Mirror Symmetry for Truncated Cluster Varieties
In the algebraic setting, cluster varieties were reformulated by Gross-Hacking-Keel as log Calabi-Yau varieties admitting a toric model. Building on the work of Shende-Treumann-Williams-Zaslow in dimension 2, we describe the mirror to the GHK construction in arbitrary dimension: given a truncated cl...
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| Цитувати: | Mirror Symmetry for Truncated Cluster Varieties. Benjamin Gammage and Ian Le. SIGMA 18 (2022), 055, 30 pages |
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| author | Gammage, Benjamin Le, Ian |
| author_facet | Gammage, Benjamin Le, Ian |
| citation_txt | Mirror Symmetry for Truncated Cluster Varieties. Benjamin Gammage and Ian Le. SIGMA 18 (2022), 055, 30 pages |
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| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | In the algebraic setting, cluster varieties were reformulated by Gross-Hacking-Keel as log Calabi-Yau varieties admitting a toric model. Building on the 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.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 055, 30 pages
Mirror Symmetry for Truncated Cluster Varieties
Benjamin GAMMAGE a and Ian LE b
a) Department of Mathematics, Harvard University, USA
E-mail: gammage@math.harvard.edu
b) Mathematical Sciences Institute, Australian National University, Australia
E-mail: ian.le@anu.edu.au
Received August 25, 2021, in final form July 15, 2022; Published online July 19, 2022
https://doi.org/10.3842/SIGMA.2022.055
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.
Key words: homological mirror symmetry; cluster varieties; almost toric fibrations
2020 Mathematics Subject Classification: 53D37; 13F60
1 Introduction
Cluster varieties are algebraic varieties which are glued together from torus charts using bira-
tional maps that are called cluster transformations. They are described by starting with a seed, 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, X and A, which have the same seed data, but use
somewhat different birational transformations to glue seed tori. For each seed s, there is a natu-
ral dual seed s∨. Fock and Goncharov conjectured that the cluster varieties X (s) and A
(
s∨
)
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 X (s) and A
(
s∨
)
.
In [16, 17], 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 P1-
bundles. This allowed them to give an algebro-geometric construction of cluster varieties: begin
with a toric variety 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 näıve result of performing this blow-up-and-delete operation described
in [16] 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.
mailto:gammage@math.harvard.edu
mailto:ian.le@anu.edu.au
https://doi.org/10.3842/SIGMA.2022.055
2 B. Gammage and I. Le
(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 X, perform Weinstein handle attachments as
indicated by the seed data. Using recent advances [12, 13, 14, 23] 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
[27, 28], subsequently made explicit in [18]; 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” [29]. The local model for this
move, its relation to cluster theory, and its higher-dimensional analogues have been previously
studied [22, 26, 30]. 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 [27], which studied cluster structures inside the Fukaya
category of a Weinstein 4-manifold W obtained from a cotangent bundle T ∗Σ by Weinstein
handle attachments. In the case when Σ = 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 W(W ) can be understood as the category of coherent
sheaves Coh
(
W∨) 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 obser-
vation is that if W∨ is mirror to a Weinstein manifold obtained from the handle attachments
described in [27], the blow-up-and-delete operations of W∨ are in bijection with the handle
attachments of W .
We prove that in arbitrary dimensions, the basic blow-up-and-delete operation of [16] is mirror
to a basic Weinstein disk attachment. A seed s, which in [16] is treated as a recipe for producing
a cluster variety U , we treat as a recipe for producing a Weinstein manifold U∨. 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 Coh(U) ∼= W
(
U∨) between the category
of coherent sheaves on the truncated cluster variety U and the wrapped Fukaya category of the
Weinstein manifold U∨.
We also mention that cluster mutations can be understood in our framework, referring to
[22, 26, 27] 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 Relation to [18]
In [18], Paul Hacking and Ailsa Keating produced an equivalence similar to the one we describe
here, in the special case where the spaces involved are 2-dimensional. As we do, they follow [16]
in beginning with a log Calabi–Yau pair (Y,D) obtained from certain non-toric blowups of
Mirror Symmetry for Truncated Cluster Varieties 3
a toric pair, which they require to be 2-dimensional. (Our U is the complement Y \D.) They
associate to this pair both a spaceM which is mirror to Y \D together with a Lefschetz fibration
f : M → C which is mirror to the compactification Y ⊃ Y \ D, and they prove a homological
mirror symmetry equivalence in each case, which we may summarize as follows:
Theorem ([18]). There is an equivalence of dg categories Coh(Y \ D) ∼= W(M) between the
category of coherent sheaves on the cluster variety Y \ D and the wrapped Fukaya category
of M . In addition, there is an equivalence Coh(Y ) ∼=W(M,f) between coherent sheaves on the
compactification and the Fukaya–Seidel category of the mirror Lefschetz fibration.
In this 2-dimensional setting, it would be useful to compare the mirror M described in [18]
to the space U∨ described in this paper. We believe it should be straightforward to check
that the Lefschetz fibration on M describes the same sequence of Weinstein handle attachments
by which we have described U∨. Their work then also indicates the changes which would
need to be made to our constructions in order to construct a mirror to the compact variety Y
(rather than just the open part Y \ D). A detailed comparison of these constructions and
the relevant homological mirror symmetry computations are interesting questions for future
work.
Conversely, our work also suggests a generalization of the results of [18], since in this paper
we allow ourselves to work in arbitrary dimensions rather than just dimension 2. Indeed, the
main idea of this paper is that so long as we restrict ourselves on the B-side to the setting
of truncated cluster varieties, there are no new phenomena besides those which have already
appeared in dimension 2. It would be very interesting to know whether this insight can be
applied to produce mirror Lefschetz fibrations in higher dimensions as well.
1.3 Future questions
As mentioned above, we restrict ourselves in this paper to varieties obtained from the construc-
tion of [16], 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 repre-
sentation theory and holomorphic symplectic geometry; in the toric hyperkähler setting, these
skeleta have been used to prove mirror-symmetry results in [10, 21].
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∨ encodes the Reeb dynamics on the contact bound-
ary of U∨. In [25], an analysis of Reeb dynamics for U∨ 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.
4 B. Gammage and I. Le
Notation
The following notation will be used throughout the paper.
� We pick dual lattices N , M , and we write NR, NC for N ⊗Z R, N ⊗Z C, and similarly
for M .
� We write NS1 , NC× for the real and complex tori NR/N , NC/N , and similarly forM . Note
that NC× = NR/N ×NR. 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 TC, respectively.
� We write Σ for a stacky fan of rays in NR. “Stacky” means that we keep track not only of
the rays ⟨ψ1⟩, . . . , ⟨ψr⟩, but of not necessarily primitive generators {diψi}ri=1, where ψi is
primitive and di ∈ Z>0.
� We denote the toric stack with fan Σ by XΣ. It has toric divisor D and dense open torus
TC := NC× , which has cocharacter lattice N and character lattice M .
Categorical conventions
Throughout this paper, we work with stable∞-categories over a field k of characteristic 0, which
we model as dg categories (so that, for instance, 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.
2 Cluster varieties from non-toric blowups
In this section, we review the Gross–Hacking–Keel construction of cluster varieties, following [19]
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 KX +D = 0. In this case we say that (X,D) is a log Calabi–Yau
pair and U = X \ D is a log Calabi–Yau variety. We write Ω 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 ′, D′) 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 ′ 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
(
X,D
)
be an n-dimensional toric variety with its toric boundary divisor.
Then
(
X,D
)
is a log Calabi–Yau pair with U = X \D ∼= (C×)n, Ω = dx1
x1
∧ · · · ∧ dxn
xn
.
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.
Mirror Symmetry for Truncated Cluster Varieties 5
Example 2.4. Let Σ ⊂ NR be a fan of cones of dimension at most 1. In other words, Σ contains
the zero cone and the rays generated by primitive vectors ψ1, . . . , ψr ∈ N . Then
(
XΣ, D
)
is a log
Calabi–Yau pair where D =
⊔r
i=1Di is a disjoint union of copies of (C×)n−1. More precisely,
the component Di corresponding to ray ⟨ψi⟩ is isomorphic to the quotient torus (N/ψi)C× . Note
that the character lattice of the torus Di is therefore the orthogonal ψ⊥
i ⊂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 Σ to be a toric stack.
For information on toric stacks, we refer to [4]. In the setting of Example 2.4, this means only
that we equip Σ with a choice of not necessarily primitive generator of each ray, or equivalently
that to each primitive generator ψi, we associate a positive integer di. The resulting toric
stack XΣ has coarse moduli space the usual toric variety associated to Σ, but XΣ has a Z/di
isotropy group along the divisor Di.
The discovery of [16, 17] is that log Calabi–Yau varieties related to the previous example in
a simple way possess cluster structure.
Construction 2.6. Let
(
XΣ, D
)
be an n-dimensional toric pair as in Example 2.4, and choose
a character χi ∈ ψ⊥
i for each component Di of D. Let H ⊂ D be the subvariety whose intersec-
tion with the component Di is equal to the linear subspace {χi = −1}. We write (X,D) for the
blowup of X along H together with the strict transform of D.
Remark 2.7. The construction of [16] more generally allows components of H to be defined by
an equation of the form χi = λ for any λ ∈ C×. We prefer to restrict to the case of λ = −1, since
this is a paper about exact symplectic geometry, and a mirror to the variety X obtained for
more general λ is expected to be non-exact. (See [18, 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 λ,
so this issue will not arise explicitly in this paper.
Observe that if the divisor D has a single component, then we can choose coordinates so that
X = (C×C×)× (C×)n−2 and H = {(0,−1)}× (C×)n−2, so that the space X resulting from the
GHK construction is
X = Bl{(0,1)}(C× C×)× (C×)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 X = C×C× and H = {(0,−1)}, as above. Then we can describe the blowup
X = BlH X as embedded in C× C× × P1 as
X ∼= {(x, y), [z : w] | xz − (y + 1)w = 0} ⊂ C× C× × P1.
The strict transform D of the toric boundary divisor D = {0} ×C× contains the point [1 : 0] in
the exceptional P1, and hence the variety U = X \D is completely contained in the affine chart
C× C× × Cz:
U ∼= {(x, y, z) | xz = y + 1} ⊂ C× C× × C.
If desired, we can also eliminate the variable y to describe U as the complement
U = C2 \ {xz = 1}.
6 B. Gammage and I. Le
Definition 2.9. Let U = X \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)→
(
X,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 χi ∈ ψ⊥
i 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 → X and the blow-up map X → X defines a birational
morphism U → X. The inverse birational map is well-defined on the dense torus T ⊂ X,
giving an open inclusion T ↪→ U , defining the toric chart on U corresponding to toric model
(X,D)→
(
X,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 s is the data of
� the lattice N ;
� a skew bilinear form σ ∈ ∧2M ;
� a basis {ψ1, . . . , ψr, ψr+1, . . . , ψn} of N , along with a designated subset {ψ1, . . . , ψr} of
unfrozen basis vectors;
� positive integers di, which can be used to modify σ to a non-skew-symmetric form ϵ, defined
by ϵ(ψi, ψj) = σ(ψi, ψj)dj =: ϵij . We say that ϵ is skew-symmetrizable.
One of the main insights of [16] 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 s be a seed as in Definition 2.11.
� The unfrozen vectors ψ1, . . . , ψr determine a fan of rays Σ in N , which can be upgraded to
a stacky fan by specifying diψi as generators of the rays in Σ. This information determines
a toric stack XΣ with dense torus TC = NC× and boundary divisors Di = (N/ψi)C×
×BZ/di.
� For i ≤ r, The pairing χi := σ(ψi,−) determines an element χi ∈ M , which by skew-
symmetry of σ is contained in ψ⊥
i and hence can be treated as a character of the (n− 1)-
torus Di. We write Hi = {χi + 1 = 0}.
� The skew bilinear form σ ∈ ∧2M ≃ H0
(
Ω2
X
(
logD
))
determines a holomorphic 2-form
on X with simple poles along D. As explained in [19, Lemma 3.4], the definition of χi
ensures that σ extends to a form in H0
(
Ω2
X(logD)
)
.
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 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, X and A. Cluster A-varieties
are quasi-affine; from our point of view, the correct variety to consider is the affine closure of
the cluster A-variety.
The variety U constructed above is, up to codimension 2, a cluster 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.
Mirror Symmetry for Truncated Cluster Varieties 7
Example 2.14. Let s be the seed data corresponding to the variety U = C2 \ {xz = 1} from
Example 2.8, and write s̃ for the product s × s of two copies of this seed data. The cluster
variety associated to s̃ ought to be
Ũ := U × U = C4 \ {(x1z1 − 1)(x2z2 − 1) = 0},
but the blow-up-and-delete construction described above produces instead the space
Ũ \ {(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 s as in Definition 2.11 and a choice of unfrozen basis
vector ψk, the mutation at ψi is a new seed µis, produced by changing the basis {ψi} to the
basis {ψ′
i} determined by the formula
ψ′
i =
{
ψi + [ϵik]+ψk for i ̸= k,
−ψk for i = k,
where we define [r]+ = max(0, r).
The geometry underlying mutations is described in [16, Section 3.1], which we can summarize
as follows:
Proposition 2.16 ([16]). Let (X,D) →
(
X,D
)
be a log Calabi–Yau variety with toric model
induced from a seed s as in Construction 2.12. Given a choice of an unfrozen vector ψi in the
seed, an elementary transformation of P1-bundles induces a new toric model (X ′, D′)→
(
X
′
, D
′)
,
which corresponds under Construction 2.12 to the mutated seed µis. The induced map of tori
TC → T
′
C is the corresponding cluster transformation.
The local model for this elementary transformation of P1-bundles is described in the following
example.
Example 2.17. Let
X = P1
[z0:z1]
× C×
w ⊃ D = {z0z1 = 0} = ({0} ∪ {∞})× C×,
and let H = {z0 = 0, w + 1 = 0}, so that
(
X,D
)
is a toric model for the log Calabi–Yau pair(
X = BlH X,D = D̃
)
discussed in Example 2.8. Let H̃ be the strict transform of the line
{w + 1 = 0} ⊂ X. Then the blowdown of H̃ can be used to define a new toric model
f : (X,D)→
(
X
′
, D
′) ∼= P1 × C×,
where now the new blowup locus H ′ ⊂ D
′
lives in {∞} × C× rather than {0} × C×. 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 des-
cribed (at least in dimension 2) by means of the toric base diagrams from [29], 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,
8 B. Gammage and I. Le
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 (C×)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 TC =
X \ 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 ([19, Example 4.4]). Consider again the basic local model described in Ex-
ample 2.8, where the log Calabi–Yau pair (X,D) is obtained from toric model
(
X,D
)
=
(C× C×, {0} × C×) by blowing up at center H = {(0,−1)}.
From the perspective of the moment-map base R≥0×R of 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 \ 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 R2.
Figure 1. The process of producing the base for a torus fibration with singularity on the blowup
Bl(0,1)(C× C×), 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 Σ
in a toric model, since each blowup in Construction 2.6 has a local model as in Example 2.18,
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 Σ be a fan of rays in R2. The SYZ base B associated to Σ is the
integral affine manifold which topologically is R2, but with singularities in the integral affine
structure as follows: for each ray v =
(
ψ1
ψ2
)
in Σ, there is a singularity at a point pv on v,
with monodromy matrix
(
1+ψ1ψ2 −ψ2
1
ψ2
2 1−ψ1ψ2
)
. After taking a branch cut at each pv pointing in
direction v, B can be embedded in R2 in the obvious way, as illustrated in Figure 2.
Proposition 2.20. Let U be the cluster surface with toric model
(
XΣ, D
)
. Then the integral
affine structure on the base B constructed in 2.19 is induced by a Lagrangian torus fibration
U → B with focus-focus singularities above the singularities of B.
Proof. Let ∆ be the moment polytope of XΣ, so that we can begin with the usual moment
fibration XΣ → ∆. 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 ∆, and that each such modification is given by the local model of
Example 2.18, modified by an integral affine-linear change of coordinates A in SL(2,Z). If ( 10 )
Mirror Symmetry for Truncated Cluster Varieties 9
Figure 2. The fan Σ for C2 \ {(0, 0)} superimposed on the SYZ base for the A2 cluster variety, where
the dashed lines emanating from singularities represent the branch cuts.
is an eigenvector for the monodromy matrix of a singularity, and the transformation A takes it
to
(
ψ1
ψ2
)
, then A must conjugate the monodromy matrix ( 1 1
0 1 ) to
(
1+ψ1ψ2 −ψ2
1
ψ2
2 1−ψ1ψ2
)
. ■
Remark 2.21. An analogue of the above proposition holds in higher dimensions, where now
we replace the local model of Example 2.18 by its product with (C×)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 Rn−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 Proposi-
tion 2.16.
Example 2.22. Consider again the local model for an elementary transformation of P1-bundles
described in Example 2.17, where X is obtained from X, X
′
= P1 ×C× by non-toric blowup at
a single point in the toric boundary divisor {0} × C× (respectively {∞} × C×). 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 R2.
Figure 3. SYZ bases for Bl(0,−1) P1 × C× and Bl(∞,−1) P1 × C×, respectively.
3 Symplectic geometry background
3.1 Liouville sectors
We begin by recalling some basic facts about Weinstein manifolds and Liouville sectors from
[5, 14].
Definition 3.1. A Liouville domain is a compact symplectic manifold (X,ω) with boundary
together with a choice of primitive ω = dλ such that λ|∂X is a contact form, or equivalently such
that the Liouville vector field V = ω−1(λ) is outwardly transversal to the boundary. A Liouville
domain X can be completed to a Liouville manifold X̂ by attaching a cylindrical end:
X̂ = X ∪ (∂X × [0,∞)),
where we extend λ to X̂ as es(λ|∂X) at the end.
10 B. Gammage and I. Le
As Liouville domains and Liouville manifolds can be obtained from each other, we will not
be careful about distinguishing them. Features of the boundary ∂X (for instance, Weinstein
hypersurfaces) can be equivalently described in terms of the ideal contact boundary of X̂.
Definition 3.2. The Liouville domain (X,ω, λ) is in addition Weinstein if it can be equipped
with an exhausting Morse–Bott function f : X → 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
λ = dcf .
Weinstein manifolds are so named in honor of the discovery, in [31], 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 ∗S1 = S1 × R is a Liouville manifold obtained as a completion
of the Liouville domain S1 × [−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 [14, 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 [5] of a Weinstein
pair.
Definition 3.4. A Weinstein pair is the data of a Weinstein domain (X,ω, λ) together with
a real hypersurface Σ ⊂ ∂X such that (Σ, dλ|Σ, λ|Σ) is itself a Weinstein manifold.
A Weinstein pair (X,Σ) gives rise to a Liouville sector in the sense of [14] by completing X
away from a standard neighborhood of Σ. Thus Σ becomes the boundary of the resulting
Liouville sector, along which we can perform gluings:
Construction 3.5. Let (X1,Σ1) and (X2,Σ2) be two Weinstein pairs, together with an iso-
morphism of Weinstein manifolds Σ1
∼= Σ2. Then the Weinstein manifolds X1 and X2 can be
glued together along a neighborhood of Σi to produce a new Weinstein manifold X1 ∪Σ X2. If
the boundary Σi 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 [12], 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 L ⊂ X, the skeleton of X.
Definition 3.7. For X a Weinstein manifold, the skeleton of X is the stable set
L :=
{
x ∈ X | lim←−
t→∞
ϕt(x) ∈ X
}
consisting of all points x ∈ X whose image under the flow ϕt of the Liouville vector field
eventually converges.
Mirror Symmetry for Truncated Cluster Varieties 11
For X a Liouville sector, we allow L to contain all those points which flow to the boundary
of X. In particular, for (X,Σ) a Weinstein pair, the skeleton L(X,Σ) of the associated Liouville
sector is the union
L(X,Σ) = LX ∪ Cone(Λ),
of the skeleton of X with the cone, under the Liouville flow, of the skeleton Λ of Σ.
Hence the skeleton of a Weinstein gluing X1 ∪Σ X2 is the gluing of the skeleta of the corre-
sponding sectors along their glued boundary:
LX1∪ΣX2 = LX1 ∪LΣ
LX2 .
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 π : X → C a global holomorphic func-
tion. We can treat a general fiber π−1(1) as a Weinstein hypersurface inside the contact boun-
dary ∂∞X and hence
(
X,π−1(1)
)
as a Weinstein pair. We will often denote the associated
Liouville sector by (X,π).
3.2 Fukaya categories and descent
In [14], a wrapped Fukaya category 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,π), this is the Fukaya–Seidel-type category associated
to the fibration π. We refer to [12, 13, 14] for details of this construction, and we summarize
here its most important properties.
Proposition 3.9 ([14]). The wrapped Fukaya category is covariantly functorial for inclusions
of Liouville sectors: a Liouville subsector i : X ⊂ Y induces a functor i : W(X)→W(Y ).
The central result of the series [12, 13, 14] 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 ([12]). The wrapped Fukaya category satisfies descent for Weinstein sectorial
covers: Let U1, . . . , Un ⊂ X be Liouville subsectors of X which cover X whose intersections
UI :=
⋂
i∈I Ui, for I ⊂ [n], are also Liouville sectors, and let P be the poset of inclusions of the
subsectors UI . Then the natural map
lim−→
I∈P
W(UI)→W(X)
from the homotopy colimit of the wrapped Fukaya categories of the UI 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 Ui 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 compu-
tations is the following (described for instance at [9, 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 Ui and X be as in Theorem 3.10, and write W∞ for the ind-completion
of the wrapped Fukaya category. Then the natural map
W∞(X)→ lim←−
I∈Pop
W∞(UI)
is an equivalence. The wrapped Fukaya category W(X) can therefore be recovered as the category
of compact objects inside the homotopy limit lim←−I∈PopW∞(UI).
12 B. Gammage and I. Le
This is particularly useful in case each Fukaya category W(UI) is equivalent to the category
Coh
(
U∨
I
)
of coherent sheaves on a smooth algebraic variety U∨
I , in which case the ind-completed
category W∞(UI) ∼= IndCoh
(
U∨
I
)
will be equivalent to the category QCoh(UI) of quasicoherent
sheaves on UI .
The computations of the Fukaya categories W(UI) of individual subsectors can be accom-
plished in the case when the subsectors UI 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 ([13, Theorem 1.1]). Let (T ∗M,Σ) be a Liouville sector, and Λ ⊂ S∗M the
skeleton of Σ inside the contact boundary of T ∗M . Write Sh∞−Λ(M) for the cocomplete dg
category of (possibly infinite-dimensional) sheaves on M whose singular support away from the
zero section is contained in −Λ. Then there is an equivalence of categories
Sh∞−Λ(M) ∼=W∞(T ∗M,Σ)
between this category of sheaves and the Fukaya category of the Liouville sector associated to
(T ∗M,Σ).
If we write Shw−Λ(M) (for “wrapped constructible sheaves”) for the category of compact objects
in Sh∞−Λ(M), then we obtain an equivalence
Shw−Λ(M) ∼=W(T ∗M,Σ).
Remark 3.13. Note that the category Shw−Λ(M) is not in general equivalent to the usual
category of constructible sheaves on M with singular support in −Λ, since it includes some
objects whose stalks are not finite-dimensional. For instance, if M = S1 and Λ is empty, then
Shw−Λ(M) includes the universal local system k[x±] on S1.
Remark 3.14. The minus sign in the theorem, denoting the anti-symplectic involution (−):
T ∗M → T ∗M which acts on the fibers as scaling by (−1), appears by convention in order to fix
the fact that Sh∞Λ (M) is actually isomorphic not to the Fukaya category W∞(T ∗M,Σ) 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 [20], summarized in Theorem 3.21 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,Σ), the
above theorems show that there exists a cosheaf of categories on the skeleton LX of X, locally
isomorphic to Shw−Λ(M), whose homotopy colimit recovers the wrapped Fukaya category of X.
We denote this cosheaf by µShw, and call it the cosheaf of wrapped microlocal sheaves along LX .
In fact, this cosheaf has now beeen constructed in [23] 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 [1].
Proposition 3.16 ([12, Section 8.3]). Let Y be a Liouville sector, and X ⊂ Y a Liouville
subdomain. Then there is a restriction functor W(Y )→W(X).
Mirror Symmetry for Truncated Cluster Varieties 13
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 XΣ be a toric variety with fan Σ of cones in NR. Let ∆ ⊂ NR be the
convex hull of the generators of the rays in Σ, and let WΣ : MC× → C be a Laurent polynomial
with Newton polytope ∆. The Landau–Ginzburg model (MC× ,W ) is a Hori–Vafa mirror to XΣ.
The skeleton of the Liouville sector (MC× ,W ) (though not originally recognized as such)
appeared implicitly in work of Bondal [3] and was studied extensively by Fang–Liu–Treumann–
Zaslow [6, 7, 8].
Definition 3.18. Let Σ be a fan of cones in NR, and note that we can treat NR as the cotangent
fiber of the cotangent bundle T ∗MS1 =MR/M×NR. The (negative) Bondal Lagrangian is a conic
Lagrangian LΣ ⊂ T ∗MS1 defined by
LΣ =
⋃
σ∈Σ
σ⊥ × σ ⊂MR/M ×NR, (3.1)
where we write σ⊥ for the image of the linear subspace σ⊥ ⊂MR in the quotient MR/M .
Example 3.19. Let dimn = 2 and Σ be the fan whose only nonzero cone is the ray R⟨e1⟩. Then
the Lagrangian L is a torus with a cylinder attached, as in Figure 4. The torus corresponds in
the decomposition of equation (3.1) to the zero cone in Σ, while the cylinder corresponds to the
cone R⟨e1⟩.
Figure 4. The skeleton LΣ for Σ the fan of C × C×, given as the union of a torus (in black) with
a cylinder (in red).
As mentioned in Remark 2.5, in the case of a cluster seed whose inner-product σ is skew-
symmetric but whose associated form ϵ is only skew-symmetrizable (i.e., when not all the inte-
gers di are equal to 1), we need to scale some of the generators of rays in Σ, making XΣ into
a toric stack. In this case, we need to make a slight modification of the above definition. For σ
a cone in Σ, write σZ := σ∩N for the sublattice defined by σ, and note that the tori σ⊥ are the
Pontrjagin duals of the quotient lattices N/σZ:
σ⊥ = Hom(N/σZ,R/Z). (3.2)
Suppose now that Σ is a stacky fan, so that the rays in Σ are equipped with not necessarily
primitive generators. If σ is a cone on rays equipped with not necessarily primitive generators
β1, . . . , βk, write σZ for the sublattice ⟨β1, . . . , βk⟩ ⊂ N . In analogy with equation (3.2), we define
a not necessarily connected torus Gσ as the Pontrjagin dual
Gσ = Hom(N/σZ,R/Z).
14 B. Gammage and I. Le
The group π0(Gσ) of connected components of Gσ takes into account the isotropy along the toric
stratum corresponding to cone σ. The definition of the Lagrangian in the stacky case proceeds
precisely in analogy to Definition 3.18:
Definition 3.20. Let Σ be a stacky fan of cones σ, and define the groups Gσ as above. Then
the stacky Bondal Lagrangian is defined as
LΣ =
⋃
σ∈Σ
Gσ × σ ⊂MR/M ×NR.
The motivation for studying the Lagrangian LΣ was the following result, studied by Bondal
and FLTZ and eventually proved in full generality by Kuwagaki (later reproved in many cases
in [33]):
Theorem 3.21 ([20]). There is an equivalence of dg categories Shw−LΣ
(MS1) ∼= Coh
(
XΣ
)
between
the category of wrapped constructible sheaves on the torus MS1 with singular support in −LΣ
and the category of coherent sheaves on the toric stack XΣ.
The theory of Liouville sectors has made it possible to relate the Lagrangian LΣ to the
Hori–Vafa Landau–Ginzburg model and hence to reframe the above in more traditional mirror-
symmetric language.
Theorem 3.22 ([11, 34]). Let ∆ ⊂ NR be a polytope containing 0, and let Σ be the fan of cones
on the faces of ∆. As above, let W be a Laurent polynomial MC× → C with Newton polytope ∆.
Then LΣ is the Lagrangian skeleton of the Liouville sector associated to (MC× ,W ).
Corollary 3.23. The dg category Shw−LΣ
(MS1) is equivalent to the wrapped Fukaya cate-
gory W(MC× ,W ) of the Liouville sector associated to the Hori–Vafa Landau–Ginzburg model
(MC× ,W ).
However, Theorem 3.22 as stated does not apply to our situation, since the hypothesis of
the theorem requires that Σ contain higher-dimensional cones, corresponding to the positive-
dimensional faces of ∆, whereas we are interested in fans which contain only one-dimensional
cones.
Nevertheless, in the case of a purely 1-dimensional fan Σ, the Lagrangian LΣ 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
(
(C×)n,
⊔r
i=1(C×)n−1
)
– and Theorem 3.21 remains true in this case, so that we must continue to understand this sector
as the mirror to the (no longer complete) toric variety XΣ. We are therefore justified in the
following notation:
Definition 3.24. We write X
∨
Σ for the Liouville sector with Lagrangian skeleton LΣ.
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 [27].
Construction 4.1. Let T = T 2 be a 2-torus and ψ1, . . . , ψr : T → S1 characters of T , defining
codimension-1 subtori S1, . . . , Sr. A choice of coorientations of the subtori Si determines lifts
of the Si to Legendrian circles Li 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 Li.
Mirror Symmetry for Truncated Cluster Varieties 15
Example 4.2. Let r = 1 and let ψ = ψ1 be the character given in coordinates by (θ1, θ2) 7→ θ1.
Then the Weinstein manifoldW has skeleton given by the union of the torus T and a Lagrangian
disk glued to T along S = {ψ = 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 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 = Z2, T = MS1 , and note that each character ψi is a vector in the dual vector space NR;
hence we can think of the collection of characters ψi as a purely one-dimensional fan Σ in NR.
Now the following is an immediate consequence of Definition 3.24:
Lemma 4.3. The Liouville sector determined by the pair
(
T ∗M,
⊔
Li
)
is the Liouville sector X
∨
Σ
described in Definition 3.24, the mirror to the (incomplete) toric variety XΣ.
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 ⊂ D an annulus, the Liouville sector T ∗D admits T ∗A as
a subsector.
Proposition 4.4. Let W and Σ as above. Then W admits a cover by the Liouville sectors X
∨
Σ
and
⊔r
i=1 T
∗D, which intersect in
⊔r
i=1 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 [27]. We now generalize
to higher dimensions by taking a product of the local model for the above disk gluing with
T ∗Tn−2 = (C×)n−2.
16 B. Gammage and I. Le
Example 4.5. Let W be the product of (C×)n−2 with the Weinstein manifold studied in Ex-
ample 4.2. Then W can be obtained as follows: starting with an n-torus T = Tn, define
a codimension-1 subtorus S by the character ψ : T → S1 given by (θ1, . . . , θn) 7→ θ1, and use
a choice of coorientation to lift S to a Legendrian L ⊂ S∗T , where now L is an (n − 1)-torus.
Then glue T ∗T to T ∗(D × Tn−2
)
along the Legendrian L.
Note that unlike in the 2-dimensional case, our construction in higher dimensions will not be
determined by the choice of Legendrian L: in addition, we need to know which of the remaining
n− 1 directions contains the disk capping off L. This will be a direction χ in the torus S.
Definition 4.6. Let χ : S1 → Tn−1 be a cocharacter of an (n − 1)-torus Tn−1, so that Tn−1
can be expressed as a product
(
S1
)
χ
×Tn−2. We write Dχ×Tn−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 MS1, and let Σ be a purely 1-dimensional fan in NR,
defining a collection of cooriented codimension-1 subtori Si ⊂ T , lifting to Legendrians Li ⊂ S∗T .
Moreover, for each Si choose a cocharacter χi : S
1 → Si of the torus Si. Choose a decomposition
Si ≃ S1
χi
× Ti, where Ti is an (n − 2)-torus. Finally, glue a copy of T ∗(Dχi × Ti) along each
Legendrian Li as in Example 4.5. We denote the resulting Weinstein manifold by U∨.
Note that the resulting Weinstein manifold does not depend on the decomposition Si =
S1
χi
× Ti. The key point here is that given two decompositions Si ≃ S1
χi
× Ti ≃ S1
χi
× T ′
i , there
is a canonical isomorphism Dχi × Ti ≃ Dχi × T ′
i .
As in Proposition 4.4, this gluing presentation of the manifold U∨ immediately gives a Liou-
ville-sectorial cover.
Proposition 4.8. The Weinstein manifold U∨ defined in Construction 4.7 is a union of the
Liouville sectors X
∨
Σ and
⊔r
i=1 T
∗(Dχi × Ti), glued along
⊔r
i=1 T
∗(S1
χi
× Ti
)
.
Proof. As in Proposition 4.4, this follows immediately from the construction of U∨: one can ob-
tain U∨ by beginning with sector X
∨
Σ and performing gluings along the r disjoint Legendrians Li
as indicated. ■
Combined with the codescent prescription of Theorem 3.10, this gives a presentation of the
wrapped Fukaya category of U∨.
Corollary 4.9. The wrapped Fukaya category W
(
U∨) of the Weinstein manifold defined by
Construction 4.7 can be presented as a homotopy pushout of the diagram
W
(⊔r
i=1 T
∗(S1
χi
× Ti
))
//
��
W
(⊔r
i=1 T
∗Dχi × Ti
)
W
(
X
∨
Σ
)
,
(4.1)
where the maps are the covariant functorialities of the wrapped Fukaya category under inclusions
of Liouville sectors.
Since the categories and maps in the pushout diagram (4.1) all have well-understood inter-
pretations 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 .
Mirror Symmetry for Truncated Cluster Varieties 17
Recall that construction also began with the construction of a fan Σ ⊂ NR of rays ⟨diψi⟩,
defining a toric variety XΣ with boundary divisor
D =
r⊔
i=1
Di
∼=
r⊔
i=1
(N/ψi)C× ×BZ/di,
along with characters χi of these quotient tori, defining a subvariety
H ∼=
r⊔
i=1
(C×)n−2 ×BZ/di ⊂ D.
We write i : D → XΣ for the inclusion of the toric boundary divisor and j : H → D for the
inclusion of H into D. Now we are ready to restate Corollary 4.9:
Corollary 4.10. The wrapped Fukaya category W
(
U∨) of U∨ can be presented as a homotopy
pushout of the diagram
Coh
(
D
) j∗ //
i∗
��
Coh(H)
Coh
(
XΣ
)
.
(4.2)
Proof. The equivalences between the categories in diagram (4.2) and those in diagram (4.1)
have already been discussed.
(
For the stacky case, where not all di are equal to 1, note that
there is an equivalence of dg categories Coh(BZ/di) ∼= Coh
(⊔di
j=1 Spec(k)
)
.
)
That the vertical
arrows agree is proved in [11, 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 func-
torialities present. Nevertheless, we will see in our main result, Theorem 4.12 below, that this
pushout does indeed present the category of coherent sheaves on the truncated cluster variety
U = X \D.
Before proceeding to that proof, we consider the most basic example.
Example 4.11. Consider again the situation of Example 4.2, where n = 2, and r = 1, so that χ
is the character of T = R2/Z2 given by (θ1, θ2) 7→ θ1. In this case, the diagram (4.2) becomes
Coh({0} × C×) //
��
Coh ({(0,−1)})
Coh(C× C×),
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 Corol-
lary 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
k[x, y±]/(x) -mod k[x, y±]/(x, y + 1) -modoo
k[x, y±] -mod
OO
(4.3)
18 B. Gammage and I. Le
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 push-
forward, is the shriek pullback functor, which on a k[x, y±]-module M is the coextension of
scalars
M 7→ Homk[x,y±]
(
k[x, y±]/(x),M
)
.
By resolving k[x, y±]/(x) as a k[x, y±]-module, we can also understand this as the functor taking
a module M to the cone of the map x : M →M .
Let us denote by C the homotopy limit of diagram (4.3). An object of C is a triple (M,N, ϕ)
consisting of a k[x, y±]-module M , a k[x, y±]/(x, y + 1)-module N , and an isomorphism
ϕ : Cone
(
M
x−→M
) ∼−→ N
of k[x, y±]/(x)-modules (regarding N as one by restriction). The isomorphism ϕ can be refor-
mulated as the imposition on Cone(x : M → M) of the equation that y + 1 acts as zero, or in
other words, it is the data of a degree-(−1) endomorphism x′ of this complex with d(x′) = y+1.
As explained in detail in [28, Proposition 4.10], if M was concentrated in degree 0, then x′ is
equivalent to the datum of a (degree-0) endomorphism of M satisfying xx′ = y + 1: pictorially,
we have
M
x //M
x′
}}||
||
||
||
M
x //M,
where the horizontal map x is the differential on Cone(x), so that xx′ = y + 1 expresses the
triviality of y + 1 acting on Cone(x).
As expected, this agrees with the category QCoh(U) = k[x, x′, y±]/(xx′ = y + 1) -mod of
quasicoherent sheaves on the mirror cluster variety U = C2 \ {xx′ = 1}.
4.2 Homological mirror symmetry
Let (X,D) be an n-dimensional log Calabi–Yau pair with toric model
(
X,D
)
, modeling the log
Calabi–Yau variety U = X \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 Σ in NR, the union of rays ⟨d1ψ1⟩, . . . , ⟨drψr⟩, where ψi ∈ N
are primitive lattice vectors, and di ∈ Z>0. As is usual in toric geometry, the ψi can be
understood as cocharacters of the torus NC× .
� For each 1 ≤ i ≤ r, an element χi ∈ ψ⊥
i = M/ψi, which we think of as a character
on the divisor Di (thought of as a complex (n − 1)-torus), defining a hypersurface Hi =
{χi = −1} ⊂ Di.
This is the same data used in Construction 4.7 to define the Weinstein manifold W∨, where
now we think of ψi ∈ N not as a cocharacter of NC× , but rather as a character of the real
n-torus MS1 , and χi as a cocharacter on the subtorus Si ⊂MS1 defined by ψi.
The main theorem of this paper is a statement of homological mirror symmetry for the
varieties U and U∨.
Theorem 4.12. There is an equivalence of dg categories
Coh(U) ∼=W
(
U∨)
between the category of coherent sheaves on U and the wrapped Fukaya category of U∨.
Mirror Symmetry for Truncated Cluster Varieties 19
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 4.11:
Theorem 4.13. Let X be a smooth variety or DM stack, D
k
↪−→ X a divisor, and H
i
↪−→ D
a smooth subvariety of codimension 2 in X. Let X = BlH X \ D̃ and let C denote the homotopy
pushout of dg categories
lim−→
(
Coh(H)
i∗←− Coh(D)
k∗−→ Coh(X)
)
. (4.4)
Then there is an equivalence
C ∼= Coh(X)
between C and the category of coherent sheaves on X.
Proof. We begin by writing down the functor C → Coh(X). From the pushout presentation
of C, such a functor may be specified by the data of functors FH : Coh(H) → Coh(X) and
FX : Coh(X) → Coh(X), together with an equivalence between the pullbacks of FH and FX to
Coh(D) along the maps in the pushout (4.4).
Consider the following diagram of all the spaces involved in our story, where E is the excep-
tional divisor of the blowup, the vertical maps are the projections, and the horizontal maps are
the inclusions:
X �
� j // BlH X
p
��
E? _
ioo
q
��
X D? _
k
oo H.? _
i
oo
The functors FH and FX will be defined as the compositions
FH = j∗i∗q
∗, FX = j∗p∗.
The restriction to D of FX is evidently given by j∗p∗k∗, and we must produce an isomorphism
between this functor and the other restriction j∗i∗q
∗i∗. Note that before applying final com-
position with j∗, these two functors differ, since p∗k∗ takes sheaves on D to their pullback to
the blowup, whereas 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 D̃ of D. The final pullback j∗ will kill this component, forcing the two functors to
agree.
Thus, we have a well-defined functor F : C → Coh(X). To understand this functor, we will
need to apply Orlov’s result [24] that p∗Coh(X) and q∗Coh(H) ⊗ OE(1) generate the cate-
gory Coh(BlH(X)) of coherent sheaves on the blowup, and in fact they form a semi-orthogonal
decomposion:
Coh(BlH(X)) = ⟨Coh(H)⊗OE(−1),Coh(X)⟩. (4.5)
Note that after pulling back to X, we can ignore the tensor product with OE(−1) in (4.5), since
deleting D̃ makes E affine. This makes clear why F is essentially surjective: the image of the
functor F contains the pullbacks to X of both components in the semi-orthogonal decomposi-
tion (4.5) and hence contains all objects in j∗Coh(BlH(X)).
20 B. Gammage and I. Le
To see why the functor F is in fact an equivalence of categories, recall that the category
Coh(X) is a localization of Coh(BlH(X)) obtained by killing the identity morphism of any
object in Coh
(
D̃
)
, and observe that this is equivalent to the localization which identifies any
object in the second term of (4.5) of the form k∗F , for some F in Coh(D), with the object
(iq)∗F ⊗OE(−1) in the first term.
Now note that the category obtained from the semi-orthogonal decomposition (4.5) by iden-
tifying the images of Coh(X) and Coh(H) is equivalent to the category obtained from the ortho-
gonal sum Coh(H)⊕Coh(X) by the same identification; but Coh(H)⊕Coh(X) is the coproduct
of the categories Coh(H) and 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
Coh(X) as the pushout (4.4), and F as the canonical equivalence C ∼= Coh(X). ■
Proof of Theorem 4.12. Recall from Corollary 4.10 that there is a commutative diagram
W
(
X
∨
Σ
)
∼
��
W
(⊔r
i=1 T
∗(S1
χi
× Ti
))
//oo
∼
��
W
(⊔r
i=1 T
∗Dχi × Ti
)
∼
��
Coh
(
X
)
Coh
(
D
)i∗oo j∗ // Coh(H)
(4.6)
whose vertical maps are equivalences, where in the bottom row, X
i←−↩ D is the toric model for
the log Calabi–Yau pair (X,D), and H
j
↪−→ D is the divisor defined by the characters specified by
the seed data; and the spaces in the top row are the Liouville sectors identified in Proposition 4.8.
Corollary 4.9 computes that the colimit of the top row of (4.6) is the wrapped Fukaya cate-
gory W
(
U∨). We now observe that the bottom row of (4.6) satisfies the hypotheses of Theo-
rem 4.13, so that we may conclude from that theorem that the colimit of the bottom row is
equivalent to the category Coh(X) of coherent sheaves on the space
X := BlH X \ D̃
obtained from the variety X by blowing up along H and then deleting the proper transform of
the toric boundary divisor D. But by definition X = U is the truncated cluster variety we are
interested in.
The vertical equivalence between rows in the diagram (4.6) induces an equivalence between
the colimits of those rows; by the above discussion, this is the equivalence of categories
Coh(U) ∼=W
(
U∨)
we wanted to establish. ■
One consequence of the equivalence from Theorem 4.12 is that the category Coh
(
TC
)
of
coherent sheaves on the torus chart TC of the cluster variety U corresponds under mirror sym-
metry to the category Loc
(
T∨) to the category of local systems on the dual torus T∨, which
is a closed subset in the skeleton LU∨ of U∨. A Weinstein neighborhood T ∗T∨ of this torus is
therefore a Liouville subdomain of the mirror cluster variety U∨, and the Viterbo restriction of
Proposition 3.16 gives a functor
W(U∨)→W
(
T ∗T∨). (4.7)
This functor is mirror to the pullback Coh(U) → Coh
(
TC
)
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 elemen-
tary transformations underlying cluster mutation should look like. This should be a procedure
Mirror Symmetry for Truncated Cluster Varieties 21
which exchanges the Weinstein presentation of L, as obtained from T ∗T∨ by Weinstein disk
attachments, to a different Weinstein presentation which begins with a different torus T ∗(T ′)∨.
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 muta-
tions have been studied in [22, 26], and in the 2-dimensional case a global description of these
mutations is given in [18, 27].
As explained in [27], from the perspective of a 2-dimensional Lagrangian skeleton L, cluster
mutation corresponds to the Lagrangian disk surgery studied in [32].
Construction 4.14. The local model for this surgery, depicted in Figure 7, begins with a La-
grangian skeleton L which can be presented as the union of a surface Σ together with a disk D
attached to Σ along a circle. There is a homotopy of Weinstein structures which collapses the
disk D and then expands it to produce a new Lagrangian skeleton L′ which is a union of D with
a surface Σ′. 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 7. The local model for the Lagrangian disk surgery mirror to cluster mutation: a disk attached
to a cylinder Σ is collapsed and then expanded to produce a new cylinder Σ′.
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 Σ also has other disks D1, . . . ,Dr glued in, along Legendrians lifting
cooriented circles S1, . . . , Sr ⊂ Σ which may intersect the boundary circle of D. If the above
surgery implements mutation, then we should expect that it should modify the boundary cir-
cles Si by a Dehn twist for each positively oriented intersection with the boundary circle of D.
The results of [27] establish that this is the case. In our language, we can restate their result as
follows:
Theorem 4.15 ([27, Theorem 5.7]). Let L be the Lagrangian skeleton of a Weinstein mani-
fold U∨ obtained as in Construction 4.1 by beginning with the cotangent bundle T ∗T of a 2-
torus T ∼= T 2 and attaching Weinstein handles along Legendrian lifts of cooriented circles
S1, . . . , Sk ⊂ T .
22 B. Gammage and I. Le
Applying the disk surgery described in Construction 4.14 to the ith disk Di results in a new
Lagrangian skeleton L′ of the same Weinstein manifold, presented by beginning with the cotan-
gent bundle T ∗T ′ of a 2-torus T ′ ∼= T 2 by handle attachments along Legendrian lifts of cooriented
circles S1, . . . , Sk ⊂ T ′, where S′
i = Si with opposite coorientation, and for j ̸= i we have
S′
j =
{
τiSj if ⟨Sj , Si⟩ > 0,
Sj otherwise,
where we write ⟨−,−⟩ for the oriented intersection number and τi for the Dehn twist about Si.
The paper [27] also checks that the respective tori T , T ′ contained in the skeleton before
and after a disk surgery are related by cluster mutation. Continuing with the notation of
Theorem 4.15, Viterbo restriction (4.7) gives a pair of functors
Coh
(
T∨
C
) ∼=W(T ∗T )←W
(
U∨)→W(T ∗T ′) ∼= Coh
(
(T ′)∨C
)
, (4.8)
which we would like to understand as mirror to a birational map T∨
C 99K (T ′)∨C between the
complexified dual tori. This map is studied in [27] by determining what it does to skyscraper
sheaves in Coh
(
T∨
C
)
, which correspond to finite-rank local systems on T . If E is a local system
on T , we can imagine E as a vector space E together with an endomorphism Eγ for each path γ
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 (4.8) are defined, so we can obtain a diagram
Tors
(
T∨
C
) ∼= Locfin(T )→W
(
U∨)← Locfin(T ′) ∼= Tors
(
(T ′)∨C
)
, (4.9)
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 W
(
U∨) represented by a local system on T ′.
Theorem 4.16 ([27, Theorem 4.16]). Let E be a finite-rank local system on T . Then E is
equivalent in W
(
U∨) to a local system E ′ on T ′ if and only if the monodromy of E around Si
does not have 1 as an eigenvalue.
Suppose this is the case. If γ ⊂ T 2 is a path not intersecting S′
i, then E ′γ = Eγ, and if γ
is a path which crosses S′
i against its coorientation, in a neighborhood of the crossing point
the holonomy is modified to E ′γ = Id−ESi, where Si is treated as an oriented path, where its
coorientation points rightward. (The identification of paths in T with paths in T ′ is given by
a Dehn twist about Si.)
These results generalize easily to a higher-dimensional Weinstein manifold U∨ obtained as
in Construction 4.7 by Weinstein handle attachments along Legendrian lifts of codimension-1
subtori S1, . . . , Sr in an n-torus T .
Construction 4.17. Let T , Si as above. Near the subtorus Si, the torus T splits as T 2×Tn−2,
where the first T 2 is spanned by Sχi (where χi is the cocharacter of Si determining the handle
attachment as in Construction 4.7) and a circle not contained in Si. We can thus take the product
with Tn−2 of the disk surgery of Construction 4.14 above to obtain an analogous operation on
T = Tn. The result is a new Lagrangian which is the union of an n-torus T ′ with Weinstein
handles attached to the Legendrian lifts of codimension-1 subtori S′
i.
Theorems 4.15 and 4.16 can be applied directly to this setting to describe the new torus T ′
and its subtori S′
i just as in the 2-dimensional case. In the language of cluster seeds, we can
summarize this discussion as follows:
Theorem 4.18. Let T , S1, . . . , Sr, as in Construction 4.7, corresponding to a seed s. Then T ′,
S′
1, . . . , S
′
r as in Construction 4.17 correspond to the ith mutation µis of s. Moreover, the
birational map defined by (4.9) agrees with the cluster transformation for this mutation.
Mirror Symmetry for Truncated Cluster Varieties 23
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 Lag-
rangian torus fibrations, in the form of the nodal trade first described in [29]. 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
µ : C2 → P , where P = (R≥0)
2 is the usual moment polytope, and note that this presentation
of C2 distinguishes an anticanonical divisor, namely the preimage of the boundary ∂P = R≥0×
{0} ∪(0,0) {0} × R≥0, which is the toric boundary {xy = 0} of C2.
The motivating question is: is it possible to change the fibration µ so as to change µ−1(∂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 ∂P is singular. In [29], 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
(
2 1
−1 0
)
.
Remark 5.1. Although the branch cut for the integral affine structure is along the ray
(
ψ1
ψ2
)
=(−1
−1
)
, 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 2.19.
After this change, P no longer embeds into R2 with its integral affine structure, but it may
still be immersed in R2 after a branch cut as in Figure 9. The boundary ∂P , which passes
through the branch cut, becomes smooth in the new integral affine structure on the base. The
result of [29] is that a fibration inducing this integral affine structure, and making ∂P smooth,
does exist.
Figure 9. The base of the new fibration on C2, 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.
Proposition 5.2 ([29]). There is a map µ′ : C2 → P whose fibers over the interior of P are
smooth tori except for a nodal torus above (1, 1), such that µ′ induces the above integral affine
structure on P and µ′−1(∂P ) = {xy = 1}.
Since the operation of replacing µ by µ′ doesn’t affect the fibration away from the branch
cut, we can perform this replacement anywhere a corner isomorphic to (R≥0)
2 appears in the
moment polygon of a toric surface:
Construction 5.3. Let µ : X∆ → ∆ be a toric surface equipped with moment polygon ∆ ⊂ R2
and let p ∈ ∆ be a vertex with an integral affine-linear identification A ∈ SL(2,Z) ⋊ R2 of
24 B. Gammage and I. Le
a neighborhood of p in ∆ with (R≥0)
2. Then using A, the moment map of X∆ can be modified
near p to introduce a singularity and branch cut in ∆ as in Proposition 5.2. This operation is
called a nodal trade.
Example 5.4. Let X∆ = Bl0C2 be the blowup of C2 at the origin, with moment polytope ∆
obtained from (R≥0)
2 by cutting out the corner. Each of the two vertices of ∆ has a neighborhood
related to the model corner (R≥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 X∆ = Bl0 C2 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 [22, 26], 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 (C×)n−2. We thus provide
the following generalization of Construction 5.3.
Definition 5.5. Let µ : X∆ → ∆ be a toric variety with moment polytope ∆ ⊂ Rn, and
let F ⊂ ∂∆ be a codimension-2 face of ∆ equipped with an affine-linear identification with
(R≥0)
2 × Rn−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 C× with the 2-dimensional
nodal trade. The bold red line is a line of singularities, over which the fibration takes the form (nodal
curve)×S1, and the blue half-plane is the “branch cut” allowing us to draw this picture in R3; it lives in
the union of monodromy eigenlines of the singular points.
Mirror Symmetry for Truncated Cluster Varieties 25
Remark 5.6. In base diagrams like Figures 10 and 11, depicting integral manifolds whose sin-
gularities 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 Rn−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 eigen-
hyperplane, 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 X∆ by performing
nodal trades and then deleting the preimage of ∂∆. The local models for this construction are
studied in detail in [26]; 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 X∆ a toric variety with moment map µ : X∆ → ∆ ⊂ Rn, we have
X∆ \ µ−1(∂∆) = X∆ \ ∂X = (C×)n,
with restricted moment map µ : (C×)n → Int(∆) ∼= Rn given by the usual moment map
(z1, . . . , zn) 7→ (log |z1|, . . . , log |zn|). For c ∈ Rn, the Morse–Bott function
∥µ− c∥2 : (z1, . . . , zn) 7→ (log |z1| − c1)2 + · · ·+ (log |zn| − cn)2 (5.1)
is a Stein Kähler potential on (C×)n whose underlying Liouville structure has Lagrangian skele-
ton the torus µ−1(c), the minimum locus of ∥µ− 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 [29]. 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 µ′ : C2 → P described in Proposi-
tion 5.2, and its restriction to a fibration
C2 \ {xy = 1} → Int(P ). (5.2)
Choose an identification Int(P ) ∼= R2 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 (5.2) is given by
F (x, y) :=
(
(log |xy − 1|), |x|2 − |y|2
)
.
We refer to [2, 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 C2 \ {xy = 1}.
As we did for the moment map µ of (C×)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
26 B. Gammage and I. Le
[26, 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 ∈ R, and define a map fc : C2 \ {xy = 1} → R by
fc(x, y) := (log |xy − 1| − c)2 +
(
|x|2 − |y|2
)2
.
As for (5.1), the function fc is Morse–Bott with manifold of minima given by the torus fi-
ber F−1(c, 0), but now, fc has one additional Morse critical point – an index 2 critical point
at (0, 0) ∈ C2. So it seems that the function fc should produce a Lagrangian skeleton of
C2 \{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 ⊂ T .
However, the function fc is not quite a Kähler potential. It is plurisubharmonic, but not
strictly so: the form i∂∂fc is positive everywhere except the point (0, 0) ∈ C2, where it vanishes.
We can fix this by adding a term χϵ
(
|x|2+ |y|2
)
to fc, where χ is a cut-off function, identically 1
near the origin (0, 0) and vanishing outside a small neighborhood of the origin, and 0 < ϵ≪ 1.
Lemma 5.7. For appropriate choices of χ and ϵ as above, the function fc + χϵ
(
|x|2 + |y|2
)
is
a Kähler potential for C2 \ {xy = 1}.
Proof. Let g(x, y) = χϵ
(
|x|2 + |y|2
)
be the correction term. In the region, where χ is identi-
cally 1, the 2-form i∂∂g is positive and therefore so is fc. This 2-form may fail to be positive in
the region where χ is decaying, but within this region it may be bounded below by compactness,
and thus for ϵ sufficiently small, we can ensure that |∂∂g| < |∂∂fc| 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 5.8. The Kähler potential of Lemma 5.7 underlies a Stein structure whose Lagrangian
skeleton L is the union of the torus fiber F−1(q) with a visible Lagrangian disk lying over the
interval [0, c] × {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
h : C2 \ {xy = 1} −→ C×, (x, y) 7−→ xy − 1,
which has general fiber C× 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 [15]), the space C2\{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| ∈ [0, c]}.
In fact, this is precisely the skeleton of the Stein structure from Lemma 5.7. First, note that
the minimum of the Kähler potential fc + 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 ̸= 0, the gradient flow of fc will flow the circle S inward along
its Lefschetz thimble, and if we require that the cut-off function χ 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 fc + 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 ∈ 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 Sv ⊂ F−1(q) dual to a tangent direction v.
We can thus reformulate the above calculation in coordinate-free way as follows:
Mirror Symmetry for Truncated Cluster Varieties 27
Corollary 5.9. Let q ∈ Int(P ) be a point lying on the monodromy eigenline for the singularity
of the integral affine base Int(P ) of the singular torus fibration
µ′ : C2 \ {xy = 1} → Int(P ).
Then the Morse–Bott function ∥µ′ − q∥2 can be modified in a neighborhood of (µ′)−1(0, 0) to
produce a Kähler potential on C2 \ {xy = 1} whose Lagrangian skeleton is the union of torus
fiber (µ′)−1(q) and a visble Lagrangian disk whose µ′-projection is a subinterval of the mon-
odromy eigenline connecting q to the singularity, glued to (µ′)−1(q) along the circle dual to this
eigendirection.
This concludes our treatment of the local model for the nodal trade. (See also [26, Sec-
tion 5.5] for further discussion of this Kähler structure and its skeleton.) From our presentation
of Corollary 5.9, 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 5.10. Let X∆ be a toric surface with moment polygon ∆, let U be the result of
applying nodal trades at vertices p1, . . . , pr and deleting the preimage of ∂∆, and write µ′ : U → ∆̊
for the resulting almost toric fibration on U . Suppose that there is a point q ∈ ∆̊ lying on the
monodromy eigenlines of all singularities in ∆̊. Then U admits a Weinstein structure with
a skeleton a union of the torus (µ′)−1(q) and the visible disks projecting to intervals between q
and the singularities of ∆̊. Each visible disk attaches to (µ′)−1(q) along the circle dual to its
eigenline.
Proof. As in Lemma 5.7, study the function ∥µ′ − q∥, which is Morse–Bott with minimum
manifold (µ′)−1(q) and r index-2 Morse critical points, and strictly plurisubharmonic away from
the Morse critical points. Performing the local modification of Lemma 5.7 near each of these
critical points produces a Kähler potential with the desired properties. ■
Corollary 5.11. Let X∆, ∆̊, U , q be as in Proposition 5.10, and let v1, . . . , vr be the directions
of rays from q to the singularities in ∆̊. Then U is isomorphic to the Weinstein manifold
described in Construction 4.1.
Example 5.12. Let X∆ = Bl0C2 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 X∆. 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 Bl0 C2
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 5.13. These diagrams are also useful for understanding cluster mutation. As described
in [26, 30], the disk surgery can be understood via the nodal slides of [29]: let µ′ : U → ∆̊ be an
almost toric fibration as in Proposition 5.10, let q ∈ ∆̊ be as described there, and let x ∈ ∆̊ be
28 B. Gammage and I. Le
a singularity. Then µ′ 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 4.14 above. See
[26, 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 X∆ be a toric variety, now of arbitrary dimension, with moment polytope ∆. As in
Proposition 5.10, we would like to perform nodal trades (in the sense of Definition 5.5) along
some codimension-2 faces F1, . . . , Fr of ∆, an then delete the smoothed boundary divisor to
obtain a space U with singular toric fibration µ′ : U → ∆̊. However, in this case we have the
additional complication that the faces Fi may intersect, in which case the toric fibration µ′ may
have more complicated singularities than those considered so far.
Example 5.14. Let X∆ = C2 × C2, and let F1, F2 be the faces corresponding to toric strata
{(0, 0)}×C2 and C2×{(0, 0)}, so that U =
(
C2 \ {xy = 1}
)2
. Thus the base of the fibration µ′
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
(
C2 \ {xy = 1}
)2
, rather than the truncated cluster variety discussed in Example 2.14.
As the above makes clear, repeating the construction of Proposition 5.10 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 ′ ⊂ U defined by
preimage under µ′ of a region ∆̊′ ⊂ ∆̊ obtained by excising neighborhoods of those singularities
in the integral affine base.
Proposition 5.15. As above, let U be obtained from a toric variety X∆ (of arbitrary dimension)
by smoothing the boundary divisor along codimension-2 faces F1, . . . , Fr of ∆ and deleting it, and
let U ′ ⊂ U be a Weinstein subdomain as above, equipped with restricted fibration µ′ : U → ∆̊′.
Suppose that there exists a point q ∈ ∆̊′ lying in all of the eigenhyperplanes of these nodal
trades. Then U ′ is isomorphic to the Weinstein manifold described in Construction 4.7, where
for the fan Σ we take the rays beginning at q and pointing toward the singularity loci in their
respective eigendirections, and the cocharacters χi arise from the defining equations of the re-
spective singularity eigenhyperplanes.
Proof. The proof is the same as it was for the 2-dimensional case, covered in Corollary 5.11:
the singular toric fibration µ′ : U ′ → ∆̊′ can be used to produce a function ∥µ′ − q∥, and (the
product of (C×)n−2 with) the local modifications of this function described in Lemma 5.7 produce
a Kähler potential, whose Lagrangian skeleton is obtained from the torus (µ′)−1(q) by Weinstein
handle attachments precisely as described in Construction 4.7. ■
Remark 5.16. Rather than passing directly to a Weinstein subdomain U ′ ⊂ 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 ′.
We conclude with a comparison of this singular torus fibration with the singular torus fibration
on the mirror truncated cluster variety.
Mirror Symmetry for Truncated Cluster Varieties 29
Corollary 5.17. Let U be a truncated cluster variety, and U∨ be its mirror Weinstein manifold.
Suppose U∨ is obtained by deleting a smoothing of a toric boundary divisor and passing to
a subdomain as in Proposition 5.15. Then the spaces U , U∨ admit dual singular torus fibrations
over the same base ∆̊.
Proof. If U is a mirror to U∨ with fan Σ, then by construction, the integral affine base of
Proposition 5.15 encodes precisely the set of nodal trades encoded by the fan Σ, as described in
Section 2.2. The torus fibrations of U and U∨ are dual, since the integral affine structure on ∆̊
encodes the character lattice of torus fibers in U (as described in Section 2.2) or the cocharacter
lattice of torus fibers in U∨ (as described in this section). ■
Acknowledgements
The authors are grateful to Denis Auroux, for explaining the construction of the Kähler potential
in Lemma 5.7; 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 [28, 27]; 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 5.16. BG is supported by an NSF postdoctoral fellowship, DMS-2001897.
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1 Introduction
1.1 Mirror construction and plan of the paper
1.2 Relation to [18]
1.3 Future questions
2 Cluster varieties from non-toric blowups
2.1 Review of the GHK construction
2.2 SYZ bases
3 Symplectic geometry background
3.1 Liouville sectors
3.2 Fukaya categories and descent
3.3 Toric mirror symmetry
4 The mirror construction
4.1 Weinstein handle attachment
4.2 Homological mirror symmetry
4.3 Mutations
5 Skeleta from nodal trades
5.1 The nodal trade construction
5.2 Mirror cluster surfaces and skeleta
5.3 Nodal trades in higher dimensions
References
|
| id | nasplib_isofts_kiev_ua-123456789-211732 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T15:55:03Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Gammage, Benjamin Le, Ian 2026-01-09T12:54:59Z 2022 Mirror Symmetry for Truncated Cluster Varieties. Benjamin Gammage and Ian Le. SIGMA 18 (2022), 055, 30 pages 1815-0659 2020 Mathematics Subject Classification: 53D37; 13F60 arXiv:2103.12232 https://nasplib.isofts.kiev.ua/handle/123456789/211732 https://doi.org/10.3842/SIGMA.2022.055 In the algebraic setting, cluster varieties were reformulated by Gross-Hacking-Keel as log Calabi-Yau varieties admitting a toric model. Building on the 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. The authors are grateful to Denis Auroux, for explaining the construction of the K¨ahler potential in Lemma5.7; 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 [28, 27]; 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 5.16. BG is supported by an NSF postdoctoral fellowship, DMS-2001897. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Mirror Symmetry for Truncated Cluster Varieties Article published earlier |
| spellingShingle | Mirror Symmetry for Truncated Cluster Varieties Gammage, Benjamin Le, Ian |
| title | Mirror Symmetry for Truncated Cluster Varieties |
| title_full | Mirror Symmetry for Truncated Cluster Varieties |
| title_fullStr | Mirror Symmetry for Truncated Cluster Varieties |
| title_full_unstemmed | Mirror Symmetry for Truncated Cluster Varieties |
| title_short | Mirror Symmetry for Truncated Cluster Varieties |
| title_sort | mirror symmetry for truncated cluster varieties |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211732 |
| work_keys_str_mv | AT gammagebenjamin mirrorsymmetryfortruncatedclustervarieties AT leian mirrorsymmetryfortruncatedclustervarieties |