MIRROR SYMMETRY AND FUKAYA CATEGORIESOF SINGULAR HYPERSURFACES

Maxim Jeffs

December 2, 2021

Abstract

We consider a definition of the Fukaya category of a singular hypersurface proposedby Auroux, given by localizing the Fukaya category of a nearby fiber at Seidel’s naturaltransformation, and show that this possesses several desirable properties. Firstly, we provean A-side analog of Orlov’s derived Knorrer periodicity theorem by showing that Auroux’category is derived equivalent to the Fukaya-Seidel category of a higher-dimensional Landau-Ginzburg model. Secondly, we describe how this definition should imply homological mirrorsymmetry at various large complex structure limits, in the context of forthcoming work ofAbouzaid-Auroux and Abouzaid-Gross-Siebert.

CONTENTS

1 Introduction 21.1 Knorrer Periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Mirror Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Outline of Proof of Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Definitions and Conventions 72.1 Cap and Cup Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Complements of Fibers 12

4 Knorrer Periodicity for Hypersurfaces 184.1 The AAK Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Proof of Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

5 Applications to Mirror Symmetry 275.1 The Tower of Pants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2 Applications to Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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1 INTRODUCTION

At a basic level, homological mirror symmetry (HMS) conjectures a relationship between theFukaya category of a Kahler manifold Y and the category of coherent sheaves on a ‘mirror’ Kahlermanifold Y of the same dimension. This is also expected to hold in the reverse direction, relatingthe Fukaya category of the mirror Y and the coherent sheaves on the original Kahler manifold Y .In many natural instances of mirror symmetry (such as when Y is non-compact), the ‘intrinsic’mirror Y is often expected to be a singular variety. However, a principled definition of the Fukayacategory of Y is lacking in this case. Some of the most important instances, discussed in detail in§5.1, are the higher-dimensional pairs of pants Πn = {x1 + · · ·+ xn+1 + 1 = 0} ⊆ (C∗)n+1, whosenaturally-constructed mirrors are given by the singular hypersurfaces {z1 · · · zn+1 = 0} ⊆ Cn+1.Using tropical techniques, it is possible to decompose other types of non-compact Kahler manifoldsinto products of pairs of pants of various dimensions; thus understanding mirror symmetry in suchcases is of crucial importance for approaches to proving HMS that rely on gluing techniques.

Moreover, one of the principal desiderata for any approach to constructing a mirror space is thatthe mirror of the mirror should be the original Kahler manifold; in other words, HMS should applyin both directions. However, this immediately runs into the problem that many ‘intrinsic’ mirrorconstructions naturally produce singular varieties, and without an understanding of their Fukayacategories, it is difficult to proceed further. While incredible progress has been made understandingmirror symmetry for smooth Kahler manifolds, outside of the orbifold case little has been writtenabout mirror symmetry for the A-model of singular varieties: discussion is included in §1.2.

In the case of singular hypersurfaces, the key insight of Auroux is that, while the symplecticgeometry of the hypersurface itself may be difficult to understand intrinsically, given a smoothingof this hypersurface, the nearby fibers are perfectly good smooth symplectic manifolds, and theirFukaya categories come equipped with extra algebraic data: a set of morphisms coming from countsof certain holomorphic sections which capture which parts of the manifold degenerate when passingto the singular fiber. In accordance with the philosophy of perverse schobers [KS14], the invariantcycles theorem suggests that performing the categorical localization at these morphisms wouldtherefore yield the ‘Fukaya category’ of the singular hypersurface. As we shall illustrate in §1.1,this technique of localization at algebraic monodromy data has a broader domain of geometricapplicability. A precise version of Auroux’ definition will be given in §2.1.

To summarize this definition, let X be a Stein manifold. For a singular symplectic fibrationf : X → C with a single singular fiber over 0, Seidel [Sei09b] defines a natural transformations : µ → id where µ is the clockwise monodromy functor acting on the wrapped Fukaya categoryW(f−1(t)) of the general fiber (see §2 for our definition). The following definition for the wrappedFukaya category of the singular fiber was proposed by Auroux:

DEFINITION 1. (Auroux) Suppose a singular symplectic fibration f : X → C has precisely onesingular fiber, over 0. Then the wrapped Fukaya category of f−1(0) is defined to be the localizationof the wrapped Fukaya category of a nearby fiber f−1(t) for sufficiently small t 6= 0 at the naturaltransformation s : µ→ id:

DW(f−1(0)) = DW(f−1(t))[s−1].

Here this is the localization of A∞-categories in the sense of [LO06] and D denotes the categoryof twisted complexes. The definition can also be extended to the case of monotone fibers, or any

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situation where Seidel’s natural transformation is defined.

In this paper, we shall prove several results that illustrate why this is indeed the correct definitionof the Fukaya category of a singular hypersurface: an A-side analog of Orlov’s derived Knorrerperiodicity theorem (Theorem 1), and several homological mirror symmetry equivalences at thelarge complex structure limit (Theorem 2), described in detail in the following sections.

We shall give a heuristic description of how Definition 1 has a very natural relation to a version ofBiran-Cornea’s Lagrangian cobordism groups in §2.1. There is also a natural analog of the ‘nearbycycles’ functor one can construct that restricts Lagrangians from the wrapped Fukaya category ofa punctured neighbourhood of the singular fiber to the wrapped Fukaya category of the singularfiber itself (cf. [Aur18]).

1.1 KNORRER PERIODICITY

The first theorem is inspired by a result of Orlov [Orl06, Corollary 3.2]:

THEOREM. (Orlov’s Derived Knorrer Periodicity) If X is a smooth quasi-projective variety,and f : X → C is a regular function with f−1(0) smooth, then there is an equivalence of categories

DbCoh(f−1(0))→ Sing(X × C, zf)

where z is the coordinate on C.

Though [Orl06] stated this theorem in the case where f−1(0) is smooth, the result holds generallyfor any hypersurface (see [Hir17, Theorem 1.2]): it hence allows us to study such singular hyper-surfaces in terms of an LG model on the smooth variety X × C. It was conjectured by Orlov in[Orl06] that the same relation should hold for the A-model: by analogy with Orlov’s result, wemight expect the wrapped Fukaya category of f−1(0) to be equivalent to the (fiberwise-wrapped)Fukaya-Seidel category of the Landau-Ginzburg model (X ×C, zf), denoted W(X ×C, zf). Thisis our main result.

THEOREM 1. (Derived Knorrer Periodicity) Suppose that X is a smooth affine variety with anembedding X → CN inducing a Stein structure on X, and suppose f : X → C is the restrictionof a polynomial function on CN . If f has a single critical fiber f−1(0) then for sufficiently smallt 6= 0 there is a quasiequivalence of A∞-categories

DπW(f−1(t))[s−1]→ DπW(X × C, zf).

Here Dπ is used to denote the idempotent-completion of the twisted complexes; we expect thatthis result should also hold before taking idempotent-completions, though our proof relies on thisstep. The hypotheses could almost certainly be weakened, but they suffice for applications tomirror symmetry.

Note that this is a different form of ‘suspension’ or ‘periodicity’ for an LG model from that usuallyconsidered in singularity theory (compare [Sei09a]).

The results above need not be limited to singular hypersurfaces. For singular complete intersec-tions, we can make the definition:

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DEFINITION 2. Suppose f1, . . . , fk : X → C are holomorphic functions on a Stein manifoldsuch that for t1, . . . , tk 6= 0, the intersection f−1

1 (t1) ∩ · · · ∩ f−1k (tk) is smooth. Then we define

DW(f−11 (0) ∩ · · · ∩ f−1

k (0)) = DW(f−11 (t1) ∩ · · · ∩ f−1

k (tk))[s−11 , . . . , s−1

k ].

where si are the natural transformations coming from the monodromy of fi around ti = 0, and tiare sufficiently small.

By iterating our proof, we also expect the main theorem to admit a simple generalization tosingular complete intersections:

CONJECTURE 1. Under appropriate hypotheses on f1, . . . , fk, we have a quasiequivalence ofA∞-categories:

DπW(f−11 (0) ∩ · · · ∩ f−1

k (0)) ' DπW(X × Ck, z1f1 + · · ·+ zkfk)

where z1, . . . , zk are coordinates on Ck.

We shall discuss below how this conjecture could be proved by an extension of the results in thispaper.

1.2 MIRROR SYMMETRY

The definition of the Fukaya category of a singular hypersurface given above manifestly dependson a choice of smoothing. This is not only desirable, but in fact crucial, for homological mirrorsymmetry purposes. Classically, mirror symmetry is a relation between a Kahler manifold and alarge complex structure limit (LCSL) family of complex manifolds. Homological mirror symmetryis expected to be an involution, so it is important to have a notion of the Fukaya category of thesingular fiber of this family. Given that the choice of the mirror depends on the entire degeneration,it is not surprising that the definition of the Fukaya category of the singular fiber should involvethe data of the smoothing.

On the other hand, since the germ of an isolated hypersurface singularity has a smooth andconnected versal deformation space [KM98, p. 144], a simple homotopy argument shows that theFukaya category (of the germ) is independent of the choice of smoothing in this case. Hence itprovides a (potentially interesting) symplectic invariant of isolated hypersurface singularities. Ingeneral, extra data must be provided for the Fukaya category to be uniquely specified: examples areprovided in §5. Expectations from mirror symmetry suggest that this extra data should take theform of a log structure on f−1(0), since in good cases this is expected to determine a smoothing off−1(0). It may be possible to formulate an intrinsic construction of the wrapped Fukaya category ofa singular hypersurface with a log structure using Parker’s theory of holomorphic curves in explodedmanifolds (certain log-schemes have the structure of exploded manifolds [Par12]). Alternatively,from the perspective of the LG model (X×C, zf), the critical locus f−1(0)×C comes with a (−1)-shifted symplectic structure: we conjecture that this extra structure is also sufficient to determinethe wrapped Fukaya category, which could be constructed intrinsically using a formalism such asJoyce’s theory of d-critical loci [Joy15].

Several papers studying mirror symmetry for the A-model of singular varieties have appearedin the literature, using Orlov’s LG model (X × C, zf) to give a definition for the A-model. Forinstance, two papers by Nadler [Nad19, Nad17], where a microlocal version of the A-model is used.

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In this case, our periodicity theorem admits another proof, which we shall sketch in §5.1, as wellas a simple proof of Nadler’s result in our terminology. We illustrate how our definition allowsthis to be generalized to the case of complements of hypersurfaces in toric varieties, appealing toforthcoming work of Abouzaid-Auroux [AA].

This is a special case of a more general mirror symmetry statement, which can be understood inthe context of the Gross-Siebert program:

THEOREM 2. Suppose B is an integral affine manifold, and let X and X be the correspondingmirror pair. Assume we are given a homological mirror symmetry equivalence between X andX that identifies the monodromy functor µ around the large complex structure limit X0 with thetensor product by a line bundle L−1 on X whose first Chern class is the negative of the Kahlerform on X. Then the large complex structure limit X0 of X is homologically mirror to the largevolume limit of X:

DπF (X0) ' DbCoh(X \ s−1(0))

where s is a section of L.

Such a homological mirror symmetry equivalence is expected to follow from combining Abouzaid’sfamily Floer theory with ideas from the Gross-Siebert program as in [AGS]. Further explanationis provided in §5.3: as not all the details of [AA, AGS] are known to us, these arguments arenecessarily somewhat heuristic, neglecting points such as local systems, brane structures, and soforth.

Theorem 2 should be taken more as an illustration that Definition 1 should yield expected mirrorsymmetry equivalences. There are however some corollaries that can be rigorously established.

One corollary of this theorem is a mirror symmetry statement between the Fukaya category of anelliptic curve with n nodes, and the derived category of coherent sheaves of an elliptic curve withn punctures. It is our understanding that these instances of homological mirror symmetry werenot previously known.

Remark 1. Many strategies for proving homological mirror symmetry work in the opposite direc-tion to Theorem 2: start by proving an equivalence for the large complex structure limit (LCSL,on the B-side) and the large volume limit (LVL, on the A-side), and then deduce HMS for thenearby fibers using a deformation theory argument. While understanding the Fukaya category ofthe LVL is easier than understanding that of the nearby fibers, the LCSL is a singular variety andso the Fukaya category is not defined conventionally but only by reference to the nearby fibers.This provides some justification for the difference in approach; of course, one can combine bothapproaches to gain an understanding of HMS at all points of the moduli space.

1.3 OUTLINE OF PROOF OF MAIN THEOREM

We will first give a proof of an upgraded form of a theorem of Abouzaid-Auroux-Katzarkov[AAK16, Corollary 7.8] in our setup, which may be of independent interest:

THEOREM 3. (Abouzaid-Auroux-Katzarkov Equivalence) Suppose that X is a smooth affinevariety with an embedding X → CN inducing a Stein structure on X, and suppose f : X → Cis the restriction of a polynomial function on CN . If f has a single critical fiber f−1(0) then for

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sufficiently small t 6= 0, we have a quasiequivalence of A∞-categories:

T :W(f−1(t))→W(X × C, z(f − t))

given by taking thimbles over admissible Lagrangians in the singular locus f−1(t).

In [AAK16] this was expected to be an equivalence of categories, assuming a generation resultof Abouzaid-Ganatra [AG] which should follow from Proposition 6. For a similar result, see[AS15, Lemma A.26]. This theorem could be considered as an open-string analog of the LG-CYcorrespondence.

There is also a relative version of this statement, for fiberwise stopped Fukaya-Seidel categories,which mirrors a result of Orlov:

THEOREM 4. Suppose that X is a smooth affine variety with an embedding X → CN inducinga Stein structure on X, and suppose f, g : X → C are restrictions of polynomial functions on CN .If f has single critical fiber f−1(0) then for sufficiently small t 6= 0 we have a fully faithful functor

W(f−1(t), g)→W(X × C, z(f − t), g).

Here the second category is fiberwise stopped with respect to g, as explained in Definition 4.

This should also imply a relative version of Theorem 1, which would then imply Conjecture 1:

CONJECTURE 2. Suppose f : X → C is a holomorphic function on a Stein manifold X havinga single critical fiber f−1(0); suppose g : X → C is another holomorphic function. Then forδ, |t| > 0 sufficiently small there is a quasiequivalence of A∞-categories

DπW(f−1(t), g)[s−1]→ DπW(X × C, zf + δg).

We shall then show in §3 that passing from (X × C, z(f − t)) to (X × C, zf) can be rephrasedas a stop-removal by carefully analyzing the Liouville geometry of the general fiber as t changes.Hence, by the stop removal theorem of [GPS19, Syl19a], the category W(X × C, zf) may beobtained as a quotient of the categoryW(X×C, z(f− t)) by a full subcategory D of linking disks.Lastly, we show in §4 that under the functor T of Theorem 3, the essential image of the cones ofthe natural transformation µ → id on W(f−1(t)) split-generates the same full subcategory as D,using a Kunneth-type argument. Our Theorem 1 then follows.

ACKNOWLEDGEMENTS

First of all I would like to thank Denis Auroux for suggesting this problem as well as for his pa-tience and important contributions throughout the process; I would also like to thank MohammedAbouzaid, David Favero, Daniel Alvarez-Gavela, Sheel Ganatra, Paul Hacking, David Nadler,John Pardon, Vivek Shende, and Zack Sylvan for helpful conversations or correspondence, as wellas Andrew Hanlon and Jeff Hicks for explaining their thesis work to me. I would also like tothank the anonymous referee for many helpful comments and suggestions. This work was par-tially supported by the Rutherford Foundation of the Royal Society of New Zealand, NSF grantDMS-1937869, and by Simons Foundation grant #385573.

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2 DEFINITIONS AND CONVENTIONS

We begin with a discussion of how to define the Fukaya-Seidel category of a Landau-Ginzburgmodel. Suppose that X is a smooth affine variety with an embedding i : X → CN ; then X becomesa Stein manifold with the Stein function φ : X → R given by the restriction of φ(z) = |z|2 on CN .Suppose that f : CN → C is a polynomial; by abuse of notation we will denote the restrictionto X also by f . Suppose for now that C carries the standard Stein structure. We would like toturn the pair (X, f) into a Landau-Ginzburg model, in particular, talk about its Fukaya-Seidelcategory.

Firstly, the function |f |2 defines a real polynomial on R2N , and X ⊆ R2N is a real affine algebraicvariety. It is a well-known result from real algebraic geometry that the set of points in R for whichthe Malgrange condition for |f |2 : X → R fails is finite (for instance see [Spo02, Remark 3] andtake the intersection with the algebraic variety X). Recall that this Malgrange condition at λ ∈ Rsays that there exists R, ε, η > 0 so that if |z| > R and ||f |2(z)− λ| ≤ ε then

|z||∇X |f |2| > η.

Without loss of generality, |f |2(0) = 0, and since we have that |∇X |f |2| > 0 for |f |2 > 0, we seethat for any λ ∈ R>0 for which the Malgrange condition holds, there exists a C > 0 (dependingon λ) so that on |f |2 = λ we have

|f |2 < C|z||∇X |f |2|.Now take δ > 0 strictly smaller than all points in R>0 where the Malgrange condition fails for |f |2on X. Then there exists some constant C so that for all 0 < |f |2 < δ we have:

|f |2 < C|z||∇X |f |2|.

Choose m ∈ N sufficiently large so that m > 4C and for any constant D > 0 define a Steinfunction via

ψ(z) = φ(z) +Dφ(z)|f |2m

which can be induced using the algebraic embedding i : X → C2N given by z 7→ (i(z),√Dfm(z)i(z)).

This has a regular homotopy to the original Stein structure given by the family with t ∈ [0, 1]

ψt(z) = φ(z) +D

(t+ φ(z)|f |2m

1 + tφ(z)|f |2m

).

PROPOSITION 1. The Liouville vector-field of ψ is outward pointing along |f |2 = δ for D > 0sufficiently large.

Proof. We study the inner product

〈∇X |f |2,∇Xψ〉 = (1 +D|f |2m)〈∇X |f |2,∇Xφ〉+mD|f |2m−2φ|∇X |f |2|2.

By the Cauchy-Schwarz inequality

(1 +D|f |2m)|〈∇X |f |2,∇Xφ〉| ≤ 2(1 +D|f |2m)|∇X |f |2||φ|1/2

and thus we will have〈∇X |f |2,∇Xψ〉 > 0

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so long as2(1 +D|f |2m) < mD|f |2m−2|φ|1/2|∇X |f |2|

since|∇Xφ| ≤ |∇|z|2| = 2|z|.

Rewriting this inequality gives

2

(1

D|f |2m−2+ |f |2

)< m|φ|1/2|∇X |f |2|.

If we take D > 1/δm then1

D|f |2m−2< δ

and hence4δ < m|z||∇X |f |2|

follows by our construction. �

Now define a stopped Liouville domain as follows. First, take the subset {|f |2 ≤ δ} ⊆ X, which byProposition 1 gives a Liouville domain, then add in a stop given by the hypersurface f−1(−δ) ⊆{|f |2 = δ}. Because this stop is a hypersurface, there is a corresponding Liouville sector whichwe denote by (X, f), and write W(X, f) for the partially wrapped Fukaya category as definedin [GPS19]. This is the (fiberwise wrapped) Fukaya-Seidel category of the Landau-Ginzburgmodel f : X → C.

Remark 2. We shall henceforth assume that this procedure has taken place and that the Steinfunction φ on X is already given by ψ. We can always equivalently consider (X, f) as a Liouvillemanifold with a stop f−1(−∞) sitting inside the contact boundary at infinity. In this case, thoughwe may informally discuss our Landau-Ginzburg models as living over all of C, their correspondingLiouville sectors are constructed as above and lie only over a small disk {|z|2 ≤ δ} ⊆ C.

Note that above all of the non-zero points where the Malgrange condition failed to hold are madeto lie outside the Liouville domain. This has several useful consequences.

LEMMA 1. Symplectic parallel transport gives exact symplectomorphisms between smooth fibersof f over 0 < |f | < ε for some ε > 0.

Proof. We follow the argument of [FSS08, §2]. By a simple calculation (cf. [Sei08b, p.213])it follows that the symplectic parallel transport gives exact symplectomorphisms whenever it isdefined. The parallel transport vector field is always a unit complex multiple of

∇XRe(f)

|∇XRe(f)|2

since ∇XIm(f) = J∇XRe(f). Because φ : X → R is a proper exhausting function, it suffices toshow that the image under φ of the flow lines γ(t) of the parallel transport vector field do notescape to infinity in finite time. Therefore consider the derivative∣∣∣∣ ∇XRe(f)

|∇XRe(f)|2φ

∣∣∣∣ =|〈∇XRe(f),∇Xφ〉||∇XRe(f)|2

≤ |∇Xφ||∇XRe(f)|

.

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Since there are only finitely many points in C where the Malgrange condition fails for the complexpolynomial f : X → C, we have a constant C > 0 and some ε > 0 so that

|f | < C|φ|1/2|dXf |

for 0 < |f | < ε, where |dXf | = |∇XRe(f)|. Observe also that on CN we have

|∇φ| = |∇|z|2| = 2|φ|1/2

and hence|∇Xφ| ≤ 2|φ|1/2.

Therefore we have that on 0 < |f | < ε, for some constant C,

φ′(t) ≤ Cφ(t)

|f(t)|

where φ(t) = φ(γ(t)). If γ(t) avoids the singular fiber of f , there is some constant β > 0 so that

1

|f(t)|< β.

Hence by Gronwall’s inequalityφ(t) ≤ φ(0)eCβt

which completes the proof. �

Remark 3. By shrinking δ > 0 further in Proposition 1, we will assume that ε from Lemma 1 hasδ < ε.

Often we shall want to have the freedom to use a different Stein function to compute Fukayacategories. We now record here a standard lemma we shall use throughout, which however onlyappears implicitly in the literature.

DEFINITION 3. Suppose X is a Liouville manifold with Liouville form λ; a smooth family λt,t ∈ [0, 1] of Liouville forms for X with λ0 = λ such that the union of the skeleta of the Liouvillestructures λt stays within a compact set, is called a simple Liouville homotopy.

For instance, this condition is satisfied by a family of Weinstein functions whose critical pointsremain in a compact set.

LEMMA 2. Given a Liouville manifold X and a simple Liouville homotopy λt, all of the Liouvillemanifolds (X,λt) are exact symplectomorphic and the wrapped Fukaya categories W(X,λt) are allquasiequivalent.

Proof. By [CE12, Proposition 11.8] for every simple Liouville homotopy there is a family of exactsymplectomorphisms φt : X → X with φ∗tλt = λ− df with f compactly supported, such that φ0 isthe identity. This gives rise to a trivial inclusion of Liouville sectors and so by [GPS19, Lemma2.6] this deformation yields a quasiequivalence of wrapped Fukaya categories between W(X,λ0)and W(X,λ1). �

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We are hence free to work with deformed Liouville structures in our proofs when consideringonly the Fukaya category up to quasi-equivalence (by [Sei08b, Corollary 1.14]). There is a similarversion of Lemma 2 also in the stopped case. A particularly useful application of this is as follows.

PROPOSITION 2. Suppose U ⊆ C is a simply-connected open set inside {0 < |z| < δ}; thenthere is a simple Liouville deformation on X (supported in a neighbourhood of f−1(U)) that makesf−1(U) exact symplectomorphic to the product f−1(δ)×U with the product 1-form, preserving the1-forms of the fibers.

The proof corresponds to the discussion around [Sei08b, Lemma 15.3]. Summarized, by Lemma 1we can trivialize f over U using symplectic parallel transport, so that smoothly

f−1(U) ∼= f−1(δ)× U.

Since the symplectic parallel transport induces exact symplectomorphisms of the fibers, up to anexact form, the symplectic form pulled back to f−1(δ)× U is given by

λ = λF + κ

where λF = λ|f−1(δ) and κ is the symplectic connection form. There is then a deformation

λs = λF + sκ+ c(1− s)π∗λU

for s ∈ [0, 1] and c > 0 sufficiently large. As in [Sei08b, Lemma 15.3] this can be supportedin a neighbourhood of f−1(U) by cutting off appropriately, and is a simple deformation since itpreserves the fibers.

2.1 CAP AND CUP FUNCTORS

Now we wish to define some functors introduced in [AS] in the language of [GPS19]. Firstly, ifF ⊆ ∂∞X is a Liouville hypersurface, then there is the Orlov functor

W(F )→W(X,F ).

given by taking small counterclockwise linking disks of the stop F [GPS19, Syl19b]. Henceforthwe shall assume that F is the fiber of a Landau-Ginzburg model f : X → C as explained abovein §2. In this case F = f−1(−δ) we call the functor W(F ) → W(X,F ) the cup functor ∪: seeFigure 1.

The cup functor has a formal adjoint given by the pullback on left Yoneda modules ModW(X,F )→ModW(F ), which we call the cap functor ∩. We then have counit and unit morphisms ε : ∪∩ → idand η : id→ ∩∪ respectively, which we may complete to exact triangles of bimodules. These exacttriangles in fact have a geometric characterization in terms of earlier work of Seidel:

THEOREM. (Abouzaid-Ganatra, [AG]) There are exact triangles of bimodules

∪∩ id

σ

ε

+1

on DπW(X,F ) and

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Figure 1: The cup functor.

∩∪ id

µ+1

η

s

on DπW(F ), where µ : W(F ) → W(F ) is the clockwise monodromy acting on the fiber andσ : W(X,F ) → W(X,F ) is the clockwise total twist acting on Lagrangians in the total space.Moreover, the natural transformation s may be identified with Seidel’s natural transformation,first introduced in [Sei09b] (see also [Sei08a, Sei17]).

Conventions for these two triangles differ: see see [Syl19b, Theorem 1.3] for the identifications ofthe twist/cotwist with the monodromy functors and [AS15, Appendix] for a proof of one triangle.Our conventions are chosen to be compatible with the counterclockwise wrapping of [GPS20] sothat there is a degree-0 natural transformation id → σ: this forces ∪, σ, µ to be clockwise and∪ the left adjoint. However, this means that in the case of a model Lefschetz fibration, µ is thenegative Dehn twist.

None of these results are logically necessary for the proof of Theorem 1 of this paper, and for theproofs following the reader may take µ, σ, s to be defined via the purely algebraic definition givenabove. Then we may make the definition:

DEFINITION 1. (Auroux) Suppose a singular symplectic fibration f : X → C has precisely onesingular fiber, over 0. Then the wrapped Fukaya category of f−1(0) is defined to be the localizationof the wrapped Fukaya category of a nearby fiber f−1(t) for sufficiently small t 6= 0 at the naturaltransformation s : µ→ id:

DW(f−1(0)) = DW(f−1(t))[s−1].

The condition that |t| 6= 0 be sufficiently small is made to exclude the possibility that t is a pointwhere the Malgrange condition fails; in our definitions earlier in §2 we define our Landau-Ginzburgmodels so as to exclude such points.

Note that this definition is equivalent to taking the quotient of the category W(f−1(t)) by thefull subcategory of the cones of the natural transformation, that is, the quotient by the essentialimage of the composition ∩∪. The following lemma is often useful for computations:

LEMMA 3. The essential image of the composition ∩∪ has the same split-closure as the essentialimage of ∩.

11

Proof. One inclusion is clear. To show the reverse inclusion, consider the two exact trianglesrelating the composition of the ∩ and ∪ functors; importantly, note that these two triangles havearrows in reverse directions with the conventions taken above, where ε and η are the counit andunit of the ∪ − ∩ adjunction respectively. We shall apply ∩ to the first triangle and precomposethe second triangle with ∩. Exactness and the identity µ∩ = ∩σ [AG] yields two triangles:

∩ ∪ ∩ ∩

µ∩

∩ε

+1

∩ ∪ ∩ ∩

µ∩+1

η∩

The unit-counit identity ∩ε ◦ η∩ = id∩ for an adjunction implies that the map µ∩ → ∩ in thesecond triangle is zero and hence that this triangle splits. Thus ∩∪∩L always has ∩L as a directsummand. �

COROLLARY 1. The category DπW(f−1(0)) is quasiequivalent to the quotient of the categoryDπW(f−1(t)) by the essential image of ∩.

Proof. By Lemma 3, the essential images of the two functors ∩∪ and ∩ have the same split-closure.Thus the resulting quotients will be quasiequivalent. �

The reader wishing to avoid [AG] may instead take Corollary 1 as a definition for the purposes ofthe proof of Theorem 1.

Definition 1 also has a particularly natural relation to Lagrangian cobordisms inside symplecticfibrations (cf. [BC17]). In place of studying Lagrangian cobordisms inside the product f−1(t)×C,we could alternatively consider Lagrangian cobordisms inside X (or several concatenated copies ofX) with ends projecting via f to rays parallel to the positive or negative real axes: see Figure 2.After applying a Hamiltonian isotopy, every such cobordism may instead be considered as havingonly positive ends; these nullcobordisms thus represent all the equivalence relations imposed onthe group of Lagrangian cobordisms. But these relations say that every complex in the imageof ∩ must be equivalent to zero, which by Corollary 1 is an equivalent description of the Fukayacategory W(f−1(0)). We therefore conjecture that this construction describes the Grothendieckgroup K0(DπW(f−1(0))).

3 COMPLEMENTS OF FIBERS

We begin with some simple observations about the geometry of the Landau-Ginzburg models(X × C, z(f − t)) for different values of t. For the purposes of this section, let ft(z) = f(z)− t.

Firstly, observe that z(f − t) has only a single critical fiber, occurring where z(f − t) = 0. Thisfiber is given set-theoretically by (f−1(t) × C) ∪ (X × {0}); the critical locus is given exactly by

12

Figure 2: A Lagrangian cobordism inside a concatenation of copies of X.

f−1(t). Observe that for t 6= 0, this critical locus is therefore smooth, and we have an explicitlocal Morse-Bott model, described in further detail below.

For ε 6= 0, the smooth fiber of z(f − t) over ε is given by {(x, z) : f(x) 6= t and z = ε/ft}, whichcan be identified set-theoretically with the complement X \ f−1(t). The corresponding Liouvillestructure on this complement comes from the restriction of the Stein function to the fiber. Usingthe Stein function |z|2n on the factor C this is hence given by

ψ(x) = φ(x) +ε2n

|ft(x)|2n.

This Stein function may be obtained by taking the embedding of X \ f−1(t) into CN+1 given

by j(z) =(i(z), ε

ft(z)

). We shall instead use a deformation-equivalent Stein structure on the

complement, given by

ψ(x) = φ(x) +Cφ(x)

|ft(x)|2n

for suitable constants C > 0 and n ∈ N. This Stein function is given by the embedding of X\f−1(t)into C2N given by z 7→ (i(z),

√Ci(z)/ft(z)n) and is Stein deformation-equivalent to the previous

under the deformation for s ∈ [0, 1] given by

ψs(x) = φ(x) +s+ φ(x)

1 + sφ(x)

C

|ft(x)|2n.

To understand how the LG models of (X × C, z(f − t)) for t 6= 0 and t = 0 differ, we need tostudy how the Liouville structure of the general fiber changes. The following elementary exampleillustrates the procedure.

Example 1. We recall how to build standard Weinstein structures on Lefschetz fibrations. Considerf : C2 → C the standard Lefschetz fibration, and equip C2 with the standard Stein structure.Consider the skeleton of the complement of the fiber f−1(t). For t = 0, we see that C2 \ f−1(0) =(C∗)2 with the standard Liouville structure, so its skeleton corresponds to the zero-section T 2 ⊆T ∗T 2 ∼= (C∗)2.

Observe that the Weinstein function coming from the Stein structure on C2 has a single criticalpoint at 0, of index 2. When we take t 6= 0, the Weinstein function on X \f−1(t) now includes this

13

Figure 3: Skeleton in Example 1 (in blue).

critical point. Therefore C2 \ f−1(t) is obtained from C2 \ f−1(0) by attaching a critical Weinsteinhandle to C2 \ f−1(0). The skeleton of C2 \ f−1(t) now includes the core of this handle, attachedto the original T 2, as in Figure 3. �

To describe the general case we shall need a dual version of Proposition 1.

PROPOSITION 3. For sufficiently large C > 0 and n ∈ N, and t sufficiently small the Liouvillevector field of ψ is inward-pointing on {|ft|2 = δ} for all δ > 0 sufficiently small.

Proof. We consider the inner product

〈∇X |ft|2,∇Xψ〉 =

(1 +

C

|ft|2n

)〈∇X |ft|2,∇Xφ〉 −

nCφ

|ft|2n+2|∇X |ft|2|2.

By applying Cauchy-Schwarz, we see that we will have

〈∇X |ft|2,∇Xψ〉 < 0

on |ft|−1(δ) whenever

2

(|ft|2n+2

C+ |ft|2

)< n|φ|1/2|∇X |ft|2|.

Again, by choosing |t|, δ > 0 sufficiently small we can ensure that the Malgrange condition issatisfied on |ft|2 = δ for some sufficiently large n ∈ N and this proposition will follow. �

Remark 4. Note that we can apply the same argument with the Stein function

ψ = φ+ Cφ|ft|2m

homotopic to the Stein function from Proposition 1 in place of φ so as to have both Proposition1 and Proposition 3 hold simultaneously (on concentric disks). The resulting inequality

C|ft|2 + |ft|2n+2 +D|ft|2m+2n+2 <φ1/2

2|∇X |ft|2|

(Cn−Dm|ft|2m+2n

)14

holds on ε ≤ |ft|2 ≤ δ for δ > ε > 0 sufficiently small and C, n chosen sufficiently large, givenD,m.

We now apply Eliashberg’s surgery theory for Weinstein manifolds to understand how the generalfiber changes as we vary t:

PROPOSITION 4. The Weinstein structure above on X \ f−1(t) is obtained from X \ f−1(0)by attaching a collection of Weinstein handles.

Proof. First we shall describe how to start with X \f−1(0) and pass to X \f−1(Bδ), both equippedwith the Stein function:

ψ0(x) = φ(x) +Cφ(x)

|f(x)|2n.

By Proposition 3, for δ > 0 sufficiently small and n,C sufficiently large, we can take Bδ a ballaround 0 so that the Liouville vector field for ψ0 on X is inward-pointing on |f |−1(δ). Possiblyshrinking δ > 0 further, there are no zeroes of the Liouville vector field in f−1(Bδ \ {0}), sincethe zeroes of φ on X are isolated under the map f : X → C. Hence we have a trivial Weinsteincobordism between X \ f−1(0) and X \ f−1(Bδ), and hence an isomorphism of Liouville manifolds[CE12].

Taking |t| < δ, we now pass from X \ f−1(Bδ) to X \ f−1(t). First, there is a deformation ofWeinstein structures on X \ f−1(Bδ(s)) (all diffeomorphic for |s| < δ) given by:

ψs(x) = φ(x) +Cφ(x)

|fs(x)|2n

for s ∈ [0, t]. For ε > 0 sufficiently small, none of the critical points of ψs outside f−1(Bδ) enterf−1(Bδ(s)) as s goes from 0 to t: hence this is a simple Liouville deformation and so ψ0 and ψtgive isomorphic Liouville structures on the manifolds X \ f−1(Bδ) and X \ f−1(Bδ(t)) (Lemma 2).

Now, since the Liouville vector field for ψt is inward-pointing along f−1(∂Bδ) note that f−1(Bδ) \f−1(t) gives a Weinstein cobordism from X \f−1(Bδ) to X \f−1(t). Hence by Eliashberg’s surgerytheory for Weinstein manifolds [CE12], we obtain X \f−1(t) by attaching Weinstein handles alongthe boundary of X \ f−1(Bδ): see Figure 4. �

Any critical handles given by Proposition 4 will be referred to as additional handles. By applyingthe stop-removal theorem of Ganatra-Pardon-Shende [GPS19, Theorem 1.16], we deduce:

PROPOSITION 5. With the Liouville structures constructed above, we have an equivalence ofcategories between W(X × C, zf) and the quotient of W(X × C, z(f − t)) by the full subcategoryD of linking disks of the stable manifolds of the additional handles in X \ f−1(t):

W(X × C, zf) ∼=W(X × C, z(f − t))/D.

PROPOSITION 6. For f a function on a Stein manifold X as above, the category W(X, f) issplit-generated by the cocores of the additional handles.

In the course of the proof we shall explain the sense in which the cocores of the additional handles,which we will denote `i, are objects of W(X, f).

15

Figure 4: The handle attachment procedure from Proposition 4: cores of additional handles arein red and cocores in blue.

Proof. We begin by studying the wrapped Fukaya category of X \f−1(Bδ) with the Stein structuregiven by:

ψ(x) = φ(x) +Cφ(x)

|f(x)|2n

with δ, C, n chosen as in the proof of Proposition 4. As in Remark 4, by changing φ we have someR > δ such that the Liouville vector field of ψ is also outward pointing along |f |−1(R). To thiswe wish to add stops as follows:

• A stop F given by f−1(−R);

• A stop Λ on |f |−1(δ) given by the framed Legendrian spheres Λi along which the additionalhandles `i are attached.

By the generation result [GPS19, Theorem 1.10],W(X\f−1(Bδ), F∪Λ) is generated by a collectionof two kinds of Lagrangians:

• Linking disks of F and of Λi;

• Lagrangians L given by the product of cocores of f−1(t) with an arc γ connecting −R to|f |−1(δ).

Using the wrapping exact triangle of [GPS19, Theorem 1.9], we can express the linking disks ofF in terms of twisted complexes of the Lagrangians L and the linking disks of Λi, as illustratedin Figure 5. Note that the arc γ with which we take the product lies below the stops F,Λ in thecomplex plane.

Now we pass to (X \ f−1(t), f) by attaching Weinstein handles as in Proposition 4, while keepingthe Liouville vector field outward-pointing along |f |−1(R). Then the linking disks of the Λi becomeexactly the cocores `i of the additional handles and so by the gluing result in [GPS19, Theorem

16

Figure 5: Producing linking disks of F from objects L (in red) and linking disks of Λ (in green).

1.20] we know thatW(X \f−1(t), f) is generated by the additional cocores `i and the LagrangiansL (cf. [GPS19, p.68]). We can take these Lagrangians L to be fibered over an arc γ from −R tof−1(t) that passes below the singular fiber of f and below the stop F .

Now we consider the Weinstein deformation given by:

ψt(x) = φ(x) +Cφ(x)

|ft(x)|2n.

Arguing similarly to Remark 4, for all sufficiently small ε > 0 we can choose φ, larger C, n, andsmaller R and δ so that there exists some R′ > R+ε so that all of the points where the Malgrangecondition fails for |f |2 lie outside of |f | ≤ R′ + ε and so that

• for 0 < |t| < R + ε the Liouville vector field of ψt on X \ f−1(t) is outward pointing along|f |−1(r) for R′ + ε > r ≥ R′;

• for R + ε/2 < |t| < R + ε, the Liouville vector field is outward pointing along |f |−1(R).

This first property prevents the skeleton of the Liouville domain from escaping outside of a compactset and hence ψt gives simple Weinstein deformations of X \ f−1(t) for 0 < |t| < R + ε. Thusthe resulting X \ f−1(t) are all exact symplectomorphic and the wrapped Fukaya categoryW(X \f−1(t), f) remains unchanged (Lemma 2).

The second property of ψt for R + ε/2 < |t| < R + ε means that, since the stop F lives inside|f |−1(−R), we have a subsector (f−1(BR(0)), f) of (X \f−1(t), f); see Figure 6. Applying Viterborestriction, the objects L and `i split-generateW(f−1(BR(0)), f) (by [Syl19b, Theorem 1.8]). But(f−1(BR(0)), f) is Weinstein deformation-equivalent to the original sector used to define W(X, f)in §2 via the deformation

ψt,s(x) = φ(x) + Cφ(x)ρs(ft(x))

for s ∈ (0, 1], where here ρs(z) = βs(|z|)/|z|2n for βs : R → [0, 1] a smooth cutoff function chosenso that

βs(r) =

{1 for r ≤ tan(πs/2)

0 for r ≥ tan(πs/2) + ε

17

Figure 6: The subdomain constructed in Proposition 6: the objects L are in red and the cocores`i in blue.

and so that ρs(ft(x)) is plurisubharmonic (here ψ1 = ψ). Moreover, the Liouville vector field ofψt,s remains outward pointing along |f |−1(r) for R + ε > |t| > r ≥ R + ε/2 for all s ∈ (0, 1] andso this in fact yields a simple Weinstein deformation.

Hence we obtain split-generators L and `i of W(X, f) (Lemma 2). But the former objects aretrivial in this category, since they may be displaced from themselves by a Hamiltonian pushoff toinfinity. Hence we obtain the desired split-generation result. �

4 KNORRER PERIODICITY FOR HYPERSURFACES

4.1 THE AAK EQUIVALENCE

We shall begin by proving the smooth case of Knorrer periodicity for hypersurfaces:

THEOREM 3. (Abouzaid-Auroux-Katzarkov Equivalence) Suppose that X is a smooth affinevariety with an embedding X → CN inducing a Stein structure on X, and suppose f : X → Cis the restriction of a polynomial function on CN . If f has a single critical fiber f−1(0) then forsufficiently small t 6= 0, we have a quasiequivalence of A∞-categories:

T :W(f−1(t))→W(X × C, z(f − t))

given by taking thimbles over admissible Lagrangians in the singular locus f−1(t).

First we shall describe the Liouville structures and almost-complex structures we will use on(X × C, z(f − t)) to prove this result. We write Ft = z(f − t) in the following. We would liketo remind the reader of the conventions in Remark 2 concerning the way we construct Liouvillestructures associated to Landau-Ginzburg models.

18

Firstly by Proposition 2 since f−1(t) is smooth for t 6= 0, we may smoothly identify V = f−1(Bδ(t))with f−1(t)×C, where Bδ(t) is a small ball around t ∈ C. Then in this open set V ×C, the mapFt : X×C→ C is given explicitly by the local Morse-Bott model f−1(t)×C2 → C via (x, y) 7→ xywhere x, y are coordinates in C2.

By Proposition 2, over an open set U where f : X → C is smoothly trivial, there exists a Liouvilledeformation λt to a product Liouville structure on f−1(U) ∼= f−1(t) × U . We can make thisdeformation constant outside a larger open set, and since this deformation preserves the fibersthis is a simple Liouville homotopy (Definition 3). We first apply this construction to U = Bδ(t)a small neighbourhood of the smooth fiber f−1(t) containing no critical points of f . Passing tothe product f−1(Bδ(t)) × C with the standard Liouville structure on C gives a deformation thatremains simple: this deformation gives an isotopy of the fiber of Ft over −∞. By Lemma 2 thisdeformation does not change the wrapped Fukaya category of (X × C, Ft) and so we are free towork with this altered Liouville structure in our proof. Note that this deformation does not changethe induced Liouville structure on f−1(t).

We apply this argument again, now to the fibration Ft : X×C→ C: over the subset {Re(z) > 0}∪{Im(z) > 0} (of course, intersected with a small disk as in §2) we may again deform the Liouvillestructure to be of product type, where we take the radial Liouville structure on the base. Notethat this deformation leaves the Liouville structure on the smooth fibers X \ f−1(t) unchanged.Once again, we may instead choose to use this Liouville structure on X × C in our arguments.

Now we can make the identification Ft : f−1(Bδ(t)) × C → C with the map f−1(t) × C2 → Cgiven by (x, y) 7→ xy compatible with the almost-complex structures. Choose a cylindrical almost-complex structure on f−1(t) and on X, and the standard complex structure on C: we begin bymodifying the complex structure on X to agree with the product almost-complex structure onf−1(t)× C inside F−1

t (D)× C for D ⊆ C a small open disk around 0, and interpolating with theusual almost-complex structure outside this open subset. Since the Liouville structure is of producttype near f−1(t), the resulting almost-complex structure on X × C can clearly also be chosen tobe cylindrical. The same construction could be repeated for any (cylindrical) almost-complexstructure on C.

Proof. (of Theorem 3) We begin by choosing a collection Li, i ∈ I of cylindrical exact SpinLagrangians inside f−1(t) representing the cocores of the induced Stein structure on f−1(t). Foreach Lagrangian, consider the thimble Ti over Li given by taking the normalized gradient flowof Re(Ft) starting along Li ⊆ f−1(t) inside X × C. Since each Li is exact, these Ti are exactLagrangian submanifolds of X × C, and the deformation retraction from Ti to Li equips themwith Spin structures: we claim that they can also be made cylindrical.

Firstly, there exists some ε > 0 sufficiently small so that over Bε(0) the thimbles Ti lie entirelyinside the Morse-Bott local model f−1(Bδ(t))×C, by the fact that the exact symplectic structureon f−1(Bδ(t)) is a product. One can check explicitly in the local Morse-Bott model that nearF−1t (0) the Ti are cylindrical at fiberwise ∞: they are given by the product of Li with the

standard Lefschetz thimble inside C2. Now we need to check at infinity in the base. Outside ofthe neighbourhood Bε(0) (where all our Lagrangians will intersect) we can instead choose to flowby the Liouville vector field rather than the parallel transport, making the resulting Lagrangiancylindrical at ∞.

Now, for each Ti choose a cofinal sequence T(t)i (generically, to ensure transversality of intersections

19

for finite totally ordered sets, see [GPS20, Definition 3.34]) using the time-t flow of the completeReeb vector field constructed in [GPS20, Lemma 3.29]: by the product form of the Liouvillestructure over {Re(z) > 0}, any Reeb flow is given by a rotation in the base: thus we may take

a compactly-supported Hamiltonian isotopy to make T(t)i lie over a ray inside Bε(0). Taking L

(t)i

to be the intersection of T(t)i with the critical locus, by [GPS20, Lemma 3.29], the L

(t)i also form

a cofinal sequence, since the quantity:∫ ∞0

min∂∞Lt

α(∂∞∂tLt)dt

can only increase when restricted to a smaller set. To define W(X ×C, Ft), choose a collection of

Lagrangians with cofinal sequences that includes the T(t)i as a subcollection.

Now consider the directed Fukaya category O given by the L(j)i s inside f−1(t), and the directed

Fukaya category A given by the T(t)i inside (X × C, Ft), as in [GPS20, Definition 3.35]. We have

that for j > j′, A(T(j)i , T

(j′)i′ ) is given by the Floer complex CF (T

(j)i , T

(j′)i′ ): in the base, T

(j)i lies

over a radial ray that is counterclockwise from T(j′)i′ and hence they intersect only in the fiber

F−1t (0), along the transverse intersection points L

(j)i ∩ L

(j′)i′ . Thus A(T

(j)i , T

(j′)i′ ) = O(L

(j)i , L

(j′)i′ ).

The additivity of the Maslov index under products, and the fact that T(j)i lies over a radial ray that

is strictly counterclockwise from T(j′)i′ means that this is an isomorphism of graded vector spaces.

For j ≤ j′ the morphism spaces obviously coincide. Next we shall match the A∞ operations onboth directed categories.

Given a collection of thimbles T(jn)i for n = 0, . . . , k with j0 > · · · > jk, choose appropriate

(depending on the domain and conformal structure) cylindrical almost-complex structures J for

f−1(t) that make the moduli spaces used to define the operation µk for L(jn)i regular (as in [GPS20,

p.41]), and extend these to cylindrical almost-complex structures on X × C as described above.With respect to such almost-complex structures, the map Ft : X × C → C is J-holomorphicin an open set containing all of the intersection points of L

(jn)i (and hence of T

(jn)i ). Therefore

for any holomorphic disk defining the operation µk for the collection T(jn)i , the projection under

Ft must be a holomorphic disk in C with respect to the standard complex structure (over theentire domain and for every conformal structure). The maximum principle then implies that theholomorphic disk must be contained in F−1

t (0). Another application of the maximum principlewith the holomorphic projection f−1(t)× C2 → C2 in the local Morse-Bott model shows that allof the holomorphic disks must be contained inside f−1(t). We claim that these must regular withthe chosen almost-complex structures on X ×C using the argument from [Sei08b, §14c]. Becauseof the product almost-complex structure on the local Morse-Bott model, the linearized Cauchy-Riemann operator splits into a direct sum of linearized Cauchy-Riemann operators on the baseC and on the fiber f−1(t). The latter is surjective, by assumption. For the former, since T

(jn+1)i

always lies over a ray in Bε(0) which is clockwise from T(jn)i , this means that the Maslov index is

0 and hence the linearized Cauchy-Riemann operator on C has index 0. Then by [Sei08b, Lemma11.5] this linearized Cauchy-Riemann operator on C is injective and hence surjective also. Thusthe moduli spaces used to define the A∞ operations in O and A can be made to coincide exactly.Moreover, the discussion of Spin structures from [AAK16, Corollary 7.8] carries over verbatim andshows that the moduli spaces of disks that appear carry the same orientations. Hence we have anequality of A∞-categories between O and A.

20

Figure 7: The relative stop.

Much the same argument as in the previous paragraph carries over to the construction of continu-ation maps T

(j+1)i → T

(j)i , so that after localization we have an equivalence of categories between

W(f−1(t)) and A[C−1]. By construction, A[C−1] includes intoW(X×C, Ft) as a full subcategory,and the composition of this inclusion with the equivalence W(f−1(t))→ A[C−1] gives the desiredfunctor.

To upgrade T to an equivalence of categories, we will need to know that the category W(X ×C, z(f− t)) is generated by thimbles. Since in a neighbourhood of the critical locus of the functionz(f − t) the Stein structure can be deformed into a product, the cocores of the index-n criticalpoints of the deformed Stein function will be given by products of cocores of critical handlesfor f−1(t) with the ordinary Lefschetz thimble of z1z2 on C2. By Proposition 6, these Morse-Bott thimbles are split-generators of W(X, z(f − t)). Compare also [Sei98, AG] for Morse-Bottgeneration by thimbles. �

Now we turn to the relative version of this theorem:

THEOREM 4. Suppose that X is a smooth affine variety with an embedding X → CN inducinga Stein structure on X, and suppose f, g : X → C are restrictions of polynomial functions on CN .If f has single critical fiber f−1(0) then for sufficiently small t 6= 0 we have a fully faithful functor

W(f−1(t), g)→W(X × C, z(f − t), g).

The second category here is fiberwise stopped with respect to g as in [AA], where admissibleLagrangians are fibered over arcs under the original map f : X → C while also being admissiblein the fibers of f for the restriction g|f−1(t).

DEFINITION 4. Given X a Stein manifold with f, g : X → C two holomorphic functions,we define the relative Fukaya-Seidel category W(X, f, g) to be the wrapped Fukaya categoryW(X,F ∪G) of X with relative stop given by the union of:

• F the relative skeleton of the sector (f−1(−∞), g), inside ∂∞X;

• G the union of (g|f−1(t))−1(−∞) ⊆ ∂∞f−1(t) ⊆ ∂∞X for t along an arc connecting −∞ to

0;

as illustrated in Figure 7.

This stop can be thought of as the boundary at infinity of the relative skeleton of the relativeskeleton. In the sequel, we shall prove that under strong assumptions on f, g and X, the relative

21

Fukaya-Seidel category is indeed equivalent to the Fukaya-Seidel category W(X, f + δg) for δ > 0sufficiently small. This equivalence certainly should not hold in general.

Because of this construction, the proof of Theorem 3 carries over exactly to show that takingthimbles over admissible Lagrangians in the critical locus gives a fully faithful functor

T :W(f−1(t), g)→W(X × C, z(f − t), g).

We expect this to be an equivalence under an appropriate modification of Proposition 6.

4.2 PROOF OF MAIN THEOREM

Since there are two Landau-Ginzburg models under consideration, it will be important to establishnotation for the corresponding cap and cup functors:

• For the Landau-Ginzburg model (X, f) we shall use ∩ : DW(X, f) → DW(f−1(t)) and∪ : DW(f−1(t))→ DW(X, f);

• For the Landau-Ginzburg model (X×C, Ft) we shall use instead the notation⋂

: DW(X×C, Ft)→ DW(X \ f−1(t)) and

⋃: DW(X \ f−1(t))→ DW(X × C, Ft).

We will regard objects of DW(X \f−1(t), f) as living inside DW(X, f) via the Viterbo restrictionmap constructed in Proposition 6.

Our result will follow by showing that under the AAK equivalence in Theorem 3, the cap functor∩ : W(X, f) → W(f−1(t)) applied to the cocore ` ⊆ X of an additional handle, is isomorphic inW(X×C, z(f−t)) to the linking disk of the core of this additional handle for the Liouville structureon X \ f−1(t). By [GPS19, §7.2], the linking disks of these handles are given by the cup functor⋃

:W(X \ f−1(t))→W(X × C, z(f − t)) applied to the cocores ` of the handles. Therefore thisisomorphism is exactly [AS15, Lemma A.28]: unfortunately the proof sketched therein is difficultto make rigorous. The isomorphism would also follow from largely formal arguments involving theadjunctions constructed in [AG]. Here we provide an alternative argument independent of [AG].

Using the Yoneda lemma for A∞-categories [Sei08b, Corollary 14.7] what we wish to show is that:

THEOREM 5. For every additional cocore `, we have an equivalence of left modules overW(f−1(t))between

T ∗Y⋃(`)∼= Y∩`

where here Y : W(f−1(t)) → W(f−1(t))Mod is the left Yoneda functor, and T ∗ denotes thepullback of left modules.

Proof. For this proof we will continue to use the Liouville structures constructed on X×C above;note that with respect to the induced Liouville structure, every fiber of Ft contains an openneighbourhood diffeomorphic to f−1(t)×C∗ where the Liouville vector field is given by the productwith the standard Liouville vector field given by:

Z = − ε

r3∂r

where r = |z| is the radial coordinate.

22

Figure 8: Identifying the modules as graded vector spaces.

Given any Lagrangian L in W(f−1(t)), the thimble TL and⋃` intersect in a single fiber of Ft,

say over 0 < δ ∈ C, where we take δ > 0 sufficiently small so that over Bδ, the Lagrangian TLwill be contained entirely inside the Morse-Bott model. Because of the product structure, in thefiber F−1

t (δ) the Lagrangian TL is given by L×K ⊆ f−1(t)×C∗ ⊆ X \ f−1(t) for K ⊆ C∗ a smallcircle around 0. By our definition of the cup functor,

⋃` in this fiber F−1

t (δ) is given exactly bythe cocore `; see Figure 8.

Observe that in the analysis of the Stein structure on X \ f−1(t) in Proposition 4 every additionalcocore must have boundary in f−1(t): they cannot remain in a compact set, since there are noindex-(n+ 1) critical points of the Stein function, and they cannot have boundary in |f − t|−1(δ)because the Liouville vector field is inward-pointing. The Liouville structure we use on X × Cto define the Fukaya category is a deformation of this Stein structure, but this exact deformationpreserves flow lines of the Liouville vector field. Because of the local form of the Liouville vectorfield on f−1(t)×C∗ ⊆ X \ f−1(t) this means that ` is fibered over C∗ and hence intersects L×Kin a single fiber of f . Inside this fiber,

⋃` is given by ∩`, by our definition of the cap functor,

and hence TL and⋃` intersect exactly along L ∩ (∩`) ⊆ f−1(t); see Figure 8. Thus we have an

isomorphism of vector spaces:

T ∗Y⋃(`)(L) = CF

(⋃`, TL

)∼= CF (∩`, L) = Y∩`(L).

Moreover, by additivity of the Maslov index under products, this is in fact an isomorphism ofgraded vector spaces. It remains to match the A∞ module operations µk.

We begin with the differential µ1. In this case, the intersection of TL and⋃` is contained entirely

inside the fiber f−1(t) inside the fiber F−1t (δ). An iterated version of the maximum principle

argument in Theorem 3 applied to the holomorphic maps Ft and f shows that the holomorphicdisks computing µ1 : T ∗Y⋃

(`)(L)→ T ∗Y⋃(`)(L) are all contained in this fiber f−1(t) ⊆ F−1

t (δ); andby a variation of the transversality argument there, we see that any almost-complex structuresused to achieve transversality for disks in the fiber f−1(t) can be extended to X×C so as to make

23

the disks in the total space for µ1 : T ∗Y⋃(`)(L)→ T ∗Y⋃

(`)(L) regular also.

Now we consider the product operation µ2 as a simpler prototype of the argument to follow;suppose L1, L2, are exact cylindrical Lagrangians in f−1(t), and let T iL(i) = T

(i)i be their thimbles

as in the proof of Theorem 3. Suppose we have intersection points y1 ∈ (∩`)∩L1, x ∈ L1∩L2 andy2 ∈ T (2)L2∩(∩`), and that we have chosen regular almost-complex structures J on f−1(t) (domain

and conformal-structure dependent) for the moduli spaces M f−1(t)J (y1, x, y2) of disks defining the

operation:CF (L2, L1)⊗ Y∩`(L2)→ Y∩`(L1).

Extend these almost-complex structures to ones on X ×C that are of product type on the neigh-bourhood f−1(t)× C as described above, which we shall also denote by J .

By Theorem 3 and the above argument, these intersection points correspond to intersection pointsy1 ∈

⋃` ∩ T (1)L1 living in F−1

t (δ), an intersection point x ∈ T (1)L1 ∩ T (2)L2 lying in F−1t (0), and

y2 ∈ T (2)L2 ∩⋃`, living in F−1

t (eiθδ) for 0 < θ < π by our construction of the thimbles T(i)i .

Consider now the moduli space of holomorphic disks MX×CJ (y1, x, y2) inside X × C associated to

the operationCF (T (2)L2, T

(1)L1)⊗ Y⋃`(T

(2)L2)→ Y⋃`(T

(1)L1).

All of these intersection points lie inside the local Morse-Bott model f−1(t) × C2 and so by themaximum principle for the standard almost-complex structure on C applied to the projectionsunder the holomorphic maps Ft and f , all of these disks must lie inside the local Morse-Bottneighbourhood f−1(t)×C2 ⊆ F−1

t (Bδ). Inside this Morse-Bott model, we have various holomorphicprojections. Firstly, projection to f−1(t) gives a surjective map

p : MX×CJ (y1, x, y2)→M f−1(t)

J (y1, x, y2)

because of how the almost-complex structures have been defined. We claim that this is in fact abijection, and that MX×C

J (y1, x, y2) is regular with respect to this almost-complex structure. Tosee this, suppose we have such a disk (u, r) ∈MX×C

J (y1, x, y2) for u : D → X×C and r the uniqueconformal class of 3-pointed disks D: then we can show that it is determined by its projection tof−1(t) as follows.

Let π : f−1(t) × C2 → C2 be the (holomorphic) projection to the second factor. We considerthe holomorphic map π ◦ u from a 3-pointed disk D to C2. Since there is a unique conformalequivalence class of 3-pointed disk, this holomorphic disk (π ◦ u, r) computes the operation µ2 :CF (U, T2) × CF (T2, T1) → CF (U, T1) where Ti, U ⊆ C2 are the Lagrangians obtained by theprojections π(T(i)Li), π(

⋃`), respectively; see Figure 9. These Ti are the thimbles for the standard

Lefschetz fibration on C2 and hence intersect transversely at the origin. The Lagrangian U is theparallel transport of the real axis R+ inside the generic fiber {z1z2 = δ} ∼= C∗ of the model Lefschetzfibration, and so intersects each of the Ti transversely in a single point.

For a generic almost-complex structure J on C2, this moduli space will consist of a single reg-ular disk computing the product of the unique point p ∈ U ∩ T1 with the identity of T1

∼= T2.Applying the previous construction of an almost-complex structure on X × C with this J inplace of the standard almost-complex structure on C2, we see that we can ensure that for any(u, r) ∈ MX×C

J (y1, x, y2), both p(u, r) and π(u, r) are regular. But this shows exactly that there

is a unique holomorphic disk in the preimage under p of any disk in M f−1(t)J (y1, x, y2). It is easy

to check that the induced orientations on these moduli spaces agree.

24

Figure 9: Identifying the µ2 operations on the modules: the holomorphic disk is in green.

For the higher µk operations, we begin by generalizing the above argument for µ2. Suppose Li,i = 1, . . . , k are exact cylindrical Lagrangians in f−1(t), and let T (i)Li = T

(i)i be their thimbles

as in the proof of Theorem 3. Suppose we have intersection points y1 ∈ (∩`) ∩ L1, xi ∈ Li ∩Li+1 for i = 1, . . . , k − 1 and y2 ∈ T (k)Lk ∩ (∩`), and that we have chosen regular almost-complex structures J on f−1(t) (domain and conformal-structure dependent) for the moduli space

M k,f−1(t)J (y1, x1, . . . , xk, y2) of disks defining the operation:

CF (L2, L1)⊗ · · ·CF (Lk, Lk−1)⊗ Y∩`(Lk)→ Y∩`(L1).

Extend these almost-complex structures to ones on X ×C that are of product type on the neigh-bourhood f−1(t)× C as described above.

By Theorem 3 and the above argument, these intersection points correspond to intersection pointsy1 ∈

⋃` ∩ T (1)L1 living in F−1

t (δ), intersection points xi ∈ T (i)Li ∩ T (i+1)Li+1 for i = 1, . . . , k − 1lying in F−1

t (0) and y2 ∈ T (k)Lk ∩⋃`, living in F−1

t (eiθδ) for 0 < θ < π by our construction of

the thimbles T(i)i . Consider now the moduli space of holomorphic disks M k,X×C

J (y1, x1, . . . , xk, y2)inside X × C associated to the operation

CF (T (2)L2, T(1)L1)⊗ · · · ⊗ CF (T (k)Lk, T

(k−1)Lk−1)⊗ Y⋃`(T

(k)Lk)→ Y⋃`(T

(1)L1).

All of these intersection points lie inside the local Morse-Bott model f−1(t) × C2 and so by themaximum principle for the standard almost-complex structure on C applied to the projectionsunder the holomorphic maps Ft and f , all of these disks must lie inside the local Morse-Bottneighbourhood f−1(t)×C2 ⊆ F−1

t (Bδ). Inside this Morse-Bott model we have various holomorphicprojections. Firstly, projection to f−1(t) gives a surjective map

p : M k,X×CJ (y1, x1, . . . , xk, y2)→M k,f−1(t)

J (y1, x1, . . . , xk, y2)

because of how the almost-complex structures have been defined. We claim that this is in facta bijection, and that M k,X×C

J (y1, x1, . . . , xk, y2) is regular with respect to this almost-complex

25

Figure 10: A one-legged tree of three-pointed holomorphic disks.

structure (as before, one can see that the induced orientations on these moduli spaces agree). Todemonstrate this, suppose we have such a disk (u, r) ∈ M k,X×C

J (y1, x1, . . . , xk, y2) for u : D →X × C and r a fixed conformal class of marked disks: then we can show that it is determined byits projection to f−1(t) as follows.

Let π : f−1(t)× C2 → C2 be the (holomorphic) projection to the second factor. We consider theholomorphic map π ◦ u and we again wish to show that there is a unique holomorphic disk in the

preimage under p of any disk in M k,f−1(t)J (y1, x1, . . . , xk, y2). Now since the conformal class r is

fixed, the disk π ◦u no longer computes the A∞ operation µk. This holomorphic disk (π ◦u, r) stillhas boundary lying along the Lagrangians Ti, U ⊆ C2 obtained by the projections π(T (i)Li), π(

⋃`),

respectively, as above; and has corners at the intersection points p1 ∈ U ∩ T1, pk ∈ U ∩ Tk and theidentity elements ai ∈ Ti ∩ Ti+1.

Now choose γ : [0, 1) → Sk a smooth simple path of conformal structures in the compactifiedmoduli space of (k+ 1)-pointed disks that travels from the fixed conformal structure r = γ(0) to-wards the deepest boundary stratum r0 = limt→1− γ(t) where r degenerates in to a one-leggedtree of 3-pointed disks r0 (see Figure 10). We now consider the parametrized moduli space

M k,C2

J (p1, a1, . . . , ak−1, pk) of J-holomorphic disks (v, s) in C2 with the same boundary conditionsas u, but with conformal structure s on the domain of v allowed to vary along the path γ. Thestandard arguments show that for a generic almost-complex structure J on C2, this moduli spacebecomes a smooth 1-manifold. Again applying the previous construction of an almost-complexstructure on X ×C with the modified J in place of the standard almost-complex structure on C2,we see that we can ensure that for any (u, r) ∈M k,X×C

J (y1, x1, . . . , xk, y2), both p(u, r) and π(u, r)are regular also.

Now we consider the Gromov compactification of M k,C2

J (p1, a1, . . . , ak−1, pk). By the maximumprinciple applied to the fibration C2 → C, disks in this moduli space may not escape to infinity andso we may apply Gromov compactness. Exactness prohibits disk and sphere bubbling, and sinceall of p1, ai, pk are in the same degree, there can be no strip breaking. Thus the only boundary

components of M k,C2

J (p1, a1, . . . , ak−1, pk) come from the conformal structures of the domain ateither end of the path γ. At 0, the conformal structure is γ(0) = r and hence one part of the

Gromov boundary of M k,C2

J (p1, a1, . . . , ak−1, pk) is given by the moduli space of disks (u, r) withfixed conformal structure r. On the other end, we have limt→1− γ(t) = r0; this component of the

26

boundary consists of disks that contribute to the iterated product operation:

µ2(µ2(· · ·µ2(a1, a2) · · · ), ak−1), pk) = p1

of pk ∈ U ∩ Tk with the identity ai of Ti ∼= Ti+1. As above, there is a unique holomorphic diskin C2 representing each one of these product operations, and thus there is only one such disk.

Hence, since M k,C2

J (p1, a1, . . . , ak−1, pk) is a 1-manifold with boundary, there must be exactly onedisk (u, r) with conformal class r under the projection π to C2. �

We may now prove the main result.

THEOREM 1. (Derived Knorrer Periodicity) Suppose that X is a smooth affine variety with anembedding X → CN inducing a Stein structure on X, and suppose f : X → C is the restrictionof a polynomial function on CN . If f has a single critical fiber f−1(0) then for sufficiently smallt 6= 0 there is a quasiequivalence of A∞-categories

DπW(f−1(t))[s−1]→ DπW(X × C, zf).

Proof. This follows by combining our previous results. To spell out the argument explicitly:

• By Corollary 1, W(f−1(0)) is the quotient of W(f−1(t)) by the subcategory D′ generatedby Lagrangians of the form ∩L;

• By Proposition 6, this subcategory D′ is the same as the subcategory generated by theLagrangians ∩` for ` additional cocores;

• By Theorem 3, W(f−1(t)) is equivalent to W(X ×C, z(f − t)), and under this equivalence,generators ∩` of the subcategory D′ are sent to the objects

⋃` of the subcategory D by

Theorem 5;

• By Proposition 5, the quotient of W(X × C, z(f − t)) by the subcategory D yields W(X ×C, zf). Thus:

DπW(f−1(0)) ∼= DπW(f−1(t))/D′ ∼= DπW(X × C, z(f − t))/D ∼= DπW(X × C, zf).

�

5 APPLICATIONS TO MIRROR SYMMETRY

For the following applications to mirror symmetry we will make use of the following lemma fromalgebraic geometry. The proof is fairly simple and a sketch can be found in [Sei08a, p.5]. SupposeX is a smooth quasiprojective algebraic variety, L is a line bundle with a section s, and letU = X \ s−1(0).

LEMMA 4. Let s : L−1⊗ → id be the natural transformation given by multiplying by the sections. Then localizing at s gives an equivalence of categories:

DbCoh(X)[s−1] ∼= DbCoh(U).

A heuristic way to interpret this lemma is that localization gives the connection between removinga divisor on one side of mirror symmetry and degenerating to a special fiber on the other.

27

5.1 THE TOWER OF PANTS

The Landau-Ginzburg model (Cn+1,Wn = z1 · · · zn+1) has particular importance in mirror sym-metry as it arises as the mirror to the higher-dimensional pair of pants which we denote byΠn−1 = {x1 + · · ·+ xn + 1 = 0} ⊆ (C∗)n: see for instance [Nad19, AAK16]. In this section, wewill sketch another proof of mirror symmetry and periodicity in this special case, drawing uponforthcoming work of Abouzaid-Auroux [AA].

In the case of this LG model, the general fiber W−1n (1) is given by the complex torus (C∗)n,

for which we know that the wrapped Fukaya category is generated by the single LagrangianL = (R+)n ⊆ (C∗)n, with endomorphism algebra given by C[x±1 , . . . , x

±n ]. We shall use the

following results announced by Abouzaid-Auroux:

THEOREM. (Abouzaid-Auroux [AA]) The category W(Cn+1, z1 · · · zn+1) is generated by the sin-gle Lagrangian ∪L. Moreover, Seidel’s natural transformation s on the general fiber of this LGmodel corresponds to a morphism L→ L given by multiplication by x1 + · · ·+ xn + 1.

Each of the terms in the sum x1 + · · · + xn + 1 in the natural transformation corresponds to acount of holomorphic curves passing through one of the hyperplanes zi = 0 in the singular fiberz1 . . . zn+1 = 0 of the LG model.

COROLLARY 2. We have quasiequivalences of categories:

DπW(W−1n (0)) ' DπW(Cn+2, z1 . . . zn+2) ' DbCoh(Πn).

Proof. Using the localization definition, the first category is given by the localization of the cat-egory of C[x±1 , . . . , x

±n ]-modules at the natural transformation id → id given by multiplication

by x1 + · · · + xn + 1. This is the same as the category of modules over the localization ofC-algebras of C[x±1 , . . . , x

±n ] at x1 + · · · + xn + 1, that is, the category of coherent sheaves on

{x1 + · · ·+ xn + 1 6= 0} ⊆ (C∗)n: but this is exactly {x1 + · · ·+ xn + 1 = −xn+1} ⊆ (C∗)n+1, thepair of pants Πn.

The second category can be given by computing the endomorphisms of the generator ∪L. Since∪L is given by applying a ∪ functor, we may push it off itself to see see that it has endomorphismalgebra given by the cone of the map End(L) → End(L) given by multiplication by the naturaltransformation x1 + · · · + xn+1 + 1 (see Figure 11). Hence W(Cn+2, z1 . . . zn+2) is given by thecategory of modules over the quotient of C[x±1 , . . . , x

±n+1] by the ideal (x1 + · · · + xn+1 + 1). But

this is exactly the category of coherent sheaves on {x1 + · · ·+ xn+1 + 1 = 0} ⊆ (C∗)n+1, that is,the pair of pants Πn. �

It should be possible to extend this proof to complements of hypersurfaces in toric varieties,following the ideas of Abouzaid-Auroux, as follows. Suppose H ⊆ V is a hypersurface in atoric variety V defined by a section s ∈ Γ(V,O(H)). The mirror to H ⊆ (C∗)n is given bya toric LG model (Y,WY ) [Aur18]; compactifying H ⊆ (C∗)n to H ⊆ V adds extra terms tothe superpotential, so that the mirror is (Y,WY + δW ): here W is considered as a secondarysuperpotential used to wrap fiberwise, as in §4.

THEOREM. (Abouzaid-Auroux [AA]) The Fukaya-Seidel category DπW(W−1Y (t),W ) of the gen-

eral fiber is quasiequivalent to DbCoh(V ). Moreover, the action of the monodromy µ on (W−1Y (t),W )

28

Figure 11: Calculating endomorphisms of a ∪-shaped object.

is mirror to tensoring by the line bundle O(−H), and the natural transformation µ→ id is mirrorto the defining section s : O(−H)→ OV .

Hence, an LG version of our definition allows us to conclude that DπW(W−1Y (0),W ) is equivalent

to DbCoh(V \H) using Lemma 4. Moreover, using Definition 2 it should be possible to generalizethis to the case of complete intersections in toric varieties considered in [AA].

5.2 APPLICATIONS TO CURVES

Definition 1 is often quite easy to compute in practice, as we can see from a simple example.

Example 2. Consider the standard Lefschetz fibration X = C2 and f = xy; let us directly computethe Fukaya categoryW({xy = 0}) of the nodal conic. By Lemma 3, to findW({xy = 0}), we justneed to quotient W({xy = t}) by the image of ∩. But in this case, the only object to which wecan apply ∩ is the standard thimble of (C2, xy), and ∩ simply gives the vanishing cycle V ⊆{xy = t}. Hence W({xy = 0}) = W({xy = t})/〈V 〉. Under mirror symmetry of DπW({xy = t})with DbCoh(C∗), V corresponds to the skyscraper sheaf of a point p 6= 0, and the quotientDbCoh(C∗)/〈Op〉 simply gives DbCoh(C∗ \ {p}), which is quasiequivalent to the coherent sheaveson the pair of pants DbCoh(Π1), as expected from the above. �

In this case, we may furthermore use the explicit formulas for the localization of A∞ categories in[LO06] to compute the full A∞ structure of W({xy = 0}). We shall have more to say about thisin future work.

Now, we consider the case of genus-1 curves with singularities. Consider the map f : X → C givenby the Tate family of elliptic curves, with f−1(0) being an elliptic curve with a single node. Weknow by [PZ01, LP12] and others that the Fukaya category of the general fiber F (f−1(t)) is derivedequivalent to the category of coherent sheaves Coh(E) on a mirror elliptic curve E. In this case, themonodromy around this singularity is given by the negative Dehn twist around the correspondingvanishing cycle, and takes the Lagrangian L ⊆ f−1(t), mirror to OE, to the slope (−1) Lagrangianµ(L) ⊆ f−1(t), mirror to a degree (−1) line bundle L−1 on E [PZ01]. Since the monodromy functoris given by translating fiberwise by µ(L), under this mirror equivalence µ is mirror to tensoring bythis line bundle L−1 (see §5.3 below for a discussion). Hence the natural transformation µ → idis mirror to a morphism L−1 → OE, that is, a section s of the dual line bundle L. Hence we maycompare the localization ofW(f−1(t)) at this natural transformation with that of Coh(E). On onehand, by Definition 1, this localization of F (f−1(t)) yields F (f−1(0)), the Fukaya category of thenodal curve. On the mirror, by Lemma 4, localization of DbCoh(E) at the natural transformation

29

L−1⊗ → id given by multiplication by a section s gives exactly the derived category of coherentsheaves of the complement, DbCoh(E \ s−1(0)). Hence, we have a mirror equivalence between thewrapped Fukaya category of the nodal elliptic curve, and the derived category of coherent sheavesof a once-punctured elliptic curve. This argument may be generalized to show that the Fukayacategories of elliptic curves with more nodes are derived equivalent to elliptic curves with thecorresponding number of punctures, by considering the monodromy of the n-nodal degenerationof elliptic curves considered by Gross-Siebert [GS03].

This can be generalized to the case of punctured elliptic curves with nodes by considering f : X →C the affine Tate family of n-punctured elliptic curves, with f−1(0) being an elliptic curve with asingle node and n punctures. Using the mirror equivalence of [LP12, LP17] for the general fiber,we can obtain mirror derived equivalences between wrapped Fukaya categories of elliptic curveswith n punctures and m nodes, and derived categories of coherent sheaves of elliptic curves withm punctures and n nodes (using a stronger version of Lemma 4). In this case, since f satisfies thehypotheses of Theorem 1 we have an additional mirror equivalence between the coherent sheavesof elliptic curves with m punctures and n nodes with the Fukaya-Seidel category of an LG mirror(X × C, zf).

5.3 GENERALIZATIONS

The above examples for elliptic curves are special cases of a more general relationship betweenlarge complex structure limits and homological mirror symmetry.

THEOREM 2. Suppose B is an integral affine manifold, and let X and X be the correspondingmirror pair. Assume we are given a homological mirror symmetry equivalence between X andX that identifies the monodromy functor µ around the large complex structure limit X0 with thetensor product by a line bundle L−1 on X whose first Chern class is the negative of the Kahlerform on X. Then the large complex structure limit X0 of X is homologically mirror to the largevolume limit of X:

DπF (X0) ' DbCoh(X \ s−1(0))

where s is a section of L.

Proof. Let X, X be a mirror pair of Kahler manifolds coming from an integral affine manifoldB. By assumption, the natural transformation µ → id given by monodromy around the largecomplex structure limit corresponds under our choice of homological mirror symmetry equivalenceto a natural transformation L−1⊗ → id given by multiplication by some section s : OX → L.Now we can compare the localizations on both sides: by Lemma 4, the localization of DbCoh(X)at this natural transformation gives the coherent sheaves on the complement, DbCoh(X \ s−1(0)).Since s is a section of L, the divisor s−1(0) is dual to the Kahler form ωX , and so X \ s−1(0) isthe large-volume limit along ωX . Since the total space of the large complex structure limit familyis smooth [GS03, p.26], on the A-side, by Definition 1, we know that the localization of F (X) atthe natural transformation given by monodromy gives F (X0), the Fukaya category of the singularcentral fiber. Therefore we have a homological mirror symmetry quasiequivalence:

DπF (X0) ' DbCoh(X \ s−1(0))

between the large complex structure limit and the large volume limit. �

30

A mirror symmetry equivalence as required in Theorem 2 is expected to follow from combiningAbouzaid’s family Floer theory with ideas from the Gross-Siebert program as in [AGS], as follows.

Remark 5. Starting with the base B with its integral affine structure, we define X to be TB/T ZBas a Kahler manifold. In this case, Gross-Siebert [GS06] construct a degeneration X with fibers Xt

over the formal disk t ∈ ∆, with Xt∼= X symplectically. Then the action of the counterclockwise

monodromy around t = 0 is equal to translation by a canonical section σ1 : B → TB that islocally given by the graph of the developing map of B [GS10, p.79]. The developing map is anintegral affine immersion δ : B →MR from the integral affine universal cover B to MR, so that itsgraph σ1 ⊆ B ×MR is given in local integral affine coordinates by (y, y) [GS06, p.13]. Under themap B ×MR 7→ TB/T ZB, this is sent to the section:

y 7→n∑i=1

yi∂

∂yi

which is exactly the canonical section σ1 described in [GS16]. In [GS16, §4], it is explained thatthe family Floer functor DπF (X) → DbCoh(X) from the forthcoming [AGS] should take σ1 tothe ample line bundle L giving the Kahler form on X. This can be seen from the fact that theLegendre transform of the developing map gives exactly the tropical affine function on the mirrordefining the Kahler form. Now, the family Floer functor [Abo17] should have the property that thefiberwise translation by a section σ is mirror to tensoring by the mirror line bundle L. Thereforethe functor µ−1 given by the counterclockwise monodromy of Xt around t = 0 should be mirrorto the functor L⊗ given by tensoring by the line bundle L.

Provided the family Floer functor of [AGS] can be extended in the same manner to the case wherethe Lagrangian torus fibration has singularities, a result similar to Theorem 2 should also hold inthis case with a suitable modification of Definition 1.

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