MIRROR SYMMETRY FOR BERGLUND-HUBSCH MILNOR FIBERS

BENJAMIN GAMMAGE

Abstract. We prove the conjecture of Yankı Lekili and Kazushi Ueda on homologicalmirror symmetry for Milnor fibers of Berglund-Hubsch invertible polynomials. The proofproceeds as usual by calculating the “very affine” Fukaya category and then deforming it,relating the local categorical deformations to a calculation of David Nadler. The proof maybe understood as a basic calculation in the deformation theory of perverse schobers.

Contents

1. Introduction 12. Geometric and categorical background 72.1. Microlocal sheaf methods 72.2. Very affine hypersurfaces 92.3. Mirrors to normal crossings 132.4. Deformation theory 163. Geometry of the Milnor fiber 173.1. Sectors from a fibration 173.2. The sector around 0 193.3. Covering spaces 224. Fukaya categories and mirror symmetry 234.1. The very affine Milnor fiber 234.2. Deformation theory 26References 27

1. Introduction

Our story begins with a quasihomogenous polynomial in n variables with n terms,

W : Cn → C, W (x1, . . . , xn) =n∑i=1

n∏j=1

xaiji(1.1)

where the exponent matrix A = (aij) is invertible. In this case we say that W is an invertiblepolynomial, and the dual polynomial W∨ is a polynomial of the same type,

(1.2) W∨(z1, . . . , zn) =n∑i=1

n∏j=1

zajii

with exponents determined by the transpose matrix At = (aji). These polynomials also definea Pontryagin dual pair of finite abelian groups G and G∨, to be defined below, acting on Cn

and for which W and W∨, respectively, are invariant.1

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2 BENJAMIN GAMMAGE

The study of such pairs of polynomials was initiated in [BH93], who conjectured a mirror-symmetry relationship between them: the Landau-Ginzburg models (Cn,W ) and (Cn/G∨,W∨)(or, equivalently, (Cn/G,W ) and (Cn,W∨), if we had a better understanding ofG-equivariantFloer theory) were expected to be dual in the sense of homological mirror symmetry. In thepast two and a half decades, features of this mirror symmetry, especially the relevant coho-mological field theories (of which the A-side theory is the FJRW theory introduced in [FJR]and the genus-0 B-side theory is K. Saito’s theory of primitive forms [Sai81]) and their manyrelations to singularity theory, have been studied in many individual cases. A general proofof homological mirror symmetry for such pairs is described in [GSc]:

Theorem ([GSc]). There is an equivalence

W(Cn,W ) ∼= MF(Cn/G∨,W∨)

between the wrapped Fukaya category of the Liouville sector associated to Landau-Ginzburgmodel (Cn,W ) and the category of G∨-equivariant matrix factorizations of W∨ on Cn.

This theorem is proved using the now-familiar strategy in mirror symmetry, pioneeredby Seidel [Sei03, Sei11] and systematized by Sheridan [She15, She16, She], of studying thecomplement of a divisor on the A-side, proving mirror symmetry there and then deformingthe resulting category by a count of holomorphic disks. In this case, this means beginningwith the equivalence

(1.3) W((C×)n,W ) ∼= Coh(Cn/G∨)

and then finding W∨ as a count of disks passing through the divisor

D = {x1 · · ·xn = 0}.

The fundamental fact which allows this trick to work is the observation that any invertiblepolynomial W can be obtained from the basic case W = x1 + · · · + xn by passing along afinite covering map

ρ : Cn → Cn, (x1, . . . , xn) 7→(∏

xa1j

j , . . . ,∏

xanjj

),(1.4)

ramified only along the divisor D. When restricted to (C×)n, this becomes a finite unramifiedcover, and by definition the group G is its Galois group. Hence the general case can be proventhrough an understanding of this much simpler case.

We use the same fact in this paper, where our concern is not the “singular category”W(Cn,W ) but rather the “nearby category,” the wrapped Fukaya category of the Milnorfiber of C. The study of this category was initiated in [LUb], and it has been described viaexplicit calculations in some special cases: for ADE singularities in [LUa], and for the 2-dimensional case in [Hab] and [CCJ]. As in [GSc], our strategy of proof in this paper allowsus to give a uniform proof for all possible invertible polynomials at once.

Let V = W−1(n) be the Milnor fiber of W in Cn. (Choosing the fiber over n as a represen-tative Milnor fiber is an irrelevant normalization we find helpful in this paper.) We study V

through its “very affine” part V = W−1(n) ∩ (C×)n. In analogy with the Landau-Ginzburgcase, we begin with a mirror symmetry isomorphism

(1.5) W(V ) ∼= Coh(∂Cn/G∨)

MIRROR SYMMETRY FOR BERGLUND-HUBSCH MILNOR FIBERS 3

between the wrapped Fukaya category of V and the coherent sheaf category of the toricboundary

∂Cn/G∨ = {z1 · · · zn = 0}/G∨

of the toric stack Cn/G∨, as described in [GSa, LP20]. As in the Landau-Ginzburg case, thisisomorphism owes its existence to the cover (1.4).

Finally, by understanding the geometry near the deleted divisor V ∩ {x1 · · ·xn = 0}, we

are able to representW(V ) as a deformation ofW(V ), proving the conjecture of Lekili-Ueda:

Theorem 1.6. The wrapped Fukaya category W(V ) is a deformation of Coh(∂Cn/G∨) byW∨. In other words, there is an equivalence of dZ/2 g-categories

W(V ) ∼= MFG∨(C× Cn, tx1 · · ·xn +W∨).

Calculating the deformation. In the proof sketch outlined above, the first step, the calculationof the Fukaya categoryW(V ) of the very affine part, has already been accomplished in [GSa].As often happens, the more difficult step consists in relating this category to the Fukayacategory of the partial compactification W(V ).

However, we observe that, unlike in the situations normally considered by Seidel and Sheri-dan, we are studying only a partial compactification, and the space V is itself a Weinsteinmanifold, so its Fukaya category can be approached using the microlocal-sheaf techniquesdeveloped in [Nad, GPS19, GPSb, GPSa, NS]. We will find that the necessary calculationsin this case have already been performed by Nadler in [Nad19].

As the partially compactified manifold V remains Weinstein, the relation of V and V maybe studied within a neighborhood of the deleted divisor

D := V \ V = V ∩ {x1 · · ·xn = 0}.As we will explain below, this can be understood as a question about the symplectic geometryof the degenerating family

V ∩ {x1 · · ·xn = t},as t varies in a small disk around 0. The results of [GSa] give a decomposition of a smoothfiber in this family into simpler Liouville sectors, and we can analyze degenerations of thosesectors individually. We find that the total space of the degeneration of each such sectoris a copy of the Liouville sector studied in [Nad19]. That paper calculated the category ofmicrolocal sheaves on the Lagrangian skeleton of this sector, and the papers [GPSa, NS]now allow us to understand the results of [Nad19] as a calculation in the wrapped Fukayacategory.

Perverse schobers. The key tool in understanding the deformation of categories discussedabove is the map

f : V → C, (x1, . . . , xn) 7→ x1 · · · xn.(1.7)

Crucially, the G-cover (1.4) happens completely within the fibers of f, so that it suffices toconsider the case when V is the (partially) “compactified pants”

P = {x1 + · · ·xn = n} ⊂ Cn,

on which the map f takes a very simple form: it has only two critical values, one nondegener-ate critical value and one degenerate critical value at 0. The questions of deformation theorytake place around the latter; we want to relate the Fukaya categories of V and V = V \f−1(0).

4 BENJAMIN GAMMAGE

(Note that the cover (1.4) is ramified over the zero fiber of f ; as we shall see, this ramifica-tion, complicating the singularity of the central fiber, accounts precisely for the form of themirror superpotential W∨.)

In other words, we are interested in the behavior of the symplectic geometry of V relativeto the holomorphic map f, an approach again pioneered by Seidel [Sei08] in the case ofLefschetz fibrations; the non-Lefschetz behavior of f near its critical value 0 requires moresophisticated categorical tools. At the same time, the fact that the base C of our fibrationf is itself Stein helps to simplify much of our analysis, allowing us to work locally on thisbase.

The appropriate language for encoding Fukaya-categorical structures relative to a base isthe theory of perverse schobers, as developed in [KS, KS16, BKS18]. A perverse schober onC with two singularities, at the points 0, 1 ∈ C, is equivalent to the data of a diagram ofcategories

(1.8) P[0,1)F // P(0,1) P(0,1],

Goo

where F and G are spherical functors. The notation is intended to suggest that PL is thecategory which P assigns to the Lagrangian L ⊂ C.

In our situation, we can produce such a datum by taking a Liouville-sectorial cover ofthe base by “left” and “right” sectors, each containing one critical value, and lifting thisdecomposition to the total space. In this case, the central category P(0,1) will be equivalent tothe Fukaya category of a general fiber f−1(1

2), and F,G are the spherical “cap” or “boundary

restriction” functors from the Fukaya categories of the two Liouville sectors stopped atf−1(1

2).

The Fukaya category W(V ) should then be calculated as the “global (co)sections” of thisperverse schober, defined as the homotopy colimit

(1.9) W(V ) ∼= P[0,1] := lim−→

(P[0,1) P(0,1)

GL //FLoo P(0,1]

),

where FL, FG are the left adjoints of F and G. (In the language of Fukaya categories, theseare the “Orlov functors” corresponding to the inclusion of f−1(1

2) as a stop in the left- or

right-hand Liouville sectors.)

Similarly, the category W(V ) is computable from a perverse schober P on C×:

(1.10) W(V ) ∼= PLC×:= lim−→

(P ( P(0,1)

//oo P( 12,1]

)where the Lagrangian LC× is the union of a small circle around 0 and a spoke connectingthe circle to the point 1.

In fact, given a sectorial cover as stipulated above, the equivalences (1.9) and (1.10) areimmediate from the [GPSb] theory of sectorial codescent.

Building on prior work [GSa] on mirror symmetry for hypersurfaces in (C×)n, we can

identify all of the pieces in the latter decomposition. Let ∂Cn be the strict transform of ∂Cn

under the blowup of Cn at the origin, and P the exceptional divisor of the blowup, which

meets ∂Cn in its toric boundary ∂P.

MIRROR SYMMETRY FOR BERGLUND-HUBSCH MILNOR FIBERS 5

Proposition. The diagram (1.10) is equivalent to a diagram of the form

(1.11) Coh(∂Cn/G∨) ∼= lim−→

(CohG

∨(∂Cn) (CohG

∨(∂P))oo // CohG

∨({0})

).

The right and middle pieces in the diagrams (1.9) and (1.10) are the same; the onlydifference is in the left-hand category. Our goal is to prove that the category P[0,1) is a

deformation of P[0,1)∼= CohG

∨(∂Cn) by the element W∨ ∈ O(∂Cn/G∨) ⊂ HHev(∂Cn/G∨).

One way to calculate the category P[0,1) is through its relation to the Fukaya category of itsstopped boundary, P(0,1)

∼= W(f−1(12)). The cap functor F , in addition to being spherical,

has the nice property of being conservative, i.e., having trivial kernel. In fact, F is monadic:the monad T = FFL obtained from composition of the cup and cap functors is sufficientto determine P[0,1), in the sense that this category can be reconstructed as the category ofT -algebras in the boundary category W(f−1(1

2)):

(1.12) P[0,1)∼=W

(f−1

(1

2

))T.

From the properties of spherical functors, the monad T can be presented as the cone

(1.13) T = Cone( µs // 1 )

of a certain natural transformation s between the monodromy automorphism µ and theidentity functor of the nearby category P(0,1).

In the case treated in this paper, a generalization of the main computation of [Nad19] shows

that µ is the endofunctor of CohG∨(∂P) given by the line bundle O(−1) on the weighted

projective stack ∂P/G∨, and the natural transformation s is the section W∨. The categoryP[0,1) is therefore equivalent to the category of coherent sheaves on the hypersurface

{W∨ = 0} ⊂ ∂P/G∨,

and a theorem of Orlov allows us to recognize this as a deformation of the category P (=CohG

∨(∂P) by W∨.

Deforming a perverse schober. Why did this work? Recall that the unit and counit of theadjunctions involving the spherical functor F , which define the monad T we studied above,are supposed to encode counts of holomorphic disks whose projections under the map fcontain the point 0 ∈ C. Thus it should be no surprise that the above monadic presentationof P[0,1) presents a deformation of the category P (. This is very natural if we imagine aperverse schober as a device which associates categories to Lagrangians: instead of studyingthe category P[0,1) which P associates to the Lagrangian [0, 1), we could equally well studythe category which it associates to the circle-with-spoke (, which should be a deformationof the category P (, obtained by taking into account the information from the monad (1.13).

The form of this deformation has nothing to do with our particular example and can bewritten in completely abstract terms: observe that P (can be presented as a category whoseobjects are pairs

(1.14)(X ∈ P(0,1), ν : µX → X

)of an object X of the nearby category P(0,1) together with an endomorphism of X twisted

by the monodromy automorphism µ ∈ Aut(P(0,1)

). The natural transformation s which

describes the monad T can be used to specify a class in HH0(P () which we denote by s

6 BENJAMIN GAMMAGE

The class s should then be the deformation of P (by the holomorphic disks with boundaryon the circle in (:

Expectation. The category P[0,1) which the perverse schober P assigns to [0, 1) is a defor-

mation of the category P (by s.

We phrase this as an “expectation” since the theory of perverse schobers and their defor-mations is still provisional, but in the situation of this paper, where P(0,1) is the category ofcoherent sheaves on an algebraic stack and the monodromy µ is given by tensoring with aline bundle, this can be proved directly from Orlov’s work on matrix factorizations.

Future directions. We have already suggested a natural framework for the deformation weconsider, but we conclude with some further suggestions on how the ad hoc constructions ofthis paper might be encapsulated as part of a more systematic theory.

The simplicity of the deformation theory involved in this paper is due largely to the factthat the coefficients of monomials in the deformation class W∨ are irrelevant, so long as theyare nonzero – and moreover, for grading reasons, these monomials are the only ones whichcan appear. (The same phenomenon occurs also in the similar situation treated in [GSc].)

This is in contrast to the situations usually considered by Seidel and Sheridan, whereone is ultimately interested in the Fukaya category of a compact symplectic manifold. Thesituation here has been simplified considerably because the total space V is Stein and thefiber D = V \ V we delete is also Stein. The coefficients in the deformation class should besensitive to the symplectic area of the deleted divisor D, but since D is not compact anddoes not have a well-defined symplectic area, we are free to scale these coefficients howeverwe like. It would be useful to have a more direct exposition of how rescalings of Liouvillestructure in such a situation affect the deformation class.

Also, we note that in the situation considered in this paper, none of the coefficients of W∨

was zero. This follows from an explicit check conducted in [Nad19], but it would be moresatsfying to have a general criterion for when this occurs. From the perspective of the base,this should fit into a more general theory of extensions of perverse schobers, categorifyingthe theory of extensions of perverse sheaves.

Finally, we note that although we were able to reproduce the data of a perverse schober asdescribed in (1.8) using the theory of Liouville sectors, it would be more satisfying to see thisstructure directly at the level of skeleta: for a perverse schober P obtained as pushforwardof a Fukaya category along a map f with Weinstein fibers, the category which P assigns

to a Lagrangian L in the base should be the category associated to a certain lift L of L to

a Lagrangian in the total space. Heuristically, L is obtained from the Lagrangian skeletonL of a fiber by parallel transport over L, collapsing vanishing cycles in L when L meetsa critical value. Such a theory can be implemented “by hand” in the case when f is aLefschetz fibration with base C; for more general singularities, or higher-dimensional bases(as considered for instance in [GMW]), a good general theory does not yet exist and requiresthe development of a better understanding of how to lift Liouville structures.

Conventions. Throughout this paper, we work with pretriangulated dg- or dZ/2 g (i.e., 2-periodic dg)-categories over the field C, in the homotopical context of derived Morita theory,as described in [Toe07] (with adaptation to the 2-periodic context described for instance in[Dyc11, Section 5]), or equivalently stable C-linear ∞-categories.

MIRROR SYMMETRY FOR BERGLUND-HUBSCH MILNOR FIBERS 7

In particular, by Coh(X) or W(X) we always mean the corresponding pretriangulated dgcategory, and all limits, colimits, and equivalences among these should be understood in theappropriate homotopical sense. For a dg category C, we write CZ/2 for the dZ/2 g-categoryobtained by 2-periodicization.

Acknowledgements. I am grateful to Jack Smith for his collaboration on the shared work[GSc], of which this paper is an offshoot. I would also like to thank David Nadler, JohnPardon, Vivek Shende, and Nick Sheridan for helpful conversations and advice.

2. Geometric and categorical background

Here we collect some results from mirror symmetry, toric and tropical geometry whichwill be necessary for the proofs of the main theorems of this paper. Section 2.1 is a reviewof wrapped Fukaya categories of Liouville sectors and their computation, and in Section2.2 we recall some features of the symplectic geometry of hypersurfaces in (C×)n and theirtoric mirrors. Homological mirror symmetry equivalences for these spaces were establishedin [GSa] using the tropical methods of Mikhalkin [Mik04], and those tropical techniques arealso necessary in this paper.

In Section 2.3, we recall a Liouville sector described in [Nad19] which will form a basicbuilding block of the spaces we study here. The monadic calculation from that paper willunderlie the deformation of categories we discuss in Section 4. Finally, in Section 2.4, weexplain how to reinterpret this monad in the language of deformation theory.

2.1. Microlocal sheaf methods. We begin by reviewing some of the results of [GPS19,GPSb, GPSa, NS] on calculation of Weinstein Fukaya categories using microlocal sheaves.The key feature of Weinstein symplectic geometry which makes computations tractable islocality: unlike in compact symplectic geometry, the symplectic behavior of a Weinsteinmanifold can be reconstructed from an open cover by simpler pieces, the Liouville sectorsdefined in [GPS19]. These are Liouville manifolds with boundary which are appropriatefor Weinstein gluings along their shared boundary. We refer to [GPS19] for details on thedefinition and technical properties of Liovuille sectors, and we will summarize here the waysin which Liouville sectors arise for us.

Definition 2.1. Let (X,ω = dλ) be a Liouville domain with boundary ∂X and completionto a Liouville manifold X.

(1) Let F ⊂ ∂X be a real hypersurface with boundary such that (F , λ) is itself a Liouvilledomain. This is a Weinstein pair in the sense of [Eli18], and we denote by (X,F ) theLiouville sector obtained from X by completing away from a standard neighborhoodof F0.

(2) Let Λ ⊂ ∂X be a closed Legendrian. This is a stop in the sense of [Syl19], and wewrite (X,Λ) for the Liouville sector obtained by completing away from a standardneighborhood of Λ.

(3) Let f : X → C be a Liouville Landau-Ginzburg model. Then we write (X, f) for thecorresponding Liouville sector.

The above constructions are all essentially equivalent; for instance, from the third, onecan obtain the first by taking F to be a general fiber of f, and the second by taking Λ tobe the skeleton of F . Recall that the skeleton LX of a Liouville manifold X is the set ofpoints in X which do not escape to infinity under the Liouville flow, and the skeleton (or

8 BENJAMIN GAMMAGE

relative skeleton) of a Liouville sector (X,Λ) is the set of points which do not escape to thecomplement of Λ under Liouville flow. In other words, the relative skeleton of (X,Λ) is theunion of the skeleton LX of X with the cone (under Liouville flow) of Λ.

One way to understand the skeleton LX of a Weinstein manifold X is as recording gluingdata describing a cover of X by simpler Weinstein sectors. If (X1, F1) and (X2, F2) aretwo Weinstein pairs equipped with an isomorphism F1

∼= F2, and we write Λ ⊂ F for theskeleton of this Weinstein manifold, then the Weinstein gluing X1 ∪F X2 will have gluedskeleton L(X1,F1) ∪Λ L(X2,F2).

One source of such gluings will provide the fundamental construction in this paper:

Example 2.2. Let f : X → C be a Liouville Landau-Ginzburg model, and let n ∈ R suchthat no critical value of f has real part n. Then the vertical line H = {z ∈ C | <(z) = n}presents the Liouville manifold X as a pair of sectors XL and XR, glued along a sectorF × T ∗R, where F = f−1(n) is a general fiber of the map f .

A generalization of such gluings along higher-codimensional strata is contained in [GPSb,Section 9.3] in terms of Liouville sectors with (sectorial) corners. The main achievementof [GPSb] consists in using such sectorial covers to establish a local-to-global principle forcalculation of wrapped Fukaya categories of Liouville manifolds.

The wrapped Fukaya category W(S) of a Liouville sector S is defined in [GPS19]. If thesectorial structure S = (X,Λ) comes from a stop, this can be understood as a partiallywrapped Fukaya category; if S = (X, f) is a Landau-Ginzburg sector, then W(S) can beunderstood as a Fukaya-Seidel type category. The main results of [GPS19, GPSb] are thefollowing:

Theorem 2.3 ([GPS19, GPSb]). The wrapped Fukaya categoryW is covariant for inclusionsof Liouville sectors: if S ′ ↪→ S is an inclusion of subsectors, then there is a functorW(S ′)→W(S). Moreover, W satisfies codescent along sectorial covers: if X is a Weinstein manifoldor sector which admits a cover X = U1 ∪ · · · ∪ Un by subsectors Ui, then the natural map

lim−→∅6=I⊂[n]

W

(⋂i∈I

Ui

)→W(X)

from the homotopy colimit of wrapped Fukaya categories of the Ui to the wrapped Fukayacategory of X is an equivalence of dg categories.

This theory heavily reduces the difficulty of computation of the wrapped Fukaya categoryW(X) – it remains only to compute the (hopefully much simpler) categories associated tosubsectors Ui. In this paper, we will barely need to do those calculations, since we willultimately reduce to sectors Ui whose Fukaya categories have already been calculated. Andthese calculations can often be performed simply using the language of microlocal sheaves. Infact, we will find that the microlocal-sheaf-theoretic calculations necessary for this paper havealready been performed in [GSa, Nad19]. Thus, we only need to recall the main theoremscomparing these calculations to wrapped Fukaya categories.

Theorem 2.4 ([NS]). Let X be a stably polarized1 Liouvile sector with skeleton L. Thereexists a cosheaf of dg categories µshW on L. If U ⊂ X is an open subset with an exactequivalence U ∼= T ∗M for some manifold M , then µshW agrees with the “wrapped microlocalsheaves cosheaf” defined in [Nad].

1Cotangent bundles and complete intersections in Cn are stably polarized.

MIRROR SYMMETRY FOR BERGLUND-HUBSCH MILNOR FIBERS 9

Theorem 2.5 ([GPSa, Theorem 1.4]). For X as above, the cosheaf µshW agrees with thecosheaf W of wrapped Fukaya categories of Liouville sectors.

Finally, we highlight one feature of the functoriality for Liouville sectors which plays amajor role in this paper.

Definition 2.6. Let S = (X,F ) be a Liouville sector, thought of as a Weinstein pair. Theinclusion of the subsector F × T ∗R into S determines a functor

(2.7) W(F × T ∗R) =W(F )→W(S),

which the Orlov functor associated to the pair (X,F ). Its left adjoint

∩ :W(S)→W(F )

is the cap functor.

Remark 2.8. We use the names Orlov and cap to suggest that functors defined above shouldagree with the functors traditionally associated with those names, although we do not proposeto develop the general theory of these functors here. It is the author’s understanding thatseveral such accounts are in preparation; the most detailed reference currently existing in theliterature is [Syl]. We content ourselves here with noting that in good cases the cap functoris spherical, in the sense of [AL17], and that this will indeed be the case, as established in[Nad19], for the horizontal cap functor discussed in Section 2.3.

2.2. Very affine hypersurfaces. We will also need some results from [GSa] on skeleta ofhypersurfaces in (C×)n. In fact, we will only be interested in two such hypersurfaces (alongwith their G-covers): the pants, and the mirror to the boundary of projective space. Webegin by recalling the abstract setup, and then we specialize to the cases of interest.

Definition 2.9. Let N be an n-dimensional lattice. To N we associate the following spaces:

• The n-dimensional real vector space NR = N ⊗ R;• The n-torus NS1 = NR/N ;• The cotangent bundle T ∗NS1 = NR/N × N∨R of this torus, whose projections to the

base and fiber we denote by Arg and Log, respectively;• The n-complex-dimensional split torus NC× = N ⊗ C×.

We also choose once and for all an inner product on N in order to identify the cotangentbundle T ∗NS1 with the complex torus NC× (which is more naturally the tangent bundle ofNS1).

An algebraic hypersurface in NC× determines a Newton polytope ∆ ⊂ N∨R , and we requirethat 0 ∈ ∆. We also choose a triangulation T of this Newton polytope whose vertices arevertices(∆)∪{0}. For the examples in this paper, we will always take T to be the triangulationwhose simplices are the cones on the faces of ∆ (and 0). We can thus interpret T equivalentlyas the fan Σ of cones on the faces of ∆.

Inside the cotangent bundle T, we specify a Lagrangian as follows:

(2.10) LΣ :=⋃σ∈Σ

σ⊥ × σ ⊂ (NR/N)×N∨R = T ∗NS1 .

The Lagrangian LΣ was first studied by Bondal in [Bon06], then later studied extensivelyby Fang-Liu-Treumann-Zaslow [FLTZ12a, FLTZ11, FLTZ12b], for the relation between thecategory Sh−LΣ

(NS1) of constructible sheaves on NS1 microsupported along −LΣ and the

10 BENJAMIN GAMMAGE

category QCoh(TΣ) of quasi-coherent sheaves on the toric stack TΣ with fan Σ. Followingthe above works, a complete statement was first obtained in [Kuw20] (followed by anotherproof in many cases in [Zho19]). In modern language, the best statement reads as follows:

Theorem 2.11 ([Kuw20]). There is an equivalence

(2.12) µshW(LΣ) ∼= Coh(TΣ)

between the dg category of wrapped microlocal sheaves on LΣ and the category of coherentsheaves on the toric variety TΣ.

This equivalence was explained in [GSa] by relating LΣ to the Landau-Ginzburg model(NC× ,WHV,Σ), where the Hori-Vafa superpotential WHV,Σ traditionally understood as themirror to the toric stack TΣ is a Laurent polynomial with Newton polytope ∆.

Theorem 2.13 ([GSa]). Suppose that the fan Σ is simplicial and that all generators ofrays in Σ lie on the boundary of the polytope ∆. Then the Lagrangian LΣ is the skeleton ofthe Landau-Ginzburg Liouville-sectorial structure defined on NC× by the function WHV,Σ :NC× → C.

Let ΛΣ = L∞Σ be the Legendrian boundary of the conic Lagrangian LΣ. The above theoremfollows from the following calculation, first performed for WHV = 1+x1 + · · ·xn in [Nad] andgeneralized to a global statement in [GSa] (and partially expanded and corrected in [Zho20]):

Theorem 2.14 ([GSa, Zho20]). With hypotheses as in 2.13, the Legendrian ΛΣ is a skeletonfor the hypersurface HΣ = {WHV,Σ = 0}, which embeds as a Weinstein hypersurface insidethe contact boundary of T ∗NS1 .

2.2.1. Tropical geometry. We will find it useful to recall an outline of the proof for the abovetheorem, although we refer to [GSa] for details. The main ingredient is tropical geometry: werefer to [Mik04] for details but recall here that given a hypersurface H ⊂ NC× whose Newtonpolytope ∆ is equipped with triangulation T , one can define a PL complex Trop(H) ⊂ N∨R ,the tropicalization of H, with the following properties:

• The amoeba Log(H) is “near” to Trop(H), in a sense to be discussed below.• The complex Trop(H) is dual to the triangulation T ; in particular, if the triangulationT comes from a complete fan Σ, then the complement N∨R \ Trop(H) will have onlyone bounded component.• The components of the complement N∨R \ Trop(H) correspond to monomials in a

defining equation for H, with the corresponence associating each region to the mono-mial dominating there.

As we require that the triangulation T = Σ be simplicial, we can reduce to the case of∆ = ∆std the basic simplex in N∨R , so that H = P is the (n − 1)-dimensional pants, andthe skeleton of this variety was calculated in [Nad]. The main technical tool involved in thecalculation is a symplectic isotopy of H, which we call “tailoring,” described first in [Mik04]and studied in detail in [Abo06]:

Proposition 2.15 ([Abo06, Section 4]). There is a symplectic isotopy {Hs}0≤s≤1 given by

Hs =

{ ∑α∈∆∩N∨

(1− sφα)cαzα = 0

},

MIRROR SYMMETRY FOR BERGLUND-HUBSCH MILNOR FIBERS 11

where φα has the property that near a face F of Trop(H), φα ≡ 1 unless the monomial cαzα

dominates in a region of N∨R \ Trop(H) adjacent to F .

In other words, near a k-face F in Trop(H) dual to a standard (n− k)-simplex in T , the

tailored hypersurface H1 is equal to a product (C×)k× Pn−1−k of (C×)k with an (n− 1− k)-dimensional pants. (If F is dual to a larger (n − k)-simplex in T , the second factor in thisproduct will be replaced by an abelian cover.) This means in particular that sufficiently farin the interior of a k-face F , the amoeba A := Log(H1) of the tailored hypersurface agreeswith Trop(H) in the directions tangent to F .

Now the strategy of proof of Theorem 2.14 can be roughly understood as follows. Drawthe fan Σ superimposed on the amoeba A of the tailored hypersurface H1, as illustrated inFigure 1. As mentioned above, the complement of A has a distinguished component, dual to

Figure 1. The fan of P2 superimposed on the tailored amoeba of its mirrorhypersurface H = {x+y+ 1

xy= 0}. Note how this tailored amoeba is precisely

“tropical” away from the vertices.

the origin of Σ, thought of as a vertex in the triangulation T . We denote by A0 the boundaryof this region, and Trop(H)0 for the corresponding subcomplex of the tropical curve.

Lemma 2.16 ([Mik04]). Suppose the coefficients in the Laurent polynomial defining M arereal positive, except for the contant term which is real negative. Then the set

H+ := H ∩ (R>0)n

of positive real points of H map to A0 under the Log map.

Now, one computes that for each k-face F in the polytope Trop(H)0, there corresponds asingle Morse-Bott critical point (indicated by the intersection of F with its dual cone in Σ)with critical locus an (n−k)-torus TF . The Lagrangian skeleton of H can then be describedas a union, over cones σ in Σ (dual to faces Fσ ⊂ Trop(H)0),

LH =⋃

06=σ∈Σ

TFσ × Log−1(σ ∩ A0),

of the downward Liouville flows of these tori. This computation completes the proof ofTheorem 2.14.

12 BENJAMIN GAMMAGE

Remark 2.17. Observe that the positive real locus H+ is a subset of LH , presented as aunion of (n− 1)-simplices corresponding to the top-dimensional cones σ of Σ. If the fan Σ iscomplete, then H+ will be a sphere S (or a disjoint union of spheres if the fan Σ is stacky).

2.2.2. Examples. Now, to simplify notation, we choose a basis N ∼= Zn, and we begin toconsider the cases of particular interest to us. The fundamental example is the following:

Example 2.18. Suppose that Σ = ΣAn is the standard fan of affine space, with rays spannedby basis vectors e1, . . . , en of Zn. If n = 1, then LΣA1 is a “circle with spoke attached”: the

union of S1 with a single conormal direction at 1 ∈ S1, as pictured on the left in Figure 2.In general, LΣAn = (LΣA1 )n; the case n = 2 is pictured on the right in Figure 2.

Figure 2. The skeleta LΣAn of the Liouville sectors mirror to An for n = 1, 2.

The sector ((C×)n, 1 +x1 + · · ·+xn) described in 2.18 is mirror to affine space Cn, but wewill be more interested in the boundary of this mirror symmetry.

Example 2.19. The boundary ∂LΣAn of the skeleton described above is a skeleton for thepants

P = {x1 + · · ·+ xn = 1},which is mirror to the toric boundary ∂Cn of affine space. This toric boundary is a union ofclosed pieces Oσ, where we write Oσ for the toric orbit corresponding to a nonzero cone σin Σ. The closed piece Oσ is itself a toric variety, with quotient fan Σ/σ. Hence in this caseit is an affine space, with skeleton as described in Example 2.18.

The skeleton of P can thus be described as follows: consider an (n− 1)-simplex ∆ (which

one can imagine as the “boundary” of the Newton polytope for P , where any face containingzero is considered part of the interior). The skeleton of P can be understood topologicallyas a union of copies of T k×F for each nonempty (n−1−k)-dimensional subsimplex F ⊂ ∆(including ∆ itself), attached according to the face poset of ∆. From the perspective of eachtorus, the attachment is through conormal tori as described in Example 2.18.

Example 2.20. The other main example of interest to us will be the mirror to the toricboundary of Pn, which is the hypersurface H = {x1 + · · ·xn + 1

x1···xn = 0}. Once again, wecan use the fact that ∂Pn is a union of toric orbit closures to write the skeleton LH of H asglued together from skeleta LΣPk

of the mirrors to Pk for k < n.The resulting skeleton for H can be described just as in the second paragraph of Example

2.19, except that instead of starting with an (n− 1)-simplex ∆, we start with n+ 1 copies of∆, glued into the boundary of an n-simplex (this time, literally the boundary of the Newtonpolytope for H), which we can understand as a triangulation of the sphere S described inRemark 2.17. The tori and subtori in LH are attached according to the combinatorics ofthis complex, just as in Example 2.19.

MIRROR SYMMETRY FOR BERGLUND-HUBSCH MILNOR FIBERS 13

2.3. Mirrors to normal crossings. Following the constructions of [Nad19], we now discussan alternative mirror symmetry equivalence for the (n− 2)-dimensional pair of pants, whichis now understood as living on the B-side.

Definition 2.21. We denote by S ′n the Liouville sector corresponding to the Landau-

Ginzburg model (Cn, x1, . . . , xn).

The general fiber F1 of the function x1 · · ·xn is a complex (n−1)-torus, which degeneratesto {x1 · · ·xn = 0} over the unique critical value 0 of this map.

T = {x1 · · ·xn = 1, |x1| = · · · = |xn|}be the unit torus in the fiber over 1. As discussed in [Nad19], the skeleton LS ′n of this sectoris the parallel transport of T over the real half-line [0,∞), which collapses T to a point inthe fiber over 0.

The wrapped Fukaya category W(S ′n) of this sector is computed in [Nad19] by studying

the spherical “cap” functor

W(S ′n)→W(F1)

and realizing this functor as mirror to the pushforward along the inclusion of a linear hyper-surface in F∨1 = (C×)n−1.

But before we explain this calculation, we will modify the sector S ′n slightly, as our

purposes here will require the A-side mirror to a hyperplane not in (C×)n−1 but rather inPn−1. To understand the appropriate modification of the sector S ′

n, we should reconceptualizeit as determined not by a Landau-Ginzburg model but by a stop at infinity.

If we imagine the general fiber F1 as part of the contact boundary of Cn, then the sectorS ′n is obtained from Cn by specifying the stop T, now thought of as a Legendrian in the

contact boundary of Cn. As a Lagrangian, the n-torus T is the mirror to (C×)n−1, and it issensible to replace it by the skeleton of the mirror to Pn−1. That is what we will do:

Definition 2.22. Let LΣPn−1 ⊂ F1 be the FLTZ skeleton corresponding to the standard

fan of Pn−1, obtained from T by attaching conormals to subtori. We will denote by Sn thesectorial structure on Cn which is specified by the stop LΣPn−1 .

Note that the stop LΣPn−1 we have placed at contact infinity itself has a boundary, makingSn into a cornered Liouville sector, with two different boundary components meeting alongthe corner. These components are easiest to describe in terms of the skeleton LSn , which asbefore is obtained from the stop LΣPn−1 by parallel transport along the ray [0,∞).

Definition 2.23. The horizontal boundary ∂hSn of the Liouville sector Sn is the boundarysector with skeleton LΣPn−1 . The vertical boundary ∂vSn of Sn is the boundary sector with

skeleton given by the parallel transport of ∂LΣPn−1 along [0,∞). Finally, the corner ∂2Sn istheir common subsector, with skeleton ∂LΣPn−1 .

Example 2.24. When n = 2, the Lagrangian LΣP1 is the union of the circle T with theconormal rays to 1 ∈ T. Hence the total skeleton LS2 is obtained from the degeneratingcircle LS ′2

by attaching a half-plane.

Remark 2.25. In the previous example, the attached half-plane does not contribute tothe calculation of the Fukaya category W(S2), since it is not affected by the degenerationat 0. (This is mirror to the fact that a general hyperplane in P1 does not intersect its

14 BENJAMIN GAMMAGE

boundary. In fact, the sector Sn was introduced in [Nad19] precisely to prove the secondassertion of Lemma 2.28 below.) In higher dimensions, the corner skeleton contains toriwhich will degenerate over 0 and hence affect the computation of W(Sn). Nevertheless,for convenience we will retain also those components of the corner skeleton which do notcontribute to W(Sn).

We will find it useful later to observe that the vertical boundary ∂vSn can be describedinductively in terms of Sk for k < n.

Lemma 2.26. Let Σ = Σ∂Pn−1 be the poset of nonzero cones in the fan for Pn−1. The verticalboundary sector ∂vSn admits a cover by subsectors indexed by Σ, where for a k-dimensionalcone σ, the corresponding subsector Sn,σ is equivalent to Sn−k × T ∗Rk, and for σ ↪→ τ, theinclusion Sn,σ ↪→ Sn,τ is as vertical boundary.

Proof. The sectorial cover {Sn,σ}σ∈Σ is induced by the sectorial cover Uσ of the cornerboundary ∂2Sn described in Example 2.20. �

Having introduced the sector Sn, we recall the results from [Nad19] about microlocalsheaves on LSn which, thanks to [GPSb, GPSa], we can now understand as computations inthe wrapped Fukaya categoryW(Sn). The first of these is the computation of the monodromyautomorphism on the Fukaya category W(∂hSn) of the horizontal boundary sector. This isthe automorphism µ obtained as the cone on the counit map of the horizontal restrictionadjunction.

Proposition 2.27 ([Nad19, Corollary 4.18]). Under the mirror symmetry identification

W(∂hSn) ∼= Coh(Pn−1),

the monodromy automorphism µ corresponds to the functor −⊗OPn−1(−1) of tensoring withthe invertible sheaf OPn−1(−1).

The calculation from [Nad19] now proceeds via the following lemma:

Lemma 2.28 ([Nad19]). (1) The horizontal cap functor

∩h :W(Sn)→W(∂hSn)

is spherical and monadic. Under the identification W(∂hSn) ∼= Coh(Pn−1), thismonad is therefore given by tensoring with the cone of a morphism

s : OPn−1(−1)→ OPn−1 .

(2) The morphism s is generic: there exist coordinates zi on Pn−1 in which s is given bythe section z1 + · · ·+ zn ∈ Γ(Pn−1,O(1)).

The above lemma establishes that the monad associated to the spherical “horizontal re-striction” functor ∩h is equivalent to the monad for the pushforward functor

i∗ : Coh(Pn−2)→ Coh(Pn−1)

for the inclusion of a generic hyperplane in Pn−1. This leads directly to the main result of[Nad19]:

MIRROR SYMMETRY FOR BERGLUND-HUBSCH MILNOR FIBERS 15

Corollary 2.29 ([Nad19, Theorem 1.4]). There is a commutative diagram with horizontalequivalences

(2.30) W(Sn)∼ //

��

Coh(Pn−2)

��W(∂hSn)

∼ // Coh(Pn−1),

where the right vertical map is the pushforward along the inclusion

i : Pn−2 = {z1 + · · ·+ zn = 0} ↪→ Pn−1.

These equivalences form one face of a commutative cube, but we will be more interested inthe other face, which can be obtained from the diagram (2.30) by restricting to the verticalboundary on the A-side, and restricting to the toric boundary on the B-side:

Corollary 2.31. There is a commutative diagram with horizontal equivalences

(2.32) W(∂vSn)∼ //

��

Coh(∂Pn−2)

��W(∂2Sn)

∼ // Coh(∂Pn−1).

Remark 2.33. Instead of taking the diagram (2.32) as obtained from (2.30) by restriction,one could also obtain (2.32) by gluing together lower-dimensional copies of the equivalencefrom (2.30), according to the sectorial covers discussed in Example 2.20 and Lemma 2.26.

Actually, we will need analogous results not only for the sector Sn, but also for a mod-ification of this sector obtained by passing to a ramified cover. Let ρ : Cn → Cn be theG-covering map, ramified along the toric boundary divisor, defined in (1.4). The sector weare interested in is the “pullback” of the sector Sn along the map ρ. We define this sectorin analogy with Definition 2.22:

Definition 2.34. We write S Gn for the sectorial structure on Cn which is specified by the

stop LΣPn−1/G.

According to standard heuristics in mirror symmetry, passing to the G-cover ρ shouldcorrespond on the mirror to passing to a G∨-quotient, where G∨ as usual is the abeliangroup dual to G. This turns out to be the case, giving us the following analogue of Corollary2.31:

Proposition 2.35. There is a commutative diagram with horizontal equivalences

(2.36) W(∂vS Gn )

∼ //

��

CohG∨(H ∩ ∂Pn−1)

��

W(∂2S Gn )

∼ // CohG∨(∂Pn−1),

where H ⊂ Pn−1 is the hypersurface defined by the polynomial W∨.

Proof. The proof proceeds exactly as in [Nad19], following the outline we have sketchedabove. The only difference is in the calculation of the monodromy autofunctor. Since the

16 BENJAMIN GAMMAGE

monodromy can be calculated in the complement of the singular fiber, this calculation canbe deduced from Nadler’s by passing along the (now unramified) cover ρ.

The monodromy autoequivalence is therefore given by tensoring with the equivariantizedline bundle OPn−1/G(−1), and the monad allowing us to compute the top row of (2.36) is thecone with a generic section of this bundle. The polynomial W∨ is such a section. �

2.4. Deformation theory. We will now reinterpret the categorical computation describedabove, so that we can understand the category W(∂vSn) as a matrix factorization category.But rather than writing categories directly in terms of matrix factorizations, we prefer to usethe language of abstract deformation theory. Recall that given an algebraic variety or stackX, and a function f ∈ O(X) ⊂ HH0(X), we can treat f as the Maurer-Cartan elementspecifying a formal deformation of the 2-periodicized category Coh(X)Z/2.

Definition 2.37. We will denote the dZ/2 g-category obtained from Coh(X) by deformingalong f ∈ O(X) by Def(X, f).

In what follows, we rely on the well-known identification of Def(X, f), for X a smoothvariety, with the singularity category Coh(f−1(0))/Perf(f−1(0)). A thorough account of thisequivalence can be found in [Pre], and an exposition of the role of curving in [Tel20, Appen-dix].

If X is smooth and W∨ has no nonzero critical values, then the category Def(X,W∨)defined above is therefore equivalent to the category MF(X,W∨) of matrix factorizations ofW∨ on X. However, even if X is not smooth, we can nevertheless relate Def(X,W∨) to atraditional matrix factorization category, using Orlov’s equivalence from [Orl06], which wecan phrase as follows:

Lemma 2.38 ([Orl06]). Let X be a hypersurface in a smooth stack Y cut out by a functionf ∈ O(Y ). Suppose moreover that W∨ ∈ O(X) is the restriction to X of a function g on Ywith no nonzero critical values. Then the category Def(X,W∨) defined in Definition 2.37 isequivalent to the category MF(Ct × Y, g + tf) of matrix factorizations on Y for the functiong + tf.

Example 2.39. Let X = ∂Cn ⊂ Cn = Y. Then an equivariant version of the above lemmagives us an equivalence

(2.40) Def(∂Cn/G∨,W∨) ∼= MF(Ct × Cn/G∨,W∨ + tx1 · · ·xn).

Note that the undeformed category Coh(∂Cn/G∨) = Def(∂Cn/G∨, 0) is equivalent to thematrix factorization category MF(Ct ×Cn, tx1 · · · xn). The category (2.40) is related to thisone as a deformation by W∨, and the main result of this paper will be to see that deformationin symplectic geometry.

Lemma 2.38 remains true in a twisted and stacky form, when the function f defining Xexists only locally and is equivariant for some group G∨. In the case where g = 0, this readsas follows:

Lemma 2.41. Let X be a hypersurface in a smooth stack Y cut out by a section s ∈ Γ(Y,L)for some line bundle L on Y . Write s ∈ O(Tot(L)) for the function on the total space of Lobtained by extending s to a function linear on fibers of L. Then there is an equivalence ofdZ/2 g-categories

Coh(X)Z/2 ∼= MF(Tot(L), s)

MIRROR SYMMETRY FOR BERGLUND-HUBSCH MILNOR FIBERS 17

between the 2-periodicized category Coh(X)Z/2 of coherent sheaves on X and the category ofmatrix factorizations for s on Tot(L).

We can apply Lemma 2.41 to understand the categoryW(∂vSn) discussed in the previoussection as a deformation.

Example 2.42. Let Y = Pn−1/G∨, and W∨ ∈ Γ(OPn−1/G∨(1)) a generic section as inProposition 2.35. Then there is an equivalence

(2.43) CohG∨({W∨ = 0})Z/2 ∼= MFG

∨(Tot(OPn−1(1)), W∨) ∼= Def(Tot(OPn−1/G∨(1), W∨)

between the 2-periodicized category of coherent sheaves on the hypersurface in Pn−1/G∨

defined by W∨ and the G∨-equivariant matrix factorization category of the fiberwise-linearextension of W∨ to a function on the total space of O(1), which as we have seen is adeformation of this equivariant total space by W∨.

By restricting to the toric boundary, we obtain a similar equivalence:

CohG∨({W∨ = 0} ∩ ∂Pn−1)Z/2 ∼= Def(Tot(O∂Pn−1/G∨(1), W∨)).

3. Geometry of the Milnor fiber

In Section 4, we will compute the wrapped Fukaya category of the Milnor fiber V = {W =1} ⊂ Cn. As preparation, we describe in this section the symplectic geometry of the spaceV. Ultimately, our goal is to decompose V into a union of several sectors such that mirrorsymmetry equivalences for these local pieces, and their relations to each other, are alreadyunderstood explicitly.

In order to simplify the exposition in this section, we begin by descending along theramified G-cover ρ : Cn → Cn defined at (1.4) in the Introduction, which restricts to aramified cover

V → P := {x1 + · · ·xn = n}.The map ρ also restricts to an unramified G-cover (C×)n → (C×)n and hence an unramified

cover of the (n− 1)-dimensional pants P ,

V → P := {x1 + · · ·+ xn = n} ⊂ (C×).

For most of Section 3, we will content ourselves with studying the spaces P, P instead ofV, V – i.e., we restrict ourselves to the case W = x1 + · · ·+xn. In Section 3.3, we will explainhow to return to the general case.

3.1. Sectors from a fibration. We will study the manifolds P, P through the map

(3.1) Cn // C, (x1 . . . , xn) � // x1 · · · xn.

The restriction of this map to P (or V ) we will denote by f , and its restriction to P (or V )

we will denote by f , or possibly also by f in cases where this will not cause confusion. Wewill need the following facts about this map.

Lemma 3.2. (1) The map f has critical values {0, 1} ⊂ C. The map f has uniquecritical value {1} ⊂ C×.

(2) The critical value 1 corresponds to a single nondegenerate critical point, at (1, . . . , 1).

18 BENJAMIN GAMMAGE

(3) The general fiber of f or f is a hypersurface in (C×)n−1 = {x1 · · ·xn = c}. Incoordinates x1, . . . , xn−1 on this (C×)n−1, this hypersurface is defined by the equationx1 + · · ·+ xn−1 + c

x1...xn−1= n.

Proof. Part (3) is clear. For parts (1) and (2), we parametrize P by x1, . . . , xn−1, so that themap f becomes

(3.3) (x1, . . . , xn−1) 7→

(n−1∏j=1

xj

)(n−

n−1∑j=1

xj

),

and hence the ith derivative of this map is given by

(3.4)∂f

∂xi= (∏j 6=i

xj)

(n− xi −

n−1∑j=1

xj

).

These derivatives all vanish simultaneously only if all xi = 0, or if all xi = 1, and the latterpoint (unlike the former) has a nondegenerate Hessian. �

The pullback along f (resp. f) of the real hypersurface {<(z) = 12} ⊂ C induces a decom-

position of P (resp. P ) into a pair of Liouville sectors PL, PR (resp. PL, PR), intersectingalong a Liouville sector of the form P 1

2× T ∗R, where P 1

2is the general fiber f−1(1

2) of P or

P .The right-hand sector PR = PR is easy to describe:

Lemma 3.5. The sector PR is equivalent to the sector T ∗D given by the cotangent bundleof a closed (n− 1)-disk.

Proof. Using the Lefschetz fibration f : PR → C<≥ 12

with a single nondegenerate critical

point, this is standard from the theory of Lefschetz fibrations. The disk D is the Lefschetzthimble for this critical point, following the vanishing path [1

2, 1] ⊂ C<≥ 1

2in the base. �

In other words, the sector PR represents a single Weinstein disk attachment to the Wein-stein manifold P 1

2, and the only further information we need in order to understand PR

as a subsector of P is a description of how this disk is attached: we need to identify theLagrangian sphere S = ∂D inside the skeleton of P 1

2.

The attachment of the disk D, or in other words the degeneration of the fiber of f over1 ∈ C, can be most easily described using tropical geometry. Recall that the complement ofthe amoeba AM has a single bounded region, whose boundary is the diffeomorphic image ofthe real positive locus M+ inside M .

Lemma 3.6. The boundary of the Lagrangian disk D attached at {1} is the real positivelocus M+ ⊂M.

Proof. Parallel transport to the fiber over 1 collapses the Lagrangian sphere M+∼= Sn−2 to

the point (1, . . . , 1), so M+ is the vanishing cycle associated to the critical point over 1. Thiscan be seen most clearly from a tropical perspective: the degeneration of the amoeba AMillustrated in Figure 3 should be read as a movie depicting the Lefschetz thimble filling inthe interior of M+. �

MIRROR SYMMETRY FOR BERGLUND-HUBSCH MILNOR FIBERS 19

Figure 3. The degeneration of the tropical hypersurface of M as it ap-proaches the critical value 1. In the figure on the left, the collapsing regionwhere the disk is to be attached is shaded.

In other words, the disk D is attached to the skeleton LP 12

of a general fiber along the

sphere S corresponding to the top-dimensional cones in the fan ΣP, as described in Remark2.17.

3.2. The sector around 0. As with the sector PR, the left-hand sector PL can be under-stood by studying a map PL → C<≤ 1

2with a single critical value. However, unlike in the

case of PR, the critical value of f |PL does not correspond to a Picard-Lefschetz singularity; infact, the singularity above 0 is not isolated, nor is it Morse-Bott. However, it is in some sensethe simplest singularity of this kind: as we will see, it is built out of the normal-crossingsdegenerations described in Section 2.3.

Since the sectorial structure on PL is defined by a fibration with a unique critical value,we will be able to give a complete description of this sector by studying the degenerationof general fiber P 1

2. We will address this question locally on P 1

2, using the sectorial cover

described in Lemma 2.26.Recall that the sectors in this cover of P 1

2are in bijection with the nonzero cones in

the fan ΣPn−1 . We will denote the subsector of P 12

corresponding to cone σ by Uσ and the

corresponding subsector of PL by Vσ.

Proposition 3.7. (1) There is an equivalence of Liouville sectors

Vσ ∼= Sn−1−k × T ∗Rk

(2) The cover of PL by subsectors Vσ is equivalent to the cover of the vertical boundarysector ∂vSn described in Lemma 2.26.

Before we proceed to the proof, we first set up some tropical geometry. Recall fromthe discussion in 2.2.1 that in proving Theorem 2.14 we found it useful to draw the fanΣPn−1 , thought of as the triangulation of the Newton polytope of P 1

2, superimposed on the

tropicalization Trop(P 12). We will find it even more useful to do so now: this fan can now be

understood as the moment-map base for the singular toric variety {x1 · · ·xn = 0}.The k-dimensional cones of ΣPn−1 , for k < n then correspond to singular strata inside{x1 · · · xn = 0}, and the smooth fiber P 1

2will degenerate precisely where it intersects these.

The singularities of the fiber P0 can therefore be seen on the tropicalization of the smoothfiber as the intersection of Trop(P 1

2) with the non-top-dimensional cones of ΣPn−1 . An exam-

ple is illustrated in Figure 4.

20 BENJAMIN GAMMAGE

Figure 4. The case n = 3. The rays of the fan for P2 are superimposed onthe tropicalization of P 1

2, and their intersection, the singular locus in P0, is

highlighted in red. The fan, usually interpreted as triangulating the Newtonpolytope for P 1

2, is reinterpreted here as a moment-map base for the singular

ambient space {xyz = 0} in which the singular fiber P0 lives.

Remark 3.8. The appearance of tropical and singular moment-map geometry in the samebase diagram occurs whenever one attempts to study large-volume and large-complex-structurelimits simultaneously, and these diagrams will play a key role in the approach to the Gross-Siebert program to be described in [GSb]. We will reserve to that paper a more completeanalysis of this situation, as the discussion so far is sufficient for the single example to beconsidered here.

Proof of Proposition 3.7. We need to show that the degenerations of the sectors Uσ over thezero fiber in PL are equivalent to the degenerations of these sectors thought of as the cornerboundary in the sector Sn defined in Definition 2.22. We can accomplish this using the“tropical tailoring” isotopy {P s}0≤s≤1 described in Proposition 2.15. Note that cones σ in Σare in bijection with subsets of monomials in W = x1 + · · ·xn; the tailoring isotopy ensuresthat in the sector Uσ, only those monomials in σ contribute to W . Let σ be a k-dimensionalcone in Σ, and F the face in the tropicalization Trop(P 1

2) dual to σ. Then in a neighborhood

of Log−1(F ), the tailored hypersurface has the form of a product

(C×)n−k−1xi /∈σ ×

{∑xi∈σ

xi = 1

}of a complex (n − k − 1)-torus with a (k − 1)-dimensional pants. There is a correspondingequivalence of Liouville sectors

Uσ ∼=((C×)n−k−1,LΣPn−1/σ

)× T ∗Rk−1.

By comparing with the tropical picture of the degeneration discussed above, we see thatthe second factor in this product does not degenerate under the parallel transport of P 1

2to

0, whereas the first factor does degenerate as Sn−k−1. �

Remark 3.9. We can combine Proposition 3.7 with Lemma 3.5 to obtain a complete de-scription of the skeleton of the Weinstein manifold P . We can think of the skeleton of thegeneral fiber P 1

2as obtained from a sphere S = ∂∆n−1, identified with the boundary of an

(n− 1)-simplex, by attaching to each k-face F of S a copy of (S1)n−k−2 × F.

MIRROR SYMMETRY FOR BERGLUND-HUBSCH MILNOR FIBERS 21

Now the two critical values of the fibration f tell us how to modify this skeleton to obtainthe skeleton of P . From the sector PR, we learn that we should cone off the sphere S, orequivalently attach a Weinstein disk with boundary on S, and from the sector PL, we learnthat we should cone off each of the copies of (S1)n−k−2 appearing in the skeleton.

Finally, we give a description of the punctured sector PL, which is significantly simpler tounderstand than PL since the map f |PL has no critical points. Let P±ε be the fibers overthis map over ±ε for some ε ∈ (0, 1), and write φ± : Pε ' P−ε for the two identifications ofthese fibers, given by parallel transport above and below 0, respectively. Also write A2 forthe Liouville sector given by a disk with three stops on the boundary.

Proposition 3.10. The sector PL is obtained from the product sector Pε × A2 by gluingtogether two of the ends using the identification (φ−)−1 ◦ φ+.

Proof. The sector PL has a map f to the sector given by an annulus with one stop on itsboundary, and this map has no critical points. Hence this sector admits a Liouville form byadding Liouville forms on the base and on the fiber Pε. If we choose a Liouville structure onthe base with skeleton the “lollipop” (, then the sector PL will have a skeleton living overthe lollipop, which has one A2 singularity with two of its ends glued together; in the fiber,the identifications on these two ends with Pε will differ by the half-monodromies φ±. �

Remark 3.11. In analogy with Remark 3.9, we can use the above description of PL tounderstand a Lagrangian skeleton for the space P . In fact, P is an (n − 1)-dimensionalpants, and we have already described its skeleton LP in Example 2.19. But the relation of

this skeleton to PL is interesting.Recall that LP has a sectorial cover indexed by the face poset of an (n−1)-simplex, where

a k-face of PL corresponds to the sector with FLTZ skeleton LΣAn−1−k × T ∗Rk, with the

interior of the simplex corresponding to sector T ∗Rn−1, which has skeleton a disk D. WriteL for the complement LP \ D of this disk in the skeleton of P .

Figure 5. The Lagrangians LM (left) and L (right) for n = 3. Note that theRk×T n−k−2 pieces of LM become copies of Rk×T n−k−1 after being swept outby the parallel transport around 0.

22 BENJAMIN GAMMAGE

Then L is precisely the Lagrangian swept out in PL by parallel transport of the skeletonof a general fiber LP 1

2

around 0, as illustrated in Figure 5. This is the Lagrangian skeleton

of the Liouville subdomain f−1(D) above a small punctured disk D around 0. The total

skeleton LP of the space P is obtained from this by the disk attachment indicated by the

critical value in PR, which attaches the interior disk to the skeleton.

3.3. Covering spaces. Thus far we have described the geometry only of the space P , andits open locus P , rather than the more general Milnor fiber V (and its very affine part V ).However, as we now explain, the case of general V immediately follows from this one.

The pants P ⊂ (C×)n is defined by the Laurent polynomial x1 + · · · + xn − n, which hasNewton polytope

Newt(x1 + · · ·+ xn − n) = ∆n := Conv(0, e1, . . . , en) ⊂ Rn

the standard simplex. The key fact we use, which played an essential role in the constructionsof [GSa], is that the Newton polytope of the function W − n is the larger simplex

Newt(W − n) = ∆W := Conv

(0,∑j

a1j, . . . ,∑j

anj

)⊂ Rn,

where as usual we write

W =n∑i=1

n∏j=1

xaiji .

The matrix A = (aij) defines a linear map Rn → Rn taking ∆n to ∆W , hence a map

ρ : (C×)n → (C×)n which restricts to an unramified G-cover V → P , as described in theIntroduction.

Passing along this cover, we can immediately lift our description of the sectorial decom-position of P to the space V . Consider the fan of cones on the faces of ∆W , and write ΣW

for its image in the quotient Rn/R∆ by the diagonal copy of Rn. Then the G-cover of the

sectorial decomposition for P described above is as follows.

Proposition 3.12. The space V has a cover by a pair of Liouville sectors VL, VR intersectingalong a shared subsector V 1

2× T ∗R.

• The central sector V 12

is a Liouville domain with skeleton the boundary FLTZ La-

grangian ∂LΣW .

• The right-hand sector VR is a disjoint union⊔|G|i=1 T

∗D of |G| copies of the cotangent

bundle of a closed disk, attached to the |G| lifts in V 12

of the real positive sphere S in

P 12.

• The left-hand sector VL can be obtained from V 12

by taking a product with the A2

sector and gluing two ends together by the monodromy isomorphism.

It is no more difficult to describe the geometry of the actual Milnor fiber V : the unramifiedG-cover (C×)n → (C×)n extends to a ramified G-cover

ρ : (C×)n → (C×)n,

MIRROR SYMMETRY FOR BERGLUND-HUBSCH MILNOR FIBERS 23

restricting to a G-cover V → P ramified only in the fiber over 0. As for P , the right andmiddle sectors VR, V 1

2agree with their very affine analogues VR, V 1

2, so we only need to give a

description of the left-hand sector. However, by comparing the covering map VL → PL withthe discussion in Section 2.3, we see that we have already introduced this sector:

Lemma 3.13. The left-hand sector VL of the Milnor fiber VL is equivalent to the normalcrossings sector S G

n constructed in Definition 2.34.

4. Fukaya categories and mirror symmetry

Having already presented the spaces V, V as covered by recognizable Liouville sectors,it remains for us only to recall the calculations of the wrapped Fukaya categories of thosesectors, and then to glue the resulting categories together. We begin with V : although wealready understand the Fukaya category W(V ) from the results of [GSa], we give here adifferent presentation as preparation for the calculation of W(V ).

4.1. The very affine Milnor fiber. The space V is an unramified G-cover of P , and themirror V ∨ is obtained from the mirror P∨ = ∂Cn by passing to a G∨-quotient. In otherwords:

Proposition 4.1 ([GSa]). There is an equivalence of categories

(4.2) W(V ) ∼= CohG∨(∂Cn)

between the wrapped Fukaya category of V and the category of G∨-equivariant coherentsheaves on the toric boundary of Cn.

The proof of Proposition 4.1 proceeds by matching the closed cover of the stack Cn/G∨

by toric orbit closures to a Liouville-sectorial cover of V . But we would like to express thecategory (4.2) in terms of a different Liouville-sectorial decomposition of V , namely the cover

by left- and right-hand sectors VL, VR discussed in Section 3. We begin with VL.

Definition 4.3. Let p : Bl0 Cn → Cn be the blowup of Cn at the origin. We write ∂Cn forthe strict transform of the toric boundary ∂Cn under this blowup, and

P := p−1(0) ∼= Pn−1

for the exceptional divisor.

Note that ∂Cn intersects P in its toric boundary ∂P. This boundary divisor plays the roleof mirror to the central Liouville sector V 1

2in our decomposition of V :

Proposition 4.4. The wrapped Fukaya category W(V 12) of the Liouville sector W(V 1

2) is

equivalent to the category CohG∨(∂P) of coherent sheaves on the toric boundary of the

weighted projective stack Pn−1/G∨.

Proof. This is a corollary of the results in [GSa], proved by matching the closed cover of

∂P/G∨ by toric orbit closures to a Liouville-sectorial cover of V 12. �

In fact, the category W(V 12) of a general fiber of the map f comes equipped with extra

structure.

24 BENJAMIN GAMMAGE

Definition 4.5. We write µ ∈ Aut(W(V 12)) for the monodromy automorphism of the cate-

gory V 12, obtained from parallel transport of the general fiber of f around 0.

The monodromy automorphism µ admits a geometric description on the mirror space ∂P.

Lemma 4.6. As an automorphism of the category CohG∨(∂P), the functor µ is given by ten-

sor product with the line bundle O∂P/G∨(1), the restriction to ∂P of the line bundle OP/G∨(1)on the weighted projective stack P/G∨.

Proof. This follows from Nadler’s calculation. �

We can use the monodromy automorphism µ to give a new method for computation ofthe wrapped Fukaya category W(VL).

Proposition 4.7. There is an equivalence

W(VL) ∼= CohG∨(∂Cn)

between the wrapped Fukaya category W(VL) and the category of G∨-equivariant coherentsheaves on the proper transform of ∂C under the blowup at 0.

Proof. The space ∂Cn/G∨ is a toric stack, so one possible proof proceeds following [GSa]as usual, matching toric orbit closures with Liouville subsectors. But we will give herea different proof, more closely associated to the description of the skeleton LVL given inProposition 3.10.

From the description in Proposition 3.10, we can see that the categoryW(VL) is equivalentto the category of pairs {(

X ∈ W(V 12), µX → X

)}of an object X in the Fukaya category of the nearby fiber and a µ-twisted endomorphism

of X, where µ ∈ Aut(W(V 1

2))

is the monodromy map. Since W(V 12) ∼= CohG

∨(∂P) with

monodromy given by the functor of tensor product −⊗O(1), we thus have an equivalence

(4.8) W(V 12) ∼=

{(F ∈ CohG

∨(∂P), ν : F(−1)→ F

)}.

Now note that ∂Cn is the total space Tot(O∂P(1)) of the bundle O(1) on ∂P, so that itcan be described equivalently as the relative Spec

∂Cn = Spec∂PSymO∂PO(−1),

and hence the category CohG∨(∂Cn) is equivalent to the category of coherent sheaves F in

CohG∨(∂Cn) equipped with the additional data of a map F ⊗ O∂P/G∨(−1) → F describing

the action of the generators of this symmetric algebra. This agrees with the description ofthe category W(V 1

2) given in (4.8). �

Now recall from Example 2.19 that the space P , the (n− 1)-dimensional pants, is mirrorto the toric boundary ∂Cn, and there is an equivalence of categories

W(P ) ∼= Coh(∂Cn).

MIRROR SYMMETRY FOR BERGLUND-HUBSCH MILNOR FIBERS 25

Accordingly, the wrapped Fukaya category of the space V , which is an unramified G-coverof P (and is called the “G-pants” in [GSa]) admits a presentation as

(4.9) W(V ) ∼= CohG∨(∂Cn).

This is not obviously identical with the presentation of V via the Liouville-sectorial coverwhich we have been discussing so far:

Lemma 4.10. The wrapped Fukaya category W(V ) of V is equivalent to the colimit

(4.11) lim−→(

CohG∨(∂Cn)← CohG

∨(∂P)→ CohG

∨({0})

),

where the maps are given by pushforwards along the inclusion P/G∨ ↪→ ∂Cn/G∨ and theprojection P/G∨ → {0}/G∨, respectively.

Proof. This colimit presentation corresponds to the Liouville-sectorial decomposition we havebeen studying in this paper. We have already proven that the categories in (4.11) match

the wrapped Fukaya categories of the sectors VL, V 12, and VR, respectively, so we only need

to check that the functors induced by the Liouville-sectorial inclusions of V 12× T ∗R into VL

and VR are as described in the lemma. (Actually, we we will check agreement on the leftadjoints of these functors, which are easier to understand. And for simplicity, we note thatit is sufficient to check the case where V = P , since the G-cover applies uniformly to allof the Liouville sectors involved and hence the G∨ equivariance applies uniformly to all thecategories involved in (4.11).)

Consider first the left cap functor

W(PL)→W(P 12).

Under the description in Proposition 4.7, this functor is given by the map which takes a pair(X, ν : µX → X) to the object Cone(ν). But in the B-side description from that proposition,the cone on F(−1)→ F is the pullback of F under the inclusion of the zero-section

∂P ↪→ Spec∂PSymO∂PO(−1)

Now consider the right cap functor

(4.12) W(PR)→W(P 12).

The wrapped category W(PR) of sector PR is equivalent to the category PerfC of finite-dimensional vector spaces, and we need to show that (4.12) sends the 1-dimensional vector

space to the structure sheaf O∂P ∈ Coh(∂P) ∼=W(P 12).

Recall that PR describes a disk attachment to P 12

with boundary sphere S ⊂ LP 12

. We

therefore need to check that the Lagrangian sphere S, equipped with trivial local system,represents the structure sheaf O∂P under the mirror symmetry equivalence

(4.13) W(P 12) ∼= Coh(∂P).

The equivalence (4.13) describes Coh(∂P) as a colimit of categories of coherent sheaves on

the toric orbit closures of ∂P, corresponding to a cover of P 12

by Liouville sectors. The

basic sectors in this cover are of the form (T ∗T n−2,ΛΣPn−2 ), and in each case the Lagrangian

mirror to the structure sheaf OPn−2 is the cotangent fiber at 1 ∈ T n−2. These cotangent fibers

26 BENJAMIN GAMMAGE

glue together to form the sphere S, matching the gluing of structure sheaves OPn−2 into thestructure sheaf OPn−1 of Pn−1. �

By comparing the equivalence (4.9) with Lemma 4.10, we can deduce a new colimit pre-

sentation of the category CohG∨(∂Cn). In fact, it is possible to prove this directly, without

any reference to mirror symmetry:

Lemma 4.14. There is an equivalence of categories

(4.15) Φ : CohG∨(∂Cn)

∼−→ lim−→(

CohG∨(∂Cn)← CohG

∨(∂P)→ CohG

∨({0})

).

Proof. The functor Φ is induced from the pullback functor

p∗ : CohG∨(∂Cn)→ CohG

∨(∂Cn).

This functor is fully faithful, hence Φ is also, and we need only to prove that Φ is essentiallysurjective. In other words, we need to check that every object of the colimit in (4.15) can

be identified with an object in CohG∨(∂Cn) which is pulled back from CohG

∨(∂Cn).

So let F be an object of CohG∨(∂Cn). By using the map p∗p∗F → F , we can reduce to the

case that F is supported on the exceptional locus P/G∨. But the colimit in (4.15) identifiesany such object with the pullback of some sheaf on ∂Cn/G∨ supported at at {0}/G∨, asdesired. �

4.2. Deformation theory. We are ready at last to proceed to the calculation of the wrappedFukaya category W(V ) of the Milnor fiber V . The main theorem of this paper is the follow-ing:

Theorem 4.16. The wrapped Fukaya categoryW(V ) is the deformation Def(∂Cn/G∨,W∨)

of the dZ/2 g-category CohG∨(∂Cn)Z/2 by the function W∨ ∈ C[∂Cn]G

∨.

Using the discussion of deformation theory in Section 2.4, we can reinterpret this in theform of a more traditional mirror symmetry equivalence, as predicted in [LUb, Conjecture1.5]:

Corollary 4.17. There is an equivalence of dZ/2 g-categories

W(V ) ∼= MFG∨(Cn, z1 · · · zn +W∨)

between the wrapped Fukaya category of the Milnor fiber V and the category of matrix fac-torizations on Cn for the function z1 · · · zn +W∨.

Proof of Theorem 4.16. The proof of Theorem 4.16 begins from the equivalence

(4.18) CohG∨(∂Cn) ∼= lim−→

(CohG

∨(∂Cn)← CohG

∨(∂P)→ CohG

∨({0})

).

proved in Lemma 4.14. The right-hand-side of (4.18) corresponds to the colimit description

of the wrapped Fukaya categoryW(V ) given by the cover of V by subsectors sectors VL, VR,

and their intersection V 12× T ∗R.

We have seen that the Weinstein manifold V has an analogous Liouville-sectorial decompo-sion, and in fact the right and middle subsectors VR, V 1

2of V are equal to the corresponding

subsectors VR, V 12

of V . Hence the categoryW(V ) admits a colimit presentation as in (4.18),

but with the category W(VL) ∼= CohG∨(∂Cn) replaced by its deformation

(4.19) W(VL) ∼=W(∂vSn) ∼= DefG∨(∂Cn,W∨),

MIRROR SYMMETRY FOR BERGLUND-HUBSCH MILNOR FIBERS 27

where the first equivalence in (4.19) comes from Proposition 3.7 and the second comes froma combination of Proposition 2.35 with Example 2.42.

Hence we need only to establish the equivalence

(4.20) MFG∨(∂Cn,W∨) ∼= lim−→

(MFG

∨(∂Cn,W∨)← CohG

∨(∂P)→ CohG

∨({0})

),

or in other words to show that the right-hand side of (4.20) is a deformation of (4.18) byW∨. This statement is proved in exactly the same manner as Lemma 4.14: the pullback map

p∗ : CohG∨(∂Cn)→ CohG

∨(∂Cn) deforms to a fully-faithful embedding

p∗,W∨

: Def(∂Cn/G∨,W∨)→ Def(∂Cn/G∨,W∨),

and every object of Def(∂Cn/G∨) is identified in the colimit with one obtained through thisdeformed pullback. �

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Dept. of Mathematics, Harvard University, 1 Oxford St., Cambridge 02138Email address: gammage@math.harvard.edu