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Mirror symmetry and Fukaya categories of singular hypersurfaces

We consider a definition of the Fukaya category of a singular hypersurface proposed by Auroux, given by localizing the Fukaya category of a nearby fiber at Seidel's natural transformation, and show that this possesses several desirable properties. Firstly, we prove an A-side analog of Orlov's derived Kn\"orrer 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 mirror symmetry at various large complex structure limits, in the context of forthcoming work of Abouzaid-Auroux and Abouzaid-Gross-Siebert.


INTRODUCTION
At a basic level, homological mirror symmetry (HMS) conjectures a relationship between the Fukaya category of a Kähler manifold Y and the category of coherent sheaves on a 'mirror' Kähler manifoldY of the same dimension. This is also expected to hold in the reverse direction, relating the Fukaya category of the mirrorY and the coherent sheaves on the original Kähler manifold Y . In many natural instances of mirror symmetry (such as when Y is non-compact), the 'intrinsic' mirrorY is often expected to be a singular variety. However, a principled definition of the Fukaya category ofY is lacking in this case. Some of the most important instances, discussed in detail in §5.1, are the higher-dimensional pairs of pants Π n = {x 1 + · · · + x n+1 + 1 = 0} ⊆ (C * ) n+1 , whose naturally-constructed mirrors are given by the singular hypersurfaces {z 1 · · · z n+1 = 0} ⊆ C n+1 . Using tropical techniques, it is possible to decompose other types of non-compact Kähler manifolds into products of pairs of pants of various dimensions; thus understanding mirror symmetry in such cases 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 that the mirror of the mirror should be the original Kähler manifold; in other words, HMS should apply in both directions. However, this immediately runs into the problem that many 'intrinsic' mirror constructions naturally produce singular varieties, and without an understanding of their Fukaya categories, it is difficult to proceed further. While incredible progress has been made understanding mirror symmetry for smooth Kähler manifolds, outside of the orbifold case little has been written about 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 symplectic geometry of the hypersurface itself may be difficult to understand intrinsically, given a smoothing of this hypersurface, the nearby fibers are perfectly good smooth symplectic manifolds, and their Fukaya categories come equipped with extra algebraic data: a set of morphisms coming from counts of certain holomorphic sections which capture which parts of the manifold degenerate when passing to the singular fiber. In accordance with the philosophy of perverse schobers [KS14], the invariant cycles theorem suggests that performing the categorical localization at these morphisms would therefore 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 geometric applicability. 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 fibration f : X → C with a single singular fiber over 0, Seidel [Sei09b] defines a natural transformation s : µ → id where µ is the clockwise monodromy functor acting on the wrapped Fukaya category W(f −1 (t)) of the general fiber (see §2 for our definition). The following definition for the wrapped Fukaya category of the singular fiber was proposed by Auroux: DEFINITION 1. (Auroux) Suppose a singular symplectic fibration f : X → C has precisely one singular fiber, over 0. Then the wrapped Fukaya category of f −1 (0) is defined to be the localization of the wrapped Fukaya category of a nearby fiber f −1 (t) for sufficiently small t = 0 at the natural transformation 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 category of twisted complexes. The definition can also be extended to the case of monotone fibers, or any situation where Seidel's natural transformation is defined.
In this paper, we shall prove several results that illustrate why this is indeed the correct definition of the Fukaya category of a singular hypersurface: an A-side analog of Orlov's derived Knörrer periodicity theorem (Theorem 1), and several homological mirror symmetry equivalences at the large 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 of Biran-Cornea's Lagrangian cobordism groups in §2.1. There is also a natural analog of the 'nearby cycles' functor one can construct that restricts Lagrangians from the wrapped Fukaya category of a punctured neighbourhood of the singular fiber to the wrapped Fukaya category of the singular fiber itself (cf. [Aur18]).

KNÖRRER PERIODICITY
The first theorem is inspired by a result of Orlov [Orl06, Corollary 3.2]: THEOREM. (Orlov's Derived Knörrer 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 where z is the coordinate on C.
Though [Orl06]  THEOREM 1. (Derived Knörrer Periodicity) Suppose that X is a smooth affine variety with an embedding X → C N inducing a Stein structure on X, and suppose f : X → C is the restriction of a polynomial function on C N . If f has a single critical fiber f −1 (0) then for sufficiently small t = 0 there is a quasiequivalence of A ∞ -categories Here D π is used to denote the idempotent-completion of the twisted complexes; we expect that this result should also hold before taking idempotent-completions, though our proof relies on this step. The hypotheses could almost certainly be weakened, but they suffice for applications to mirror symmetry.
Note that this is a different form of 'suspension' or 'periodicity' for an LG model from that usually considered in singularity theory (compare [Sei09a]).
The results above need not be limited to singular hypersurfaces. For singular complete intersections, we can make the definition: DEFINITION 2. Suppose f 1 , . . . , f k : X → C are holomorphic functions on a Stein manifold such that for t 1 , . . . , t k = 0, the intersection f −1 where s i are the natural transformations coming from the monodromy of f i around t i = 0, and t i are sufficiently small.
By iterating our proof, we also expect the main theorem to admit a simple generalization to singular complete intersections: Under appropriate hypotheses on f 1 , . . . , f k , we have a quasiequivalence of A ∞ -categories: where z 1 , . . . , z k are coordinates on C k .
We shall discuss below how this conjecture could be proved by an extension of the results in this paper.

MIRROR SYMMETRY
The definition of the Fukaya category of a singular hypersurface given above manifestly depends on a choice of smoothing. This is not only desirable, but in fact crucial, for homological mirror symmetry purposes. Classically, mirror symmetry is a relation between a Kähler manifold and a large complex structure limit (LCSL) family of complex manifolds. Homological mirror symmetry is expected to be an involution, so it is important to have a notion of the Fukaya category of the singular 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 involve the data of the smoothing.
On the other hand, since the germ of an isolated hypersurface singularity has a smooth and connected versal deformation space [KM98,p. 144], a simple homotopy argument shows that the Fukaya category (of the germ) is independent of the choice of smoothing in this case. Hence it provides a (potentially interesting) symplectic invariant of isolated hypersurface singularities. In general, extra data must be provided for the Fukaya category to be uniquely specified: examples are provided in §5. Expectations from mirror symmetry suggest that this extra data should take the form of a log structure on f −1 (0), since in good cases this is expected to determine a smoothing of f −1 (0). It may be possible to formulate an intrinsic construction of the wrapped Fukaya category of a singular hypersurface with a log structure using Parker's theory of holomorphic curves in exploded manifolds (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 determine the wrapped Fukaya category, which could be constructed intrinsically using a formalism such as Joyce's theory of d-critical loci [Joy15].
Several papers studying mirror symmetry for the A-model of singular varieties have appeared in the literature, using Orlov's LG model (X × C, zf ) to give a definition for the A-model. For instance, two papers by Nadler [Nad19, Nad17], where a microlocal version of the A-model is used.
In this case, our periodicity theorem admits another proof, which we shall sketch in §5.1, as well as a simple proof of Nadler's result in our terminology. We illustrate how our definition allows this to be generalized to the case of complements of hypersurfaces in toric varieties, appealing to forthcoming work of Abouzaid-Auroux [AA].
This is a special case of a more general mirror symmetry statement, which can be understood in the context of the Gross-Siebert program: THEOREM 2. Suppose B is an integral affine manifold, and let X andX be the corresponding mirror pair. Assume we are given a homological mirror symmetry equivalence between X anď X that identifies the monodromy functor µ around the large complex structure limit X 0 with the tensor product by a line bundle L −1 onX whose first Chern class is the negative of the Kähler form onX. Then the large complex structure limit X 0 of X is homologically mirror to the large volume limit ofX: where s is a section of L.
Such a homological mirror symmetry equivalence is expected to follow from combining Abouzaid's family Floer theory with ideas from the Gross-Siebert program as in [AGS]. Further explanation is provided in §5.3: as not all the details of [AA, AGS] are known to us, these arguments are necessarily somewhat heuristic, neglecting points such as local systems, brane structures, and so forth.
Theorem 2 should be taken more as an illustration that Definition 1 should yield expected mirror symmetry 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 an elliptic curve with n nodes, and the derived category of coherent sheaves of an elliptic curve with n punctures. It is our understanding that these instances of homological mirror symmetry were not previously known.
Remark 1. Many strategies for proving homological mirror symmetry work in the opposite direction 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 the nearby fibers using a deformation theory argument. While understanding the Fukaya category of the LVL is easier than understanding that of the nearby fibers, the LCSL is a singular variety and so 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 both approaches to gain an understanding of HMS at all points of the moduli space.

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 affine variety with an embedding X → C N inducing a Stein structure on X, and suppose f : X → C is the restriction of a polynomial function on C N . If f has a single critical fiber f −1 (0) then for sufficiently small t = 0, we have a quasiequivalence of A ∞ -categories: 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 result of 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-CY correspondence.
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 → C N inducing a Stein structure on X, and suppose f, g : X → C are restrictions of polynomial functions on C N . If f has single critical fiber f −1 (0) then for sufficiently small t = 0 we have a fully faithful functor 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 having a single critical fiber f −1 (0); suppose g : X → C is another holomorphic function. Then for δ, |t| > 0 sufficiently small there is a quasiequivalence of A ∞ -categories We shall then show in §3 that passing from (X × C, z(f − t)) to (X × C, zf ) can be rephrased as 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 be obtained as a quotient of the category W(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 of the natural transformation µ → id on W(f −1 (t)) split-generates the same full subcategory as D, using a Künneth-type argument. Our Theorem 1 then follows.

DEFINITIONS AND CONVENTIONS
We begin with a discussion of how to define the Fukaya-Seidel category of a Landau-Ginzburg model. Suppose that X is a smooth affine variety with an embedding i : X → C N ; then X becomes a Stein manifold with the Stein function φ : X → R given by the restriction of φ(z) = |z| 2 on C N . Suppose that f : C N → C is a polynomial; by abuse of notation we will denote the restriction to X also by f . Suppose for now that C carries the standard Stein structure. We would like to turn the pair (X, f ) into a Landau-Ginzburg model, in particular, talk about its Fukaya-Seidel category.
Firstly, the function |f | 2 defines a real polynomial on R 2N , and X ⊆ R 2N is a real affine algebraic variety. It is a well-known result from real algebraic geometry that the set of points in R for which the Malgrange condition for |f | 2 : X → R fails is finite (for instance see [Spo02, Remark 3] and take the intersection with the algebraic variety X). Recall that this Malgrange condition at λ ∈ R says that there exists R, ε, η > 0 so that if |z| > R and ||f | 2 (z) − λ| ≤ ε then Without loss of generality, |f | 2 (0) = 0, and since we have that |∇ X |f | 2 | > 0 for |f | 2 > 0, we see that for any λ ∈ R >0 for which the Malgrange condition holds, there exists a C > 0 (depending on 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 | 2 on X. Then there exists some constant C so that for all 0 < |f | 2 < δ we have: Choose m ∈ N sufficiently large so that m > 4C and for any constant D > 0 define a Stein function via ψ(z) = φ(z) + Dφ(z)|f | 2m which can be induced using the algebraic embeddingĩ : X → C 2N given by z → (i(z), √ Df m (z)i(z)). This has a regular homotopy to the original Stein structure given by the family with t ∈ [0, 1] PROPOSITION 1. The Liouville vector-field of ψ is outward pointing along |f | 2 = δ for D > 0 sufficiently large.
Proof. We study the inner product By the Cauchy-Schwarz inequality and thus we will have ∇ X |f | 2 , ∇ X ψ > 0 so long as Rewriting this inequality gives follows by our construction.
Now define a stopped Liouville domain as follows. First, take the subset {|f | 2 ≤ δ} ⊆ X, which by Proposition 1 gives a Liouville domain, then add in a stop given by the hypersurface Because this stop is a hypersurface, there is a corresponding Liouville sector which we denote by (X, f ), and write W(X, f ) for the partially wrapped Fukaya category as defined in [GPS19]. This is the (fiberwise wrapped) Fukaya-Seidel category of the Landau-Ginzburg model f : X → C.
Remark 2. We shall henceforth assume that this procedure has taken place and that the Stein function φ on X is already given by ψ. We can always equivalently consider (X, f ) as a Liouville manifold with a stop f −1 (−∞) sitting inside the contact boundary at infinity. In this case, though we may informally discuss our Landau-Ginzburg models as living over all of C, their corresponding Liouville 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 made to lie outside the Liouville domain. This has several useful consequences.
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 is defined. The parallel transport vector field is always a unit complex multiple of Because φ : X → R is a proper exhausting function, it suffices to show that the image under φ of the flow lines γ(t) of the parallel transport vector field do not escape to infinity in finite time. Therefore consider the derivative Since there are only finitely many points in C where the Malgrange condition fails for the complex polynomial f : X → C, we have a constant C > 0 and some ε > 0 so that Therefore we have that on 0 < |f | < ε, for some constant C, Hence by Grönwall's inequality which completes the proof.
Often we shall want to have the freedom to use a different Stein function to compute Fukaya categories. We now record here a standard lemma we shall use throughout, which however only appears implicitly in the literature.
DEFINITION 3. Suppose X is a Liouville manifold with Liouville form λ; a smooth family λ t , t ∈ [0, 1] of Liouville forms for X with λ 0 = λ such that the union of the skeleta of the Liouville structures λ t 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 points remain in a compact set.
LEMMA 2. Given a Liouville manifold X and a simple Liouville homotopy λ t , all of the Liouville manifolds (X, λ t ) are exact symplectomorphic and the wrapped Fukaya categories W(X, λ t ) are all quasiequivalent.
Proof. By [CE12, Proposition 11.8] for every simple Liouville homotopy there is a family of exact symplectomorphisms φ t : X → X with φ * t λ t = λ − df with f compactly supported, such that φ 0 is the identity. This gives rise to a trivial inclusion of Liouville sectors and so by [GPS19, Lemma 2.6] this deformation yields a quasiequivalence of wrapped Fukaya categories between W(X, λ 0 ) and W(X, λ 1 ).
We are hence free to work with deformed Liouville structures in our proofs when considering only the Fukaya category up to quasi-equivalence (by [Sei08b, Corollary 1.14]). There is a similar version 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}; then there is a simple Liouville deformation on X (supported in a neighbourhood of f −1 (U )) that makes f −1 (U ) exact symplectomorphic to the product f −1 (δ) × U with the product 1-form, preserving the 1-forms of the fibers.
The proof corresponds to the discussion around [Sei08b, Lemma 15.3]. Summarized, by Lemma 1 we can trivialize f over U using symplectic parallel transport, so that smoothly Since the symplectic parallel transport induces exact symplectomorphisms of the fibers, up to an exact form, the symplectic form pulled back to f −1 (δ) × U is given by and κ is the symplectic connection form. There is then a deformation for s ∈ [0, 1] and c > 0 sufficiently large. As in [Sei08b,Lemma 15.3] this can be supported in a neighbourhood of f −1 (U ) by cutting off appropriately, and is a simple deformation since it preserves the fibers.

CAP AND CUP FUNCTORS
Now we wish to define some functors introduced in [AS] in the language of [GPS19]. Firstly, if F ⊆ ∂ ∞ X is a Liouville hypersurface, then there is the Orlov functor given by taking small counterclockwise linking disks of the stop F [GPS19, Syl19b]. Henceforth we shall assume that F is the fiber of a Landau-Ginzburg model f : X → C as explained above in §2. In this case F = f −1 (−δ) we call the functor W(F ) → W(X, F ) the cup functor ∪: see Figure 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 : ∪∩ → id and η : id → ∩∪ respectively, which we may complete to exact triangles of bimodules. These exact triangles in fact have a geometric characterization in terms of earlier work of Seidel: THEOREM. (Abouzaid-Ganatra, [AG]) There are exact triangles of bimodules on D π W(X, F ) and ∩∪ id is the clockwise monodromy acting on the fiber and σ : 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 of the 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] so that 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 the negative Dehn twist.
None of these results are logically necessary for the proof of Theorem 1 of this paper, and for the proofs following the reader may take µ, σ, s to be defined via the purely algebraic definition given above. Then we may make the definition: Suppose a singular symplectic fibration f : X → C has precisely one singular fiber, over 0. Then the wrapped Fukaya category of f −1 (0) is defined to be the localization of the wrapped Fukaya category of a nearby fiber f −1 (t) for sufficiently small t = 0 at the natural transformation s : µ → id: The condition that |t| = 0 be sufficiently small is made to exclude the possibility that t is a point where the Malgrange condition fails; in our definitions earlier in §2 we define our Landau-Ginzburg models so as to exclude such points.
Note that this definition is equivalent to taking the quotient of the category W(f −1 (t)) by the full subcategory of the cones of the natural transformation, that is, the quotient by the essential image 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 essential image of ∩.
Proof. One inclusion is clear. To show the reverse inclusion, consider the two exact triangles relating the composition of the ∩ and ∪ functors; importantly, note that these two triangles have arrows in reverse directions with the conventions taken above, where ε and η are the counit and unit of the ∪ − ∩ adjunction respectively. We shall apply ∩ to the first triangle and precompose the second triangle with ∩. Exactness and the identity µ∩ = ∩σ [AG] yields two triangles: The unit-counit identity ∩ε • η∩ = id ∩ for an adjunction implies that the map µ∩ → ∩ in the second triangle is zero and hence that this triangle splits. Thus ∩ ∪ ∩L always has ∩L as a direct summand.
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 of the proof of Theorem 1.
Definition 1 also has a particularly natural relation to Lagrangian cobordisms inside symplectic fibrations (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 of X) 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 having only positive ends; these nullcobordisms thus represent all the equivalence relations imposed on the group of Lagrangian cobordisms. But these relations say that every complex in the image of ∩ must be equivalent to zero, which by Corollary 1 is an equivalent description of the Fukaya category W(f −1 (0)). We therefore conjecture that this construction describes the Grothendieck group K 0 (D π W(f −1 (0))).

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 f t (z) = f (z) − t.
Firstly, observe that z(f − t) has only a single critical fiber, occurring where z(f − t) = 0. This fiber is given set-theoretically by (f −1 (t) × C) ∪ (X × {0}); the critical locus is given exactly by f −1 (t). Observe that for t = 0, this critical locus is therefore smooth, and we have an explicit local Morse-Bott model, described in further detail below.
The corresponding Liouville structure on this complement comes from the restriction of the Stein function to the fiber. Using the Stein function |z| 2n on the factor C this is hence given by This Stein function may be obtained by taking the embedding of X \ f −1 (t) into C N +1 given by j(z) = i(z), ε ft(z) . We shall instead use a deformation-equivalent Stein structure on the complement, given by for suitable constants C > 0 and n ∈ N. This Stein function is given by the embedding of X \f −1 (t) into C 2N given by z → (i(z), √ Ci(z)/f t (z) n ) and is Stein deformation-equivalent to the previous under the deformation for s ∈ [0, 1] given by To understand how the LG models of (X × C, z(f − t)) for t = 0 and t = 0 differ, we need to study how the Liouville structure of the general fiber changes. The following elementary example illustrates the procedure.
Example 1. We recall how to build standard Weinstein structures on Lefschetz fibrations. Consider f : C 2 → C the standard Lefschetz fibration, and equip C 2 with the standard Stein structure. Consider the skeleton of the complement of the fiber f −1 (t). For t = 0, we see that C 2 \ 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 C 2 has a single critical point at 0, of index 2. When we take t = 0, the Weinstein function on X \ f −1 (t) now includes this critical point. Therefore C 2 \ f −1 (t) is obtained from C 2 \ f −1 (0) by attaching a critical Weinstein handle to C 2 \ f −1 (0). The skeleton of C 2 \ f −1 (t) now includes the core of this handle, attached to 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 Liouville vector field of ψ is inward-pointing on {|f t | 2 = δ} for all δ > 0 sufficiently small.
Proof. We consider the inner product By applying Cauchy-Schwarz, we see that we will have Again, by choosing |t|, δ > 0 sufficiently small we can ensure that the Malgrange condition is satisfied on |f t | 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 homotopic to the Stein function from Proposition 1 in place of φ so as to have both Proposition 1 and Proposition 3 hold simultaneously (on concentric disks). The resulting inequality holds on ε ≤ |f t | 2 ≤ δ for δ > ε > 0 sufficiently small and C, n chosen sufficiently large, given D, m.
We now apply Eliashberg's surgery theory for Weinstein manifolds to understand how the general fiber changes as we vary t: 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 (B δ ), both equipped with the Stein function: By Proposition 3, for δ > 0 sufficiently small and n, C sufficiently large, we can take B δ a ball around 0 so that the Liouville vector field for ψ 0 on X is inward-pointing on |f | −1 (δ). Possibly shrinking δ > 0 further, there are no zeroes of the Liouville vector field in f −1 (B δ \ {0}), since the zeroes of φ on X are isolated under the map f : X → C. Hence we have a trivial Weinstein cobordism between X \ f −1 (0) and X \ f −1 (B δ ), and hence an isomorphism of Liouville manifolds [CE12].
Taking |t| < δ, we now pass from First, there is a deformation of Weinstein structures on X \ f −1 (B δ (s)) (all diffeomorphic for |s| < δ) given by: For ε > 0 sufficiently small, none of the critical points of ψ s outside f −1 (B δ ) enter f −1 (B δ (s)) as s goes from 0 to t: hence this is a simple Liouville deformation and so ψ 0 and ψ t give isomorphic Liouville structures on the manifolds X \ f −1 (B δ ) and X \ f −1 (B δ (t)) (Lemma 2). Now, since the Liouville vector field for ψ t is inward-pointing along f −1 (∂B δ ) note that f −1 (B δ ) \ f −1 (t) gives a Weinstein cobordism from X \ f −1 (B δ ) to X \ f −1 (t). Hence by Eliashberg's surgery theory for Weinstein manifolds [CE12], we obtain X \ f −1 (t) by attaching Weinstein handles along the boundary of X \ f −1 (B δ ): see Figure 4. Any critical handles given by Proposition 4 will be referred to as additional handles. By applying the 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 of categories between W(X × C, zf ) and the quotient of W(X × C, z(f − t)) by the full subcategory D of linking disks of the stable manifolds of the additional handles in X \ f −1 (t): PROPOSITION 6. For f a function on a Stein manifold X as above, the category W(X, f ) is split-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 ). Proof. We begin by studying the wrapped Fukaya category of X \f −1 (B δ ) with the Stein structure given by: with δ, C, n chosen as in the proof of Proposition 4. As in Remark 4, by changing φ we have some R > δ such that the Liouville vector field of ψ is also outward pointing along |f | −1 (R). To this we 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 Λ i along which the additional handles i are attached.
By the generation result [GPS19, Theorem 1.10], W(X \f −1 (B δ ), F ∪Λ) is generated by a collection of two kinds of Lagrangians: • Linking disks of F and of Λ i ; • 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 of F in terms of twisted complexes of the Lagrangians L and the linking disks of Λ i , as illustrated in Figure 5. Note that the arc γ with which we take the product lies below the stops F, Λ in the complex plane. 1.20] we know that W(X \ f −1 (t), f ) is generated by the additional cocores i and the Lagrangians L (cf. [GPS19, p.68]). We can take these Lagrangians L to be fibered over an arc γ from −R to f −1 (t) that passes below the singular fiber of f and below the stop F . Now we consider the Weinstein deformation given by: Arguing similarly to Remark 4, for all sufficiently small ε > 0 we can choose φ, larger C, n, and smaller R and δ so that there exists some R > R + ε so that all of the points where the Malgrange condition fails for |f | 2 lie outside of |f | ≤ R + ε and so that • for 0 < |t| < R + ε the Liouville vector field of ψ t 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 compact set and hence ψ t gives simple Weinstein deformations of X \ f −1 (t) for 0 < |t| < R + ε. Thus the resulting X \ f −1 (t) are all exact symplectomorphic and the wrapped Fukaya category W(X \ f −1 (t), f ) remains unchanged (Lemma 2).

THE AAK EQUIVALENCE
We shall begin by proving the smooth case of Knörrer periodicity for hypersurfaces: THEOREM 3. (Abouzaid-Auroux-Katzarkov Equivalence) Suppose that X is a smooth affine variety with an embedding X → C N inducing a Stein structure on X, and suppose f : X → C is the restriction of a polynomial function on C N . If f has a single critical fiber f −1 (0) then for sufficiently small t = 0, we have a quasiequivalence of A ∞ -categories: 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 F t = z(f − t) in the following. We would like to remind the reader of the conventions in Remark 2 concerning the way we construct Liouville structures associated to Landau-Ginzburg models.
Firstly by Proposition 2 since f −1 (t) is smooth for t = 0, we may smoothly identify V = f −1 (B δ (t)) with f −1 (t) × C, where B δ (t) is a small ball around t ∈ C. Then in this open set V × C, the map F t : X × C → C is given explicitly by the local Morse-Bott model f −1 (t) × C 2 → C via (x, y) → xy where x, y are coordinates in C 2 .
By Proposition 2, over an open set U where f : X → C is smoothly trivial, there exists a Liouville deformation λ t to a product Liouville structure on f −1 (U ) ∼ = f −1 (t) × U . We can make this deformation constant outside a larger open set, and since this deformation preserves the fibers this is a simple Liouville homotopy (Definition 3). We first apply this construction to U = B δ (t) a small neighbourhood of the smooth fiber f −1 (t) containing no critical points of f . Passing to the product f −1 (B δ (t)) × C with the standard Liouville structure on C gives a deformation that remains simple: this deformation gives an isotopy of the fiber of F t over −∞. By Lemma 2 this deformation does not change the wrapped Fukaya category of (X × C, F t ) and so we are free to work with this altered Liouville structure in our proof. Note that this deformation does not change the induced Liouville structure on f −1 (t).
We apply this argument again, now to the fibration F t : 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 Liouville structure to be of product type, where we take the radial Liouville structure on the base. Note that 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.
given by (x, y) → xy compatible with the almost-complex structures. Choose a cylindrical almostcomplex structure on f −1 (t) and on X, and the standard complex structure on C: we begin by modifying the complex structure on X to agree with the product almost-complex structure on f −1 (t) × C inside F −1 t (D) × C for D ⊆ C a small open disk around 0, and interpolating with the usual almost-complex structure outside this open subset. Since the Liouville structure is of product type near f −1 (t), the resulting almost-complex structure on X × C can clearly also be chosen to be cylindrical. The same construction could be repeated for any (cylindrical) almost-complex structure on C.
Proof. (of Theorem 3) We begin by choosing a collection L i , i ∈ I of cylindrical exact Spin Lagrangians inside f −1 (t) representing the cocores of the induced Stein structure on f −1 (t). For each Lagrangian, consider the thimble T i over L i given by taking the normalized gradient flow of Re(F t ) starting along L i ⊆ f −1 (t) inside X × C. Since each L i is exact, these T i are exact Lagrangian submanifolds of X × C, and the deformation retraction from T i to L i equips them with Spin structures: we claim that they can also be made cylindrical.
Firstly, there exists some ε > 0 sufficiently small so that over B ε (0) the thimbles T i lie entirely inside the Morse-Bott local model f −1 (B δ (t)) × C, by the fact that the exact symplectic structure on f −1 (B δ (t)) is a product. One can check explicitly in the local Morse-Bott model that near F −1 t (0) the T i are cylindrical at fiberwise ∞: they are given by the product of L i with the standard Lefschetz thimble inside C 2 . Now we need to check at infinity in the base. Outside of the neighbourhood B ε (0) (where all our Lagrangians will intersect) we can instead choose to flow by the Liouville vector field rather than the parallel transport, making the resulting Lagrangian cylindrical at ∞. Now, for each T i choose a cofinal sequence T 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 on both directed categories.
Given a collection of thimbles T (jn) i for n = 0, . . . , k with j 0 > · · · > j k , 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 µ k for L ). Therefore for any holomorphic disk defining the operation µ k for the collection T (jn) i , the projection under F t must be a holomorphic disk in C with respect to the standard complex structure (over the entire domain and for every conformal structure). The maximum principle then implies that the holomorphic disk must be contained in F −1 t (0). Another application of the maximum principle with the holomorphic projection f −1 (t) × C 2 → C 2 in the local Morse-Bott model shows that all of the holomorphic disks must be contained inside f −1 (t). We claim that these must regular with the chosen almost-complex structures on X × C using the argument from [Sei08b,§14c]. Because of 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 base C and on the fiber f −1 (t). The latter is surjective, by assumption. For the former, since T (j n+1 ) i always lies over a ray in B ε (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,Lemma 11.5] this linearized Cauchy-Riemann operator on C is injective and hence surjective also. Thus the 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 and shows that the moduli spaces of disks that appear carry the same orientations. Hence we have an equality of A ∞ -categories between O and A. Much the same argument as in the previous paragraph carries over to the construction of continuation maps T 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 into W(X × C, F t ) as a full subcategory, and the composition of this inclusion with the equivalence W(f −1 (t)) → A[C −1 ] gives the desired functor.
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 function z(f − t) the Stein structure can be deformed into a product, the cocores of the index-n critical points of the deformed Stein function will be given by products of cocores of critical handles for f −1 (t) with the ordinary Lefschetz thimble of z 1 z 2 on C 2 . By Proposition 6, these Morse-Bott thimbles are split-generators of W(X, z(f − t)). Compare also [Sei98,AG] for Morse-Bott generation 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 → C N inducing a Stein structure on X, and suppose f, g : X → C are restrictions of polynomial functions on C N . If f has single critical fiber f −1 (0) then for sufficiently small t = 0 we have a fully faithful functor The second category here is fiberwise stopped with respect to g as in [AA], where admissible Lagrangians are fibered over arcs under the original map f : X → C while also being admissible in 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 category W(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 relative skeleton. In the sequel, we shall prove that under strong assumptions on f, g and X, the relative Fukaya-Seidel category is indeed equivalent to the Fukaya-Seidel category W(X, f + δg) for δ > 0 sufficiently small. This equivalence certainly should not hold in general.
Because of this construction, the proof of Theorem 3 carries over exactly to show that taking thimbles over admissible Lagrangians in the critical locus gives a fully faithful functor We expect this to be an equivalence under an appropriate modification of Proposition 6.

PROOF OF MAIN THEOREM
Since there are two Landau-Ginzburg models under consideration, it will be important to establish notation 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, F t ) we shall use instead the notation : We will regard objects of DW(X \ f −1 (t), f ) as living inside DW(X, f ) via the Viterbo restriction map 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 in W(X ×C, z(f −t)) to the linking disk of the core of this additional handle for the Liouville structure on 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 this isomorphism is exactly [AS15, Lemma A.28]: unfortunately the proof sketched therein is difficult to make rigorous. The isomorphism would also follow from largely formal arguments involving the adjunctions 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 over W(f −1 (t)) between Mod is the left Yoneda functor, and T * denotes the pullback 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 F t contains an open neighbourhood diffeomorphic to f −1 (t)×C * where the Liouville vector field is given by the product with the standard Liouville vector field given by: where r = |z| is the radial coordinate. Given any Lagrangian L in W(f −1 (t)), the thimble T L and intersect in a single fiber of F t , say over 0 < δ ∈ C, where we take δ > 0 sufficiently small so that over B δ , the Lagrangian T L will be contained entirely inside the Morse-Bott model. Because of the product structure, in the fiber F −1 t (δ) the Lagrangian T L is given by for K ⊆ C * a small circle around 0. By our definition of the cup functor, in this fiber F −1 t (δ) is given exactly by the cocore ; see Figure 8.
Observe that in the analysis of the Stein structure on X \ f −1 (t) in Proposition 4 every additional cocore must have boundary in f −1 (t): they cannot remain in a compact set, since there are no index-(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 × C to define the Fukaya category is a deformation of this Stein structure, but this exact deformation preserves flow lines of the Liouville vector field. Because of the local form of the Liouville vector field on f −1 (t) × C * ⊆ X \ f −1 (t) this means that is fibered over C * and hence intersects L × K in a single fiber of f . Inside this fiber, is given by ∩ , by our definition of the cap functor, and hence T L and intersect exactly along L ∩ (∩ ) ⊆ f −1 (t); see Figure 8. Thus we have an isomorphism of vector spaces: Moreover, by additivity of the Maslov index under products, this is in fact an isomorphism of graded vector spaces. It remains to match the A ∞ module operations µ k .
We begin with the differential µ 1 . In this case, the intersection of T L and is contained entirely inside the fiber f −1 (t) inside the fiber F −1 t (δ). An iterated version of the maximum principle argument in Theorem 3 applied to the holomorphic maps F t and f shows that the holomorphic disks computing µ 1 : T * Y ( ) (L) → T * Y ( ) (L) are all contained in this fiber f −1 (t) ⊆ F −1 t (δ); and by a variation of the transversality argument there, we see that any almost-complex structures used to achieve transversality for disks in the fiber f −1 (t) can be extended to X × C so as to make the disks in the total space for µ 1 : T * Y ( ) (L) → T * Y ( ) (L) regular also. Now we consider the product operation µ 2 as a simpler prototype of the argument to follow; suppose L 1 , L 2 , are exact cylindrical Lagrangians in f −1 (t), and let T i L (i) = T (i) i be their thimbles as in the proof of Theorem 3. Suppose we have intersection points y 1 ∈ (∩ ) ∩ L 1 , x ∈ L 1 ∩ L 2 and y 2 ∈ T (2) L 2 ∩(∩ ), 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 (y 1 , x, y 2 ) of disks defining the operation: Extend these almost-complex structures to ones on X × C that are of product type on the neighbourhood 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 points y 1 ∈ ∩ T (1) L 1 living in F −1 Figure 9: Identifying the µ 2 operations on the modules: the holomorphic disk is in green.
For the higher µ k operations, we begin by generalizing the above argument for µ 2 . Suppose L i , i = 1, . . . , k are exact cylindrical Lagrangians in f −1 (t), and let be their thimbles as in the proof of Theorem 3. Suppose we have intersection points y 1 ∈ (∩ ) ∩ L 1 , x i ∈ L i ∩ L i+1 for i = 1, . . . , k − 1 and y 2 ∈ T (k) L k ∩ (∩ ), and that we have chosen regular almostcomplex structures J on f −1 (t) (domain and conformal-structure dependent) for the moduli space M k,f −1 (t) J (y 1 , x 1 , . . . , x k , y 2 ) of disks defining the operation: Extend these almost-complex structures to ones on X × C that are of product type on the neighbourhood f −1 (t) × C as described above.
By Theorem 3 and the above argument, these intersection points correspond to intersection points y 1 ∈ ∩ T (1) L 1 living in F −1 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). To demonstrate this, suppose we have such a disk (u, r) ∈ M k,X×C J (y 1 , x 1 , . . . , x k , y 2 ) for u : D → X × C and r a fixed conformal class of marked disks: then we can show that it is determined by its projection to f −1 (t) as follows.
Let π : f −1 (t) × C 2 → C 2 be the (holomorphic) projection to the second factor. We consider the holomorphic 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 (y 1 , x 1 , . . . , x k , y 2 ). Now since the conformal class r is fixed, the disk π • u no longer computes the A ∞ operation µ k . This holomorphic disk (π • u, r) still has boundary lying along the Lagrangians T i , U ⊆ C 2 obtained by the projections π(T (i) L i ), π( ), respectively, as above; and has corners at the intersection points p 1 ∈ U ∩ T 1 , p k ∈ U ∩ T k and the identity elements a i ∈ T i ∩ T i+1 . Now choose γ : [0, 1) →S k a smooth simple path of conformal structures in the compactified moduli space of (k + 1)-pointed disks that travels from the fixed conformal structure r = γ(0) towards the deepest boundary stratum r 0 = lim t→1 − γ(t) where r degenerates in to a one-legged tree of 3-pointed disks r 0 (see Figure 10). We now consider the parametrized moduli spacẽ M k,C 2 boundary consists of disks that contribute to the iterated product operation: µ 2 (µ 2 (· · · µ 2 (a 1 , a 2 ) · · · ), a k−1 ), p k ) = p 1 of p k ∈ U ∩ T k with the identity a i of T i ∼ = T i+1 . As above, there is a unique holomorphic disk in C 2 representing each one of these product operations, and thus there is only one such disk. Hence, sinceM k,C 2 J (p 1 , a 1 , . . . , a k−1 , p k ) is a 1-manifold with boundary, there must be exactly one disk (u, r) with conformal class r under the projection π to C 2 .
We may now prove the main result.
THEOREM 1. (Derived Knörrer Periodicity) Suppose that X is a smooth affine variety with an embedding X → C N inducing a Stein structure on X, and suppose f : X → C is the restriction of a polynomial function on C N . If f has a single critical fiber f −1 (0) then for sufficiently small t = 0 there is a quasiequivalence of A ∞ -categories 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 generated by Lagrangians of the form ∩L; • By Proposition 6, this subcategory D is the same as the subcategory generated by the Lagrangians ∩ 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:

APPLICATIONS TO MIRROR SYMMETRY
For the following applications to mirror symmetry we will make use of the following lemma from algebraic geometry. The proof is fairly simple and a sketch can be found in [Sei08a,p.5]. Suppose X is a smooth quasiprojective algebraic variety, L is a line bundle with a section s, and let U = X \ s −1 (0).
LEMMA 4. Let s : L −1 ⊗ → id be the natural transformation given by multiplying by the section s. Then localizing at s gives an equivalence of categories: A heuristic way to interpret this lemma is that localization gives the connection between removing a divisor on one side of mirror symmetry and degenerating to a special fiber on the other.

THE TOWER OF PANTS
The Landau-Ginzburg model (C n+1 , W n = z 1 · · · z n+1 ) has particular importance in mirror symmetry as it arises as the mirror to the higher-dimensional pair of pants which we denote by Π n−1 = {x 1 + · · · + x n + 1 = 0} ⊆ (C * ) n : see for instance [Nad19, AAK16]. In this section, we will sketch another proof of mirror symmetry and periodicity in this special case, drawing upon forthcoming work of Abouzaid-Auroux [AA].
In the case of this LG model, the general fiber W −1 n (1) is given by the complex torus (C * ) n , for which we know that the wrapped Fukaya category is generated by the single Lagrangian L = (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(C n+1 , z 1 · · · z n+1 ) is generated by the single Lagrangian ∪L. Moreover, Seidel's natural transformation s on the general fiber of this LG model corresponds to a morphism L → L given by multiplication by x 1 + · · · + x n + 1.
Each of the terms in the sum x 1 + · · · + x n + 1 in the natural transformation corresponds to a count of holomorphic curves passing through one of the hyperplanes z i = 0 in the singular fiber z 1 . . . z n+1 = 0 of the LG model.
COROLLARY 2. We have quasiequivalences of categories: Proof. Using the localization definition, the first category is given by the localization of the category of C[x ± 1 , . . . , x ± n ]-modules at the natural transformation id → id given by multiplication by x 1 + · · · + x n + 1. This is the same as the category of modules over the localization of C-algebras of C[x ± 1 , . . . , x ± n ] at x 1 + · · · + x n + 1, that is, the category of coherent sheaves on {x 1 + · · · + x n + 1 = 0} ⊆ (C * ) n : but this is exactly {x 1 + · · · + x n + 1 = −x n+1 } ⊆ (C * ) n+1 , the pair of pants Π n .
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 endomorphism algebra given by the cone of the map End(L) → End(L) given by multiplication by the natural transformation x 1 + · · · + x n+1 + 1 (see Figure 11). Hence W(C n+2 , z 1 . . . z n+2 ) is given by the category of modules over the quotient of C[x ± 1 , . . . , x ± n+1 ] by the ideal (x 1 + · · · + x n+1 + 1). But this is exactly the category of coherent sheaves on {x 1 + · · · + x n+1 + 1 = 0} ⊆ (C * ) n+1 , that is, the pair of pants Π n .
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 a toric variety V defined by a section s ∈ Γ(V, O(H)). The mirror to H ⊆ (C * ) n is given by a toric LG model (Y, W Y ) [Aur18]; compactifying H ⊆ (C * ) n to H ⊆ V adds extra terms to the superpotential, so that the mirror is (Y, W Y + δW ): here W is considered as a secondary superpotential used to wrap fiberwise, as in §4. Hence, an LG version of our definition allows us to conclude that D π W(W −1 Y (0), W ) is equivalent to D b Coh(V \ H) using Lemma 4. Moreover, using Definition 2 it should be possible to generalize this to the case of complete intersections in toric varieties considered in [AA].

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 = C 2 and f = xy; let us directly compute the Fukaya category W({xy = 0}) of the nodal conic. By Lemma 3, to find W({xy = 0}), we just need to quotient W({xy = t}) by the image of ∩. But in this case, the only object to which we can apply ∩ is the standard thimble of (C 2 , xy), and ∩ simply gives the vanishing cycle V ⊆ {xy = t}. Hence W({xy = 0}) = W({xy = t})/ V . Under mirror symmetry of D π W({xy = t}) with D b Coh(C * ), V corresponds to the skyscraper sheaf of a point p = 0, and the quotient D b Coh(C * )/ O p simply gives D b Coh(C * \ {p}), which is quasiequivalent to the coherent sheaves on the pair of pants D b Coh(Π 1 ), 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 this in future work. Now, we consider the case of genus-1 curves with singularities. Consider the map f : X → C given by the Tate family of elliptic curves, with f −1 (0) being an elliptic curve with a single node. We know by [PZ01,LP12] and others that the Fukaya category of the general fiber F (f −1 (t)) is derived equivalent to the category of coherent sheaves Coh(E) on a mirror elliptic curve E. In this case, the monodromy around this singularity is given by the negative Dehn twist around the corresponding vanishing cycle, and takes the Lagrangian L ⊆ f −1 (t), mirror to O E , to the slope (−1) Lagrangian µ(L) ⊆ f −1 (t), mirror to a degree (−1) line bundle L −1 on E [PZ01]. Since the monodromy functor is given by translating fiberwise by µ(L), under this mirror equivalence µ is mirror to tensoring by this line bundle L −1 (see §5.3 below for a discussion). Hence the natural transformation µ → id is mirror to a morphism L −1 → O E , that is, a section s of the dual line bundle L. Hence we may compare the localization of W(f −1 (t)) at this natural transformation with that of Coh(E). On one hand, by Definition 1, this localization of F (f −1 (t)) yields F (f −1 (0)), the Fukaya category of the nodal curve. On the mirror, by Lemma 4, localization of D b Coh(E) at the natural transformation L −1 ⊗ → id given by multiplication by a section s gives exactly the derived category of coherent sheaves of the complement, D b Coh(E \ s −1 (0)). Hence, we have a mirror equivalence between the wrapped Fukaya category of the nodal elliptic curve, and the derived category of coherent sheaves of a once-punctured elliptic curve. This argument may be generalized to show that the Fukaya categories of elliptic curves with more nodes are derived equivalent to elliptic curves with the corresponding number of punctures, by considering the monodromy of the n-nodal degeneration of 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 a single 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 curves with n punctures and m nodes, and derived categories of coherent sheaves of elliptic curves with m punctures and n nodes (using a stronger version of Lemma 4). In this case, since f satisfies the hypotheses of Theorem 1 we have an additional mirror equivalence between the coherent sheaves of elliptic curves with m punctures and n nodes with the Fukaya-Seidel category of an LG mirror (X × C, zf ).

GENERALIZATIONS
The above examples for elliptic curves are special cases of a more general relationship between large complex structure limits and homological mirror symmetry.
THEOREM 2. Suppose B is an integral affine manifold, and let X andX be the corresponding mirror pair. Assume we are given a homological mirror symmetry equivalence between X anď X that identifies the monodromy functor µ around the large complex structure limit X 0 with the tensor product by a line bundle L −1 onX whose first Chern class is the negative of the Kähler form onX. Then the large complex structure limit X 0 of X is homologically mirror to the large volume limit ofX: where s is a section of L.
Proof. Let X,X be a mirror pair of Kähler manifolds coming from an integral affine manifold B. By assumption, the natural transformation µ → id given by monodromy around the large complex structure limit corresponds under our choice of homological mirror symmetry equivalence to 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 D b Coh(X) at this natural transformation gives the coherent sheaves on the complement, D b Coh(X \ s −1 (0)). Since s is a section of L, the divisor s −1 (0) is dual to the Kähler form ωX, and soX \ s −1 (0) is the large-volume limit along ωX. Since the total space of the large complex structure limit family is smooth [GS03,p.26], on the A-side, by Definition 1, we know that the localization of F (X) at the natural transformation given by monodromy gives F (X 0 ), the Fukaya category of the singular central fiber. Therefore we have a homological mirror symmetry quasiequivalence: Coh(X \ s −1 (0)) between the large complex structure limit and the large volume limit.
A mirror symmetry equivalence as required in Theorem 2 is expected to follow from combining Abouzaid'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 T B/T Z B as a Kähler manifold. In this case, Gross-Siebert [GS06] construct a degeneration X with fibers X t over the formal disk t ∈ ∆, with X t ∼ = X symplectically. Then the action of the counterclockwise monodromy around t = 0 is equal to translation by a canonical section σ 1 : B → T B that is locally given by the graph of the developing map of B [GS10, p.79]. The developing map is an integral affine immersion δ :B → M R from the integral affine universal coverB to M R , so that its graph σ 1 ⊆B × M R is given in local integral affine coordinates by (y, y) [GS06,p.13]. Under the mapB × M R → T B/T Z B, this is sent to the section: which is exactly the canonical section σ 1 described in [GS16]. In [GS16,§4], it is explained that the family Floer functor D π F (X) → D b Coh(X) from the forthcoming [AGS] should take σ 1 to the ample line bundle L giving the Kähler form onX. This can be seen from the fact that the Legendre transform of the developing map gives exactly the tropical affine function on the mirror defining the Kähler form. Now, the family Floer functor [Abo17] should have the property that the fiberwise translation by a section σ is mirror to tensoring by the mirror line bundle L. Therefore the functor µ −1 given by the counterclockwise monodromy of X t around t = 0 should be mirror to the functor L⊗ given by tensoring by the line bundle L.