*}k_{*}$ takes sheaves on
$D$ to their pullback to the blowup, whereas $\overline{i}_{*}q^{*}i^{*}$
takes sheaves on $D$ to their component lying along the exceptional divisor
$E.$ In other words, the difference between these two functors is a sheaf on
the proper transform $\widetilde{D}$ of $D.$ The final pullback $j^{*}$ will
kill this component, forcing the two functors to agree.

Thus, we have a well-defined functor
$F:\mathcal{C}\to\textup{Coh}(\mathfrak{X}).$ To understand this functor, we
will need to apply Orlov’s result ([Orl93]) that $p^{*}\textup{Coh}(X)$ and
$q^{*}\textup{Coh}(H)\otimes\mathcal{O}_{E}(1)$ generate the category
$\textup{Coh}(\operatorname{Bl}_{H}(X))$ of coherent sheaves on the blowup,
and in fact they form a semi-orthogonal decomposion:

(4.18)
$\textup{Coh}(\operatorname{Bl}_{H}(X))=\langle\textup{Coh}(H)\otimes\mathcal{O}_{E}(-1),\textup{Coh}(X)\rangle.$

Note that after pulling back to $\mathfrak{X},$ we can ignore the tensor
product with $\mathcal{O}_{E}(-1)$ in (4.18), since deleting $\widetilde{D}$
makes $E$ affine. This makes clear why $F$ is essentially surjective: the
image of the functor $F$ contains the pullbacks to $\mathfrak{X}$ of both
components in the semi-orthogonal decomposition (4.18) and hence contains all
objects in $j^{*}\textup{Coh}(\operatorname{Bl}_{H}(X)).$

To see why the functor $F$ is in fact an equivalence of categories, recall
that the category $\textup{Coh}(\mathfrak{X})$ is a localization of
$\textup{Coh}(\operatorname{Bl}_{H}(X))$ obtained by killing the identity
morphism of any object in $\textup{Coh}(\widetilde{D}),$ and observe that this
is equivalent to the localization which identifies any object in the second
term of (4.18) of the form $k_{*}\mathcal{F},$ for some $\mathcal{F}$ in
$\textup{Coh}(D),$ with the object
$(iq)^{*}\mathcal{F}\otimes\mathcal{O}_{E}(-1)$ in the first term.

Now note that the category obtained from the semi-orthogonal decomposition
(4.18) by identifying the images of $\textup{Coh}(X)$ and $\textup{Coh}(H)$ is
equivalent to the category obtained from the orthogonal sum
$\textup{Coh}(H)\oplus\textup{Coh}(X)$ by the same identification; but
$\textup{Coh}(H)\oplus\textup{Coh}(X)$ is the coproduct of the categories
$\textup{Coh}(H)$ and $\textup{Coh}(X),$ and the identification just described
is the coequalizer of the respective maps from $D$ to these categories. In
other words, this presentation exhibits $\textup{Coh}(\mathfrak{X})$ as the
pushout (4.17), and $F$ as the canonical equivalence
$\mathcal{C}\cong\textup{Coh}(\mathfrak{X}).$ ∎

###### Proof of Theorem 4.15.

From Corollary 4.11, we have a presentation of the wrapped Fukaya category
$\mathcal{W}(U)$ as a colimit as in Theorem 4.16; that theorem identifies the
colimit with the category $\textup{Coh}(U^{\vee}).$ ∎

One consequence of the equivalence from Theorem 4.15 is that the category
$\textup{Coh}(\overline{T}_{\mathbb{C}})$ of coherent sheaves on the torus
chart $\overline{T}_{\mathbb{C}}$ of the cluster variety $U$ corresponds under
mirror symmetry to the category $\textup{Loc}(T^{\vee})$ to the category of
local systems on the dual torus $T^{\vee},$ which is a closed subset in the
skeleton $\mathbb{L}_{U^{\vee}}$ of $U^{\vee}.$ A Weinstein neighborhood
$T^{*}T^{\vee}$ of this torus is therefore a Liouville subdomain of the mirror
cluster variety $U^{\vee},$ and the Viterbo restriction of Proposition 3.16
gives a functor

(4.19) $\mathcal{W}(U^{\vee})\to\mathcal{W}(T^{*}T^{\vee}).$

This functor is mirror to the pullback
$\textup{Coh}(U)\to\textup{Coh}(\overline{T}_{\mathbb{C}})$ of coherent
sheaves on the cluster variety $U$ to a toric chart.

From this description, one can immediately guess what the symplectic mirror to
the elementary transformations underlying cluster mutation should look like.
This should be a procedure which exchanges the Weinstein presentation of
$\mathbb{L},$ as obtained from $T^{*}T^{\vee}$ by Weinstein disk attachments,
to a different Weinstein presentation which begins with a different torus
$T^{*}(T^{\prime})^{\vee}.$ We will see below that this guess is essentially
correct.

###  4.3. Mutations

We now discuss the effect of mutations in symplectic geometry. Local models
for these mutations have been studied in [PT20, Nad], and in the 2-dimensional
case a global description of these mutations is given in [STW, HK].

As explained in [STW], from the perspective of a 2-dimensional Lagrangian
skeleton $\mathbb{L}$, cluster mutation corresponds to the Lagrangian disk
surgery studied in [Yau].

######  Construction 4.20.

The local model for this surgery, depicted in Figure 7,

Figure 7. The local model for the Lagrangian disk surgery mirror to cluster
mutation: a disk attached to a cylinder $\Sigma$ is collapsed and then
expanded to produce a new cylinder $\Sigma^{\prime}.$

begins with a Lagrangian skeleton $\mathbb{L}$ which can be presented as the
union of a surface $\Sigma$ together with a disk $\mathbb{D}$ attached to
$\Sigma$ along a circle. There is a homotopy of Weinstein structures which
collapses the disk $\mathbb{D}$ and then expands it to produce a new
Lagrangian skeleton $\mathbb{L}^{\prime}$ which is a union of $\mathbb{D}$
with a surface $\Sigma^{\prime}.$ Alternatively, if one folds up the bottom
half of the cylinder to make a plane with the central disk, this
transformation can be pictured via a cooriented circle in the plane shrinking
to a point and then reexpanding with its opposite coorientation, as in Figure
8.

Figure 8. The Lagrangian from Figure 7 is obtained by taking the cone on the
Legendrian lift of the cooriented circle–or, in the central diagram, the whole
conormal circle of the point. The coorientation of the circle changes as it
shrinks to a point and regrows, although the Legendrian circles at infinity
are all related by a contact isotopy.

Suppose that the surface $\Sigma$ also has other disks
$\mathbb{D}_{1},\ldots,\mathbb{D}_{r}$ glued in, along Legendrians lifting
cooriented circles $S_{1},\ldots,S_{r}\subset\Sigma$ which may intersect the
boundary circle of $\mathbb{D}.$ If the above surgery implements mutation,
then we should expect that it should modify the boundary circles $S_{i}$ by a
Dehn twist for each positively oriented intersection with the boundary circle
of $\mathbb{D}.$ The results of [STW] establish that this is the case. In our
language, we can restate their result as follows:

######  Theorem 4.21 ([STW, Theorem 5.7]).

Let $\mathbb{L}$ be the Lagrangian skeleton of a Weinstein manifold $U^{\vee}$
obtained as in Construction 4.1 by beginning with the cotangent bundle
$T^{*}T$ of a 2-torus $T\cong T^{2}$ and attaching Weinstein handles along
Legendrian lifts of cooriented circles $S_{1},\ldots,S_{k}\subset T.$

Applying the disk surgery described in Construction 8 to the $i$th disk
$\mathbb{D}_{i}$ results in a new Lagrangian skeleton $\mathbb{L}^{\prime}$ of
the same Weinstein manifold, presented by beginning with the cotangent bundle
$T^{*}T^{\prime}$ of a 2-torus $T^{\prime}\cong T^{2}$ by handle attachments
along Legendrian lifts of cooriented circles $S_{1},\ldots,S_{k}\subset
T^{\prime},$ where $S_{i}^{\prime}=S_{i}$ with opposite coorientation, and for
$j\neq i$ we have

$S_{j}^{\prime}=\begin{cases}\tau_{i}S_{j}&\text{if }\langle
S_{j},S_{i}\rangle>0,\\\ S_{j}&\text{otherwise},\end{cases}$

where we write $\langle-,-\rangle$ for the oriented intersection number and
$\tau_{i}$ for the Dehn twist about $S_{i}.$

The paper [STW] also checks that the respective tori $T,T^{\prime}$ contained
in the skeleton before and after a disk surgery are related by cluster
mutation. Continuing with the notation of Theorem 4.21, Viterbo restriction
(4.19) gives a pair of functors

(4.22)
$\textup{Coh}(T_{\mathbb{C}}^{\vee})\cong\mathcal{W}(T^{*}T)\leftarrow\mathcal{W}(U^{\vee})\to\mathcal{W}(T^{*}T^{\prime})\cong\textup{Coh}((T^{\prime})^{\vee}_{\mathbb{C}}),$

which we would like to understand as mirror to a birational map
$T_{\mathbb{C}}^{\vee}\dashrightarrow(T^{\prime})^{\vee}_{\mathbb{C}}$ between
the complexified dual tori. This map is studied in [STW] by determining what
it does to skyscraper sheaves in $\textup{Coh}(T_{\mathbb{C}}^{\vee}),$ which
correspond to finite-rank local systems on $T.$ If $\mathcal{E}$ is a local
system on $T,$ we can imagine $\mathcal{E}$ as a vector space $\mathcal{E}$
together with an endomorphism $\mathcal{E}_{\gamma}$ for each path $\gamma$ in
$T$.

Now note that on finite-rank local systems (i.e., objects of the compact
Fukaya category of $T^{*}T$), the right adjoints in (4.22) are defined, so we
can obtain a diagram

(4.23)
$\operatorname{Tors}(T^{\vee}_{\mathbb{C}})\cong\textup{Loc}^{fin}(T)\to\mathcal{W}(U^{\vee})\leftarrow\textup{Loc}^{fin}(T^{\prime})\cong\operatorname{Tors}((T^{\prime})^{\vee}_{\mathbb{C}}),$

and we can ask when a local system on $T$ maps into the image of the right-
hand functor – i.e., to an object of $\mathcal{W}(U^{\vee})$ represented by a
local system on $T^{\prime}.$

######  Theorem 4.24 ([STW, Theorem 4.16]).

Let $\mathcal{E}$ be a finite-rank local system on $T.$ Then $\mathcal{E}$ is
equivalent in $\mathcal{W}(U^{\vee})$ to a local system $\mathcal{E}^{\prime}$
on $T^{\prime}$ if and only if the monodromy of $\mathcal{E}$ around $S_{i}$
does not have 1 as an eigenvalue.

Suppose this is the case. If $\gamma\subset T^{2}$ is a path not intersecting
$S_{i}^{\prime},$ then $\mathcal{E}^{\prime}_{\gamma}=\mathcal{E}_{\gamma},$
and if $\gamma$ is a path which crosses $S_{i}^{\prime}$ against its
coorientation, in a neighborhood of the crossing point the holonomy is
modified to
$\mathcal{E}^{\prime}_{\gamma}=\operatorname{Id}-\mathcal{E}_{S_{i}},$ where
$S_{i}$ is treated as an oriented path where its coorientation points
rightward. (The identification of paths in $T$ with paths in $T^{\prime}$ is
given by a Dehn twist about $S_{i}.$)

These results generalize easily to a higher-dimensional Weinstein manifold
$U^{\vee}$ obtained as in Construction 4.7 by Weinstein handle attachments
along Legendrian lifts of codimension-1 subtori $S_{1},\ldots,S_{r}$ in an
$n$-torus $T$.

######  Construction 4.25.

Let $T,S_{i}$ as above. Near the subtorus $S_{i},$ the torus $T$ splits as
$T^{2}\times T^{n-2},$ where the first $T^{2}$ is spanned by $S{\chi_{i}}$
(where $\chi_{i}$ is the cocharacter of $S_{i}$ determining the handle
attachment as in Construction 4.7) and a circle not contained in $S_{i}$. We
can thus take the product with $T^{n-2}$ of the disk surgery of Construction 8
above to obtain an analogous operation on $T=T^{n}.$ The result is a new
Lagrangian which is the union of an n-torus $T^{\prime}$ with Weinstein
handles attached to the Legendrian lifts of codimension-1 subtori
$S_{i}^{\prime}.$

Theorems 4.21, 4.24 can be applied directly to this setting to describe the
new torus $T^{\prime}$ and its subtori $S_{i}^{\prime}$ just as in the
2-dimensional case. In the language of cluster seeds, we can summarize this
discussion as follows:

######  Theorem 4.26.

Let $T,S_{1},\ldots,S_{r},$ as in Construction 4.7, corresponding to a seed
$\mathfrak{s}.$ Then $T^{\prime},S_{1}^{\prime},\ldots,S_{r}^{\prime}$ as in
Construction 4.25 correspond to the $i$th mutation $\mu_{i}\mathfrak{s}$ of
$\mathfrak{s}.$ Moreover, the birational map defined by (4.23) agrees with the
cluster transformation for this mutation.

##  5\. Skeleta from nodal trades

###  5.1. The nodal trade construction

In two dimensions, the disk gluing described in Section 4.1 has an avatar in
the realm of Lagrangian torus fibrations, in the form of the nodal trade first
described in [Sym03]. The Lagrangian torus fibrations we describe play the
role of the SYZ fibrations in our mirror-symmetry setup, although we do not
use that fact explicitly here.

One way to motivate the nodal trade construction is to begin with the moment-
map fibration $\mu:\mathbb{C}^{2}\to P,$ where $P=(\mathbb{R}_{\geq 0})^{2}$
is the usual moment polytope, and note that this presentation of
$\mathbb{C}^{2}$ distinguishes an anticanonical divisor, namely the preimage
of the boundary $\partial P=\mathbb{R}_{\geq
0}\times\\{0\\}\cup_{(0,0)}\\{0\\}\times\mathbb{R}_{\geq 0},$ which is the
toric boundary $\\{xy=0\\}$ of $\mathbb{C}^{2}.$

The motivating question is: is it possible to change the fibration $\mu$ so as
to change $\mu^{-1}(\partial P)$ to a different anticanonical divisor –
namely, the smooth quadric ${xy=1}$? At first glance, the answer seems to be
no, since the quadric ${xy=1}$ is smooth while $\partial P$ is singular. In
[Sym03], Symington resolves this difficulty by introducing a focus-focus
singularity into the fibration, as pictured in Figure 9. The cocharacter
lattices of smooth fibers equip the base of this fibration (away from the
singular point) with the structure of an integral affine manifold, in which
the monodromy around the singular point is given by $\begin{pmatrix}2&1\\\
-1&0\end{pmatrix}.$

######  Remark 5.1.

Although the branch cut for the integral affine structure is along the ray
$\begin{pmatrix}\psi_{1}\\\ \psi_{2}\end{pmatrix}=\begin{pmatrix}-1\\\
-1\end{pmatrix},$ we are here taking the integral affine structure determined
by cocharacter (rather than character) lattices of torus fibers, so the
monodromy matrix is given by the transpose of that calculated in Construction
2.

After this change, $P$ no longer embeds into $\mathbb{R}^{2}$ with its
integral affine structure, but it may still be immersed in $\mathbb{R}^{2}$
after a branch cut as in Figure 9.

Figure 9. The base of the new fibration on $\mathbb{C}^{2},$ with singularity
marked in red and branch cut emanating from it. The fibers over interior
points are still smooth 2-tori, except for the fiber over the singular point,
which is a nodal 2-torus.

The boundary $\partial P,$ which passes through the branch cut, becomes smooth
in the new integral affine structure on the base. The result of [Sym03] is
that a fibration inducing this integral affine structure, and making $\partial
P$ smooth, does exist.

######  Proposition 5.2 ([Sym03]).

There is a map $\mu^{\prime}:\mathbb{C}^{2}\to P$ whose fibers over the
interior of $P$ are smooth tori except for a nodal torus above $(1,1),$ such
that $\mu^{\prime}$ induces the above integral affine structure on $P$ and
$\mu^{\prime-1}(\partial P)=\\{xy=1\\}.$

Since the operation of replacing $\mu$ by $\mu^{\prime}$ doesn’t affect the
fibration away from the branch cut, we can perform this replacement anywhere a
corner isomorphic to $(\mathbb{R}_{\geq 0})^{2}$ appears in the moment polygon
of a toric surface:

######  Construction 5.3.

Let $\mu:\overline{X}_{\Delta}\to\Delta$ be a toric surface equipped with
moment polygon $\Delta\subset\mathbb{R}^{2}$ and let $p\in\Delta$ be a vertex
with an integral affine-linear identification
$A\in\textup{SL}(2,\mathbb{Z})\rtimes\mathbb{R}^{2}$ of a neighborhood of $p$
in $\Delta$ with $(\mathbb{R}_{\geq 0})^{2}.$ Then using $A$, the moment map
of $\overline{X}_{\Delta}$ can be modified near $p$ to introduce a singularity
and branch cut in $\Delta$ as in Proposition 5.2. This operation is called a
nodal trade.

######  Example 5.4.

Let $\overline{X}_{\Delta}=\operatorname{Bl}_{0}\mathbb{C}^{2}$ be the blowup
of $\mathbb{C}^{2}$ at the origin, with moment polytope $\Delta$ obtained from
$(\mathbb{R}_{\geq 0})^{2}$ by cutting out the corner. Each of the two
vertices of $\Delta$ has a neighborhood related to the model corner
$(\mathbb{R}_{\geq 0})^{2}$ by an integral affine-linear translation rotating
the branch cut to the horizontal or vertical, respectively. The result of
applying nodal trades at these vertices is pictured in Figure 10.

Figure 10. The integral affine base of
$\overline{X}_{\Delta}=\operatorname{Bl}_{0}\mathbb{C}^{2}$ after a pair of
nodal trades.

In general, we will also be interested in higher-dimensional toric varieties.
There exists a higher-dimensional analogue of the nodal trade, studied in
[Nad, PT20], situated at a higher-codimension face of a moment polytope, but
cluster theory as traditionally understood pertains only to the nodal trade as
described above, or products of it with $(\mathbb{C}^{\times})^{n-2}.$ We thus
provide the following generalization of Construction 5.3.

######  Definition 5.5.

Let $\mu:\overline{X}_{\Delta}\to\Delta$ be a toric variety with moment
polytope $\Delta\subset\mathbb{R}^{n},$ and let $F\subset\partial\Delta$ be a
codimension-2 face of $\Delta$ equipped with an affine-linear identification
with $(\mathbb{R}_{\geq 0})^{2}\times\mathbb{R}^{n-2}.$ Then we can perform a
nodal trade on the first factor of this product and identify the result with a
neighborhood of $F$. The result of this construction is our higher-dimensional
analogue of Construction 5.3, which we will continue to refer to as a nodal
trade.

This higher-dimensional setting for nodal trades has not been studied as much
as the 2-dimensional case, largely because it is more difficult to draw–but we
emphasize that since it is only a product with a 2-dimensional nodal trade, it
is fundamentally no more difficult than that case. We depict the case $n=3$ in
Figure 11.

Figure 11. The base of a nodal trade in 3 dimensions, i.e., the product of
$\mathbb{C}^{\times}$ with the 2-dimensional nodal trade. The bold red line is
a line of singularities, over which the fibration takes the form (nodal
curve)$\times S^{1},$ and the blue half-plane is the “branch cut” allowing us
to draw this picture in $\mathbb{R}^{3}$; it lives in the union of monodromy
eigenlines of the singular points.

######  Remark 5.6.

In base diagrams like Figures 10 and 11, depicting integral manifolds whose
singularities are obtained from a fixed local model by an integral affine-
linear change of coordinates, the singularities are determined by their
monodromy-invariant direction. In dimension 2, we refer to this line, which
contains the branch cut associated to the singularity, as its eigenline (or
eigenray if we want to specify only half of it.) As the higher-dimensional
bases we consider are always obtained locally as product of the 2-dimensional
case with $\mathbb{R}^{n-2}$, the singularities in these bases will be the
same but will now occur in families, living along $(n-2)$-dimensional affine-
linear subspaces. Such a family is therefore determined by its monodromy-
invariant eigenhyperplane, as in Figure 11. (This hyperplane is the union of
an eigenline emerging from each point in the codimension-2 family of
singularities.)

###  5.2. Mirror cluster surfaces and skeleta

We will now consider the Weinstein manifold obtained from a toric variety
$\overline{X}_{\Delta}$ by performing nodal trades and then deleting the
preimage of $\partial\Delta.$ The local models for this construction are
studied in detail in [PT20]; here, we summarize some of their results and then
deduce a global description of the mirror to a general cluster variety.

As a warm-up, first consider the case where we do not perform a nodal trade.
In this case, for $\overline{X}_{\Delta}$ a toric variety with moment map
$\mu:\overline{X}_{\Delta}\to\Delta\subset\mathbb{R}^{n},$ we have

$\overline{X}_{\Delta}\setminus\mu^{-1}(\partial\Delta)=\overline{X}_{\Delta}\setminus\partial\overline{X}=(\mathbb{C}^{\times})^{n},$

with restricted moment map
$\mu:(\mathbb{C}^{\times})^{n}\to\operatorname{Int}(\Delta)\cong\mathbb{R}^{n}$
given by the usual moment map
$(z_{1},\ldots,z_{n})\mapsto(\log|z_{1}|,\ldots,\log|z_{n}|).$ For
$c\in\mathbb{R}^{n},$ the Morse-Bott function

(5.7)
$\|\mu-c\|^{2}:(z_{1},\ldots,z_{n})\mapsto(\log|z_{1}|-c_{1})^{2}+\cdots+(\log|z_{n}|-c_{n})^{2}$

is a Stein Kähler potential on $(\mathbb{C}^{\times})^{n}$ whose underlying
Liouville structure has Lagrangian skeleton the torus $\mu^{-1}(c),$ the
minimum locus of $\|\mu-c\|^{2}.$

A similar result will turn out to be true for the manifold obtained by
performing a nodal trade, but in this case the skeleton will also contain a
Lagrangian disk projecting to the interval between $0$ and the singularity of
the integral affine base. Such a Lagrangian, projecting to an interval under
an almost toric fibration, is called a visible Lagrangian in [Sym03]. Away
from the singularity, this visible Lagrangian lives in each torus fiber as a
circle in homology class $(1,-1),$ and the whole Lagrangian is obtained by
following this circle as it collapses to the singular point of the nodal
fiber.

Consider the local “nodal trade” almost toric fibration
$\mu^{\prime}:\mathbb{C}^{2}\to P$ described in Proposition 5.2, and its
restriction to a fibration

(5.8) $\mathbb{C}^{2}\setminus\\{xy=1\\}\to\operatorname{Int}(P).$

Choose an identification $\operatorname{Int}(P)\cong\mathbb{R}^{2}$ which has
the singularity of the integral affine structure at the origin and the
eigenline along the $x$-axis. Then a model for the restricted singular torus
fibration (5.8) is given by

$F(x,y):=\left((\log|xy-1|),|x|^{2}-|y|^{2}\right).$

We refer to [Aur07, Section 5] for a more detailed study of the map $F$. We
now explain how this map can be used to produce our favorite Lagrangian
skeleton of $\mathbb{C}^{2}\setminus\\{xy=1\\}.$

As we did for the moment map $\mu$ of $(\mathbb{C}^{\times})^{n},$ we would
like to produce a Kähler potential by composing $F$ with the distance function
from a point on the base. This won’t work if the point lies off the monodromy
eigenline of the singularity (because the torus fiber will not be exact – see
[PT20, Section 4.5] for a more detailed discussion), but we will see that we
can produce a skeleton containing any given torus fiber on the eigenline.

So let $c\in\mathbb{R},$ and define a map
$f_{c}:\mathbb{C}^{2}\setminus\\{xy=1\\}\to\mathbb{R}$ by

(5.9)
$f_{c}(x,y):=\left(\log|xy-1|-c\right)^{2}+\left(|x|^{2}-|y|^{2}\right)^{2}.$

As for (5.7), the function $f_{c}$ is Morse-Bott with manifold of minima given
by the torus fiber $F^{-1}(c,0),$ but now, $f_{c}$ has one additional Morse
critical point – an index 2 critical point at $(0,0)\in\mathbb{C}^{2}.$ So it
seems that the function $f_{c}$ should produce a Lagrangian skeleton of
$\mathbb{C}^{2}\setminus\\{xy=1\\}$ which is the union of the torus
$T=F^{-1}(c,0)$ and a disk (coming from the index-2 critical point) glued to
$T$ along a circle $S\subset T$.

However, the function $f_{c}$ is not quite a Kähler potential. It is
plurisubharmonic, but not strictly so: the form
$i\partial\overline{\partial}f_{c}$ is positive everywhere except the point
$(0,0)\in\mathbb{C}^{2},$ where it vanishes. We can fix this by adding a term
$\chi\epsilon(|x|^{2}+|y|^{2})$ to $f_{c}$, where $\chi$ is a cut-off
function, identically 1 near the origin $(0,0)$ and vanishing outside a small
neighborhood of the origin, and $0<\epsilon\ll 1.$

######  Lemma 5.10.

For appropriate choices of $\chi$ and $\epsilon$ as above, the function
$f_{c}+\chi\epsilon(|x|^{2}+|y|^{2})$ is a Kähler potential for
$\mathbb{C}^{2}\setminus\\{xy=1\\}.$

###### Proof.

Let $g(x,y)=\chi\epsilon(|x|^{2}+|y|^{2})$ be the correction term. In the
region where $\chi$ is identically 1, the 2-form
$i\partial\overline{\partial}g$ is positive and therefore so is $f_{c}.$ This
2-form may fail to be positive in the region where $\chi$ is decaying, but
within this region it may be bounded below by compactness, and thus for
$\epsilon$ sufficiently small, we can ensure that
$|\partial\overline{\partial}g|<|\partial\overline{\partial}f_{c}|$ within
this region, since the latter term has a uniform lower bound outside any
neighborhood of the origin. ∎

We would like to be specific about the Lagrangian skeleton associated to this
Kähler structure.

######  Lemma 5.11.

The Kähler potential of Lemma 5.10 underlies a Stein structure whose
Lagrangian skeleton $\mathbb{L}$ is the union of the torus fiber $F^{-1}(q)$
with a visible Lagrangian disk lying over the interval $[0,c]\times\\{0\\}$
and which is attached to the torus $F^{-1}(q)$ along the circle
$\\{|x|^{2}=|y|^{2},xy-1=-\exp(c)\\}.$

###### Proof.

Consider the Lefschetz fibration

(5.12) $\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
37.21362pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{h:\mathbb{C}^{2}\setminus\\{xy=1\\}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
61.21362pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
61.21362pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{\mathbb{C}^{\times},}$}}}}}}}{\hbox{\kern
105.56918pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{(x,y)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
134.76826pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@stopper}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
158.76825pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
158.76825pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{xy-1}$}}}}}}}\ignorespaces}}}}\ignorespaces,$

which has general fiber $\mathbb{C}^{\times}$ and a single critical value at
$-1$ with fiber $\\{xy=0\\}.$ Inside the fiber $h^{-1}(-\exp(c)),$ the
vanishing cycle $S$ is determined by the equation $|x|=|y|.$ By the general
theory of Lefschetz fibrations (see for instance [GP17]), the space
$\mathbb{C}^{2}\setminus\\{xy=1\\}$ admits a Lagrangian skeleton which is the
union of the parallel transport of $S$ around the circle $\\{\log|xy-1|=c\\}$
in the base together with the Lefschetz thimble obtained by parallel
transporting $S$ over the vanishing path $\\{\log|xy-1|\in[0,c]\\}.$

In fact, this is precisely the skeleton of the Stein structure from Lemma
$\ref{lem:potential}.$ First, note that the minimum of the Kähler potential
$f_{c}+g$ was the torus fiber $F^{-1}(0,c),$ which is equal to the parallel
transport of $S$ around the radius $\exp(c)$ circle in the base of the
$h$-fibration. Now, away from the region where $g\neq 0,$ the gradient flow of
$f_{c}$ will flow the circle $S$ inward along its Lefschetz thimble, and if we
require that the cut-off function $\chi$ depends only on the quantity
$|x|^{2}+|y|^{2},$ then the gradient flow will continue to flow the circle $S$
into the critical point $(0,0)$ of $f_{c}+g$ even where $g$ is nonzero.
Therefore this Lefschetz thimble is the piece of the skeleton corresponding to
the index-2 critical point at $(0,0).$ ∎

Recall that the rational tangent directions at a nonsingular point
$q\in\operatorname{Int}(P)$ of the base of the Lagrangian torus fibration $F$
correspond to characters of the torus fiber $F^{-1}(q),$ so that it makes
sense to speak of the codimension-1 subtorus $S_{v}\subset F^{-1}(q)$ dual to
a tangent direction $v$. We can thus reformulate the above calculation in
coordinate-free way as follows:

######  Corollary 5.13.

Let $q\in\operatorname{Int}(P)$ be a point lying on the monodromy eigenline
for the singularity of the integral affine base $\operatorname{Int}(P)$ of the
singular torus fibration

$\mu^{\prime}:\mathbb{C}^{2}\setminus\\{xy=1\\}\to\operatorname{Int}(P).$

Then the Morse-Bott function $\|\mu^{\prime}-q\|^{2}$ can be modified in a
neighborhood of $(\mu^{\prime})^{-1}(0,0)$ to produce a Kähler potential on
$\mathbb{C}^{2}\setminus\\{xy=1\\}$ whose Lagrangian skeleton is the union of
torus fiber $(\mu^{\prime})^{-1}(q)$ and a visble Lagrangian disk whose
$\mu^{\prime}$-projection is a subinterval of the monodromy eigenline
connecting $q$ to the singularity, glued to $(\mu^{\prime})^{-1}(q)$ along the
circle dual to this eigendirection.

This concludes our treatment of the local model for the nodal trade. (See also
[PT20, Section 5.5] for further discussion of this Kähler structure and its
skeleton.) From our presentation of Corollary 5.13, it is clear how the above
calculation should be globalized to a general toric surface, with several
nodal trades. The following result, while known to experts, does not currently
seem to appear in the literature.

######  Proposition 5.14.

Let $\overline{X}_{\Delta}$ be a toric surface with moment polygon $\Delta,$
let $U$ be the result of applying nodal trades at vertices
$p_{1},\ldots,p_{r}$ and deleting the preimage of $\partial\Delta,$ and write
$\mu^{\prime}:U\to\mathring{\Delta}$ for the resulting almost toric fibration
on $U$. Suppose that there is a point $q\in\mathring{\Delta}$ lying on the
monodromy eigenlines of all singularities in $\mathring{\Delta}.$ Then $U$
admits a Weinstein structure with a skeleton a union of the torus
$(\mu^{\prime})^{-1}(q)$ and the visible disks projecting to intervals between
$q$ and the singularities of $\mathring{\Delta}.$ Each visible disk attaches
to $(\mu^{\prime})^{-1}(q)$ along the circle dual to its eigenline.

###### Proof.

As in Lemma 5.10, study the function $\|\mu^{\prime}-q\|,$ which is Morse-Bott
with minimum manifold $(\mu^{\prime})^{-1}(q)$ and $r$ index-2 Morse critical
points, and strictly plurisubharmonic away from the Morse critical points.
Performing the local modification of Lemma 5.10 near each of these critical
points produces a Kähler potential with the desired properties. ∎

######  Corollary 5.15.

Let $\overline{X}_{\Delta},\mathring{\Delta},U,q$ be as in Proposition 5.14,
and let $v_{1},\ldots,v_{r}$ be the directions of rays from $q$ to the
singularities in $\mathring{\Delta}.$ Then $U$ is isomorphic to the Weinstein
manifold described in Construction 4.1.

######  Example 5.16.

Let $\overline{X}_{\Delta}=\operatorname{Bl}_{0}\mathbb{C}^{2}$ and perform
nodal trades at both vertices as in Example 5.4, so that $U$ is the result of
deleting a smoothing of the toric boundary of $\overline{X}_{\Delta}.$ Then
the Weinstein manifold $U$ admits a Lagrangian skeleton obtained from $T^{2}$
by attaching disks to homology classes $(1,0)$ and $(0,1),$ as in Figure 12.

Figure 12. The base projection of the skeleton (in blue) of the Weinstein
manifold obtained from $\operatorname{Bl}_{0}\mathbb{C}^{2}$ by deleting a
smoothing of its toric boundary. Compare the image of the skeleton itself, as
depicted in Figure 6, and the B-side SYZ base, as depicted in Figure 2.

######  Remark 5.17.

These diagrams are also useful for understanding cluster mutation. As
described in [Via16, PT20], the disk surgery can be understood via the nodal
slides of [Sym03]: let $\mu^{\prime}:U\to\mathring{\Delta}$ be an almost toric
fibration as in Proposition 5.14, let $q\in\mathring{\Delta}$ be as described
there, and let $x\in\mathring{\Delta}$ be a singularity. Then $\mu^{\prime}$
can be modified by sliding $x$ along its eigenline until it passes $q.$ As $x$
passes $q$, the disk in the Lagrangian skeleton attached to the torus fiber is
collapsed and then reexpanded precisely according to the local model described
at Construction 8 above. See [PT20, Section 5.3] for more details.

###  5.3. Nodal trades in higher dimensions

Finally, we would like to generalize the above results to the case of higher-
dimensional nodal trades defined in Definition 5.5.

Let $\overline{X}_{\Delta}$ be a toric variety, now of arbitrary dimension,
with moment polytope $\Delta.$ As in Proposition 5.14, we would like to
perform nodal trades (in the sense of Definition 5.5) along some codimension-2
faces $F_{1},\ldots,F_{r}$ of $\Delta,$ an then delete the smoothed boundary
divisor to obtain a space $U$ with singular toric fibration
$\mu^{\prime}:U\to\mathring{\Delta}.$ However, in this case we have the
additional complication that the faces $F_{i}$ may intersect, in which case
the toric fibration $\mu^{\prime}$ may have more complicated singularities
than those considered so far.

######  Example 5.18.

Let $\overline{X}_{\Delta}=\mathbb{C}^{2}\times\mathbb{C}^{2},$ and let
$F_{1},F_{2}$ be the faces corresponding to toric strata
$\\{(0,0)\\}\times\mathbb{C}^{2}$ and $\mathbb{C}^{2}\times\\{(0,0)\\},$ so
that $U=(\mathbb{C}^{2}\setminus\\{xy=1\\})^{2}.$ Thus the base of the
fibration $\mu^{\prime}$ will have two planes of focus-focus singularities as
before, but also a product singularity at the intersection point of those
planes. The space $U$ so constructed is mirror to the actual cluster variety
$(\mathbb{C}^{2}\setminus\\{xy=1\\})^{2},$ rather than the truncated cluster
variety discussed in Example 2.14.

As the above makes clear, repeating the construction of Proposition 5.14
without change will produce the mirror to an actual rather than a truncated
cluster variety. This is desirable, and we will return to it in future work,
but for the purposes of this paper, we will prefer to ignore these deeper
singularities, which we can do by passing to a Weinstein subdomain
$U^{\prime}\subset U$ defined by preimage under $\mu^{\prime}$ of a region
$\mathring{\Delta}^{\prime}\subset\mathring{\Delta}$ obtained by excising
neighborhoods of those singularities in the integral affine base.

######  Proposition 5.19.

As above, let $U$ be obtained from a toric variety $\overline{X}_{\Delta}$ (of
arbitrary dimension) by smoothing the boundary divisor along codimension-2
faces $F_{1},\ldots,F_{r}$ of $\Delta$ and deleting it, and let
$U^{\prime}\subset U$ be a Weinstein subdomain as above, equipped with
restricted fibration $\mu^{\prime}:U\to\mathring{\Delta}^{\prime}.$

Suppose that there exists a point $q\in\mathring{\Delta}^{\prime}$ lying in
all of the eigenhyperplanes of these nodal trades. Then $U^{\prime}$ is
isomorphic to the Weinstein manifold described in Construction 4.7, where for
the fan $\Sigma$ we take the rays beginning at $q$ and pointing toward the
singularity loci in their respective eigendirections, and the cocharacters
$\chi_{i}$ arise from the defining equations of the respective singularity
eigenhyperplanes.

###### Proof.

The proof is the same as it was for the 2-dimensional case, covered in
Corollary 5.15: the singular toric fibration
$\mu^{\prime}:U^{\prime}\to\mathring{\Delta}^{\prime}$ can be used to produce
a function $\|\mu^{\prime}-q\|,$ and (the product of
$(\mathbb{C}^{\times})^{n-2}$ with) the local modifications of this function
described in 5.10 produce a Kähler potential, whose Lagrangian skeleton is
obtained from the torus $(\mu^{\prime})^{-1}(q)$ by Weinstein handle
attachments precisely as described in Construction 4.7. ∎

######  Remark 5.20.

Rather than passing directly to a Weinstein subdomain $U^{\prime}\subset U$ as
above, one could also work throughout with $U$ but keeping track of which
critical points of the Kähler potential correspond to the deeper singularities
in the integral affine base. These critical points correspond to certain
pieces in the skeleton of $U$, and by throwing out those pieces one obtains
the skeleton of the subdomain $U^{\prime}.$

We conclude with a comparison of this singular torus fibration with the
singular torus fibration on the mirror truncated cluster variety.

######  Corollary 5.21.

Let $U$ be a truncated cluster variety, and $U^{\vee}$ be its mirror Weinstein
manifold. Suppose $U^{\vee}$ is obtained by deleting a smoothing of a toric
boundary divisor and passing to a subdomain as in Proposition 5.19. Then the
spaces $U,U^{\vee}$ admit dual singular torus fibrations over the same base
$\mathring{\Delta}$.

###### Proof.

If $U$ is a mirror to $U^{\vee}$ with fan $\Sigma,$ then by construction, the
integral affine base of Proposition 5.19 encodes precisely the set of nodal
trades encoded by the fan $\Sigma,$ as described in Section 2.2. The torus
fibrations of $U$ and $U^{\vee}$ are dual, since the integral affine structure
on $\mathring{\Delta}$ encodes the character lattice of torus fibers in $U$
(as described in Section 2.2) or the cocharacter lattice of torus fibers in
$U^{\vee}$ (as described in this section). ∎

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