Homological mirror symmetry for generalized Greene-Plesser mirrors

We prove Kontsevich's homological mirror symmetry conjecture for certain mirror pairs arising from Batyrev-Borisov's `dual reflexive Gorenstein cones' construction. In particular we prove HMS for all Greene-Plesser mirror pairs (i.e., Calabi-Yau hypersurfaces in quotients of weighted projective spaces). We also prove it for certain mirror Calabi-Yau complete intersections arising from Borisov's construction via dual nef partitions, and also for certain Calabi-Yau complete intersections which do not have a Calabi-Yau mirror, but instead are mirror to a Calabi-Yau subcategory of the derived category of a higher-dimensional Fano variety. The latter case encompasses Kuznetsov's `K3 category of a cubic fourfold', which is mirror to an honest K3 surface; and also the analogous category for a quotient of a cubic sevenfold by an order-3 symmetry, which is mirror to a rigid Calabi-Yau threefold.


Toric mirror constructions
One of the first constructions of mirror pairs of Calabi-Yau varieties was due to Greene and Plesser [GP90]. They considered Calabi-Yau hypersurfaces in weighted projective spaces and their quotients. They were interested in the three-dimensional case, but their construction works just as well in any dimension.
Batyrev generalized this to a construction of mirror pairs of Calabi-Yau hypersurfaces in toric varieties [Bat94]. In Batyrev's construction one considers dual reflexive lattice polytopes ∆ and∆, corresponding to toric varieties Y andY . Batyrev conjectured that Calabi-Yau hypersurfaces in Y andY ought to be mirror. His construction reduces to Greene-Plesser's in the case that ∆ and∆ are simplices. Borisov generalized Batyrev's construction to encompass mirror pairs of Calabi-Yau complete intersections in toric varieties [Bor93].
However, these constructions did not encompass certain examples in the literature, which suggested that some Calabi-Yau complete intersection might not admit any Calabi-Yau mirror, but might nevertheless be mirror in some generalized sense to a higher-dimensional Fano variety, which one considers to be a 'generalized Calabi-Yau'. 1 For example, Candelas, Derrick and Parkes considered a certain rigid Calabi-Yau threefold, and showed that it should be mirror to a quotient of a cubic sevenfold by an order-3 symmetry group [CDP93].
Batyrev and Borisov succeeded in generalizing their constructions to include these generalized Calabi-Yau varieties. They constructed mirror pairs of Landau-Ginzburg models, depending on dual pairs of 'reflexive Gorenstein cones' [BB97]. They showed that a reflexive Gorenstein cone equipped with a 'complete splitting' determines a Calabi-Yau complete intersection in a toric variety; and Borisov's previous construction was equivalent to the new one in the case that both cones were completely split. 1 It has since been understood that in these contexts, the derived category of the higher-dimensional Fano variety admits a semi-orthogonal decomposition, one interesting component of which is Calabi-Yau, which means it looks like it could be the derived category of some honest Calabi-Yau variety (see [Kuz15]). We think of the 'generalized Calabi-Yau variety' as having derived category equal to this Calabi-Yau category.
However it may happen that a reflexive Gorenstein cone is completely split, but its dual is not; in this case the dual will correspond to some generalized Calabi-Yau variety.
In this paper, we prove that certain special cases of Batyrev-Borisov's construction (which we call generalized Greene-Plesser mirrors) satisfy an appropriate version of Kontsevich's homological mirror symmetry conjecture. The rest of the introduction is organized as follows: we give the construction of generalized Greene-Plesser mirrors in § § 1.2-1.4; we formulate a version of Kontsevich's homological mirror symmetry conjecture for generalized Greene-Plesser mirrors in § 1.5; we state our main result, which constitutes a proof of the conjecture (contingent in some cases on certain technical assumptions), in §1.6; and we describe some explicit examples in § 1.7. In particular, we remark that generalized Greene-Plesser mirrors include all Greene-Plesser mirrors, and work through the case of the quartic surface and its mirror (in the 'reverse' direction from that considered in [Sei14c]); we also consider some examples which do not arise from the Greene-Plesser construction, including the rigid Calabi-Yau threefold mentioned above, as well as a certain K3 surface which is mirror to Kuznetsov's 'K3 category associated to the cubic fourfold' [Kuz10].

Toric data
We start by giving the toric data on which our construction of generalized Greene-Plesser mirrors depends.
Let I 1 , . . . , I r be finite sets with |I j | ≥ 3 for all j, and let I := I 1 . . . I r . Let d ∈ (Z >0 ) I be a tuple of positive integers such that i∈I j 1/d i = 1 for all j. We denote d := lcm(d i ), and let q ∈ (Z >0 ) I be the vector with entries q i := d/d i . Let e i be the standard basis of Z I , and denote e K := i∈K e i .
Let M ⊂ Z I be a sublattice such that • M contains d i e i for all i and e I j for all j.
• d| q, m for all m ∈ M .
We explain how these data give rise to a pair of dual reflexive Gorenstein cones.
The dual lattice to M is The dual cone isσ = (R ≥0 ) I ⊂ R I , equipped with the dual lattice N . It is Gorenstein because it is generated by the vectors e i , which lie on the hyperplane {n : mσ, n = 1} where mσ := e I . Therefore σ andσ are dual reflexive Gorenstein cones, of index n σ , mσ = r.
Our constructions will depend on one further piece of data, which is a vector λ ∈ (R >0 ) Ξ 0 . This is now the complete set of data on which our constructions depend: the sublattice M and the vector λ (we will later put additional conditions on the data).
The coneσ is completely split by [BN08, Corollary 2.5], because mσ = j e I j . Therefore it determines a Calabi-Yau complete intersection in a toric variety, in accordance with [BB97]. This complete intersection may be singular, but under certain hypotheses the vector λ determines a maximal projective crepant desingularization X (in the sense of [Bat94]), together with a Kähler form ω λ . We describe the construction explicitly in § 1.3.
We associate a graded Landau-Ginzburg model (S, W) to the reflexive Gorenstein cone σ . This cone is completely split if and only if the nef-partition condition holds (see Definition 1.1 below); in that case we have an associated Calabi-Yau complete intersectionX . The nef-partition condition is automatic if r = 1. If r > 1, then whether or not the nef-partition condition holds we have an associated Fano hypersurfaceŽ (of dimension greater than that of X ). We describe the constructions explicitly in § 1.4. Now let us explain when the reflexive Gorenstein cone σ is completely split. We define a map ι : Definition 1.1 We say that the nef-partition condition holds if ι(e I j ) ∈ N for all j.
When the nef-partition condition holds, we have n σ = ι(e I ) = j ι(e I j ), so σ is completely split by [BN08, Corollary 2.5]. It is not difficult to show that this is the only way that σ can be completely split.
In this situation we have a symmetry of our data which exchanges M ↔ N , where N is regarded as a sublattice of Z I ∼ = im(ι).
Remark 1.2 The case r = 1 will be the Greene-Plesser construction, which we recall is a special case of Batyrev's construction in terms of dual reflexive polytopes. The reflexive polytopes in this case are the simplex ∆ (with lattice M ∩ ∆) and its polar dual ∇ := {n ∈σ : mσ, n = 1} (with lattice N ∩ ∇). The nef-partition condition always holds in this case.
Definition 1.3 Let V ⊂ Z I be the set of vertices of the unit hypercube [0, 1] I . We say that the embeddedness condition holds if Remark 1.4 The embeddedness condition always holds in the Greene-Plesser case r = 1. That is because, if e K ∈ M ∩ V , we have n σ , e K ∈ Z because e K ∈ M ; on the other hand, for ∅ K I we have (1-6) 0 = n σ , e ∅ < n σ , e K < n σ , e I = 1.

Symplectic construction
The elements e I j define an embedding (1-7) Z r → M → Z I , which induces an embedding Note thatM =M 1 × . . . ×M r whereM j := Z I j /e I j . EachM j supports a complete fanΣ j whose rays are generated by the images of the basis vectors e i for i ∈ I j . The corresponding toric variety isỸ j ∼ = P |I j |−1 . We denote the product fan inM byΣ , which is the fan of the product of projective spacesỸ :=Ỹ 1 × . . . ×Ỹ r .
Let π : R I →M R denote the projection. Let ψ λ :M R → R be the smallest convex function such that ψ λ (t · π(p)) ≥ −t · λ p for all t ∈ R ≥0 and all p ∈ Ξ 0 . The decomposition ofM R into domains of linearity of ψ λ induces a fanΣ λ .
Definition 1.5 We say that the MPCP condition holds ifΣ λ is a projective simplicial refinement of Σ whose rays are generated by the projections of elements of Ξ 0 . λ lies in the interior of a top-dimensional cone cpl(Σ λ ) of the secondary fan (or Gelfand-Kapranov-Zelevinskij decomposition) associated to π(Ξ 0 ) ⊂M R , whereΣ λ is a projective simplicial refinement ofΣ whose rays are generated by π(Ξ 0 ). Now we consider the fans Σ and Σ λ , which are the same asΣ andΣ λ except we equip the vector spaceM R with the lattice M rather thanM .
Remark 1.7 If r = 1 and the MPCP condition holds, Σ λ is called a simplified projective subdivision of Σ in [CK99, Definition 6.2.5].
We have morphisms of fans Σ λ → Σ →Σ , the first being a refinement and the second being a change of lattice. It follows that we have toric morphisms Y λ → Y →Ỹ , the first being a blowdown and the second being a branched cover with covering group G :=M/M . We consider the hyperplane (1-9)X j := for all j, and denoteX := jX j ⊂ jỸ j . We let X ⊂ Y be the pre-image ofX , and X ⊂ Y λ the proper transform of X . The intersection of X with each toric stratum is a product of hypersurfaces of Fermat type, and in particular smooth; so if the MPCP condition holds then X is a maximal projective crepant partial desingularization (hence the name of the condition). Observe that the topology of X may depend on λ, but we will omit this from the notation.
Observe that even if the MPCP condition holds, Y λ may have finite quotient singularities, since Σ λ is only assumed to be simplicial. Therefore X may also have finite quotient singularities. We would like to understand when X is in fact smooth. Observe that for each j we have a morphism Y λ →Ỹ →Ỹ j , where the last map is projection. We denote the union (over all j) of the pre-images of toric fixed points inỸ j by Y λ,0 ⊂ Y λ , and we observe that X avoids Y λ,0 becauseX j avoids the toric fixed points ofỸ j .
Definition 1.8 We say that the MPCS condition holds if the MPCP condition holds, and furthermore Y λ is smooth away from Y λ,0 . MPCS stands for Maximal Projective Crepant Smooth desingularization.
We remark that the MPCS and MPCP conditions are equivalent when dim C (Y λ ) ≤ 4 (see [Bat94,§2.2]). If the MPCS condition holds, then X avoids the non-smooth locus of Y λ , so X is in fact a smooth complete intersection. The fact thatσ is reflexive Gorenstein of index r means that X is Calabi-Yau by [BB97, Corollary 3.6] (in the weak sense that c 1 (TX) = 0). We denote the toric boundary divisor of Y λ by D Y . Note that it has irreducible components D Y p indexed by p ∈ Ξ 0 . Let D Y λ := p∈Ξ 0 λ p · D Y p be the toric R-Cartier divisor with support function ψ λ . Because ψ λ is strictly convex (by our assumption that its domains of linearity are the cones of the fan Σ λ ), this divisor is ample, so the first Chern class of the corresponding line bundle is represented in de Rham cohomology by an orbifold Kähler form (see discussion in [AGM93,§ 4]). We denote the restriction of this orbifold Kähler form to X by ω λ . Because X avoids the non-smooth locus of Y λ , ω λ is an honest Kähler form. Its cohomology class is Poincaré dual

Algebraic construction
We work over the universal Novikov field: (1-10) It is an algebraically closed field extension of C. It has a valuation val : We consider the graded polynomial ring S Λ := Λ[z i ] i∈I , where |z i | = q i . We consider polynomials The dual to the group G introduced in § 1.3 is G * ∼ = hom(Z I /M, G m ), which acts torically on A I . The action preserves W b , because all monomials z p appearing in W b satisfy p ∈ M . Thus we have a Landau-Ginzburg model ([A I /G * ], W b ).
Remark 1.9 Observe that we have a correspondence (1-14) monomial z p of W b ↔ divisor D p ⊂ X . This correspondence is called the 'monomial-divisor mirror map' (see [AGM93]).
Because W b is weighted homogeneous, its vanishing locus defines a hypersurface inside the weighted projective stack WP(q). The action of G * descends to an action of Γ := G * /(Z/d) on WP(q), preserving the hypersurface. We denoteV := [WP(q)/Γ], andŽ b := {W b = 0} ⊂V . Now suppose that the nef-partition condition holds. Then the vectors ι(e I j ) ∈ N define a map M Z r splitting the inclusion Z r → M , so we have M ∼ = Z r ⊕ M . We denote ∆ j := {m ∈ ∆ : ι(e I k ), m = δ jk }, (1-15) We denote the toric stack corresponding to the polytope ∆ byY , and the divisor corresponding to ∆ j byĎ j . We define a section W j It is a Calabi-Yau complete intersection by [BB97, Corollary 3.6], and it corresponds to the hypersurfaceŽ b in accordance with [BB97, Section 3].

Statement of homological mirror symmetry
On the B-side of mirror symmetry we consider the category of Γ-equivariant graded matrix factorizations of W b , which we denote GrMF Γ (S Λ , W b ) (the precise definition is reviewed in § 4.4). It is a Λ-linear Z-graded cohomologically unital A ∞ (in fact, DG) category.
On the A-side of mirror symmetry we consider the Fukaya A ∞ category F(X, ω λ ). More precisely, we recall that the Fukaya category may be curved, i.e., it may have non-vanishing µ 0 and therefore not be an honest A ∞ category. Therefore we consider the version whose objects are bounding cochains on objects of the Fukaya category, which we denote by F(X, ω λ ) bc . It is another Λ-linear Z-graded cohomologically unital A ∞ category.
One part of Kontsevich's homological mirror symmetry conjecture for generalized Greene-Plesser mirrors then reads: In order to relate the category of graded matrix factorizations with a more manifestly geometric category, we recall the following theorems. The first is proved by Favero and Kelly [FK16], and employs a theorem which is due independently to Isik and Shipman [Isi13,Shi12] (extending a theorem of Orlov which applies in the case r = 1 [Orl09]): Theorem 1.10 (Favero-Kelly, Isik, Shipman, Orlov) If the nef-partition condition holds, then we have a quasi-equivalence where the right-hand side denotes a DG enhancement of the stacky derived category ofX b (which is unique by [LO10,CS15]).
The second is due to Orlov [Orl09], and does not depend on the nef-partition condition: Theorem 1.11 (Orlov) We have a quasi-equivalence b is a certain full subcategory of D b Coh(Ž b ) (which is in fact Calabi-Yau, see [Kuz15]). Thus we see that Conjecture A implies: Corollary B If the nef-partition condition holds (recall that this is true, in particular, in the Greene-Plesser case r = 1), then there is a quasi-equivalence Even if the nef-partition condition does not hold, there is a quasi-equivalence (1-23) D π F(X, ω λ ) bc AŽ b .

Main results
In order for Conjecture A to make sense, one needs a definition of the Fukaya category F(X, ω λ ). Unfortunately a general definition is not available at the time of writing, although it is expected that one will be given in the work-in-preparation [AFO + ], following [FOOO10]. However, the Fukaya category of a compact Calabi-Yau symplectic manifold of complex dimension ≤ 2 has been defined using classical pseudoholomorphic curve theory in [Sei14c]. Using this definition of the Fukaya category, we prove: Theorem C Conjecture A holds when dim C (X) ≤ 2 and the embeddedness and MPCS conditions hold.
If furthermore the 'no bc condition' below holds, then Conjecture A holds even if we remove the 'bc' from the Fukaya category.
It is not possible at present to give a complete proof of Conjecture A when dim C (X) ≥ 3, because we don't have a construction of the Fukaya category in that case. Nevertheless we have: Theorem D If we assume that the MPCS condition holds, and that the Fukaya category F(X, ω λ ) bc has the properties stated in § 2.5, then Conjecture A holds.
If furthermore the 'no bc condition' below holds, then Conjecture A holds even if we remove the 'bc' from the Fukaya category.
Definition 1.12 We say the no bc condition holds if there does not exist any K ⊂ I such that e K ∈ M and (1-24) This is the case, in particular, for all Greene-Plesser mirrors with dim C (X) ≥ 2.
Remark 1.13 In the case that X is a Calabi-Yau hypersurface in projective space, Theorems C and D were proved in [Sei14c] and [She15b] respectively.
Remark 1.14 If the embeddedness condition does not hold, then one must work with a version of the Fukaya category that includes certain specific immersed Lagrangians (see Lemma 3.7). It may well be the case that it is easier to include these specific immersed Lagrangians as objects of the Fukaya category, than to include general immersed Lagrangians (compare [AJ10]).
Remark 1.15 One might hope that the MPCS condition could be replaced by the weaker MPCP condition in Theorem D, and that the proof would go through with minimal changes. In that case (X, ω λ ) would be a symplectic orbifold, so the definition of the Fukaya category would need to be adjusted accordingly (compare [CP14]).
Remark 1.16 The properties of the Fukaya category outlined in § 2.5 should be thought of as axioms, analogous to the Kontsevich-Manin axioms for Gromov-Witten theory (without any claim to completeness however). They are structural, rather than being specific to the symplectic manifold X . Using these axioms, we reduce the problem of proving Conjecture A to certain computations in the Fukaya category of an exact symplectic manifold. Thus we have separated the proof of Conjecture A into two parts: one foundational and general, about verifying that the axioms of § 2.5 hold; and one computational and specific to X , taking place within a framework where foundational questions are unproblematic. The present work addresses the first (foundational) part in the setting of Theorem C, but not in the setting of Theorem D; and it addresses the second (computational) part in full generality.
Remark 1.17 The properties of the Fukaya category outlined in § 2.5 will be verified for a certain substitute for F(X, ω λ ) bc in the works-in-preparation [PS, GPS]. Namely, they will be verified for the relative Fukaya category specialised to the Λ-point corresponding to ω λ (see [She17,§ 5.4]). This will allow us to prove Theorem D for a specific version of the Fukaya category. However, this version of the Fukaya category is not so useful if one wants to study the symplectic topology of X , which is one of our intended applications. For example, it does not help one to study arbitrary Lagrangians in X : the only objects it admits are exact Lagrangians in the complement of a certain divisor D ⊂ X . On the other hand, the results of [PS, GPS] combined with the present work and [GPS15] are sufficient to compute rational Gromov-Witten invariants of X via mirror symmetry, so the substitute is good for this purpose.
Remark 1. 18 We can refine Conjecture A by giving a specific formula for the mirror map b(λ) (formulae in the Greene-Plesser case r = 1 can be found, for example, in [CK99, § 6.3.4]). We can also prove this refined version if we assume additional structural results about the cyclic open-closed map and its mirror, which are stated in [GPS15] and will be proved in [GPS]  Remark 1.20 There is a Fano version of the generalized Greene-Plesser mirror construction: the main difference is that one should assume i∈I j 1/d i > 1 for all j, rather than i∈I j 1/d i = 1. It should be straightforward to adapt the arguments of this paper to prove the Fano version. The technical aspects are easiest if one works with the monotone symplectic form, which means λ = e I . In that case the assumptions of § 2.5 can be shown to hold in any dimension (see [She15a]), so one does not need to impose caveats as in the statement of Theorem D. The situation is simpler than the Calabi-Yau case because the mirror map is trivial: one may take b p = q for all p ∈ Ξ 0 . However one slight difference arises in the Fano index 1 case: a constant term needs to be added to the superpotential W b (compare [She15a]).

Examples
We collect some interesting cases of our main theorems: first we consider the Greene-Plesser case r = 1, and describe the case of the mirror quartic in detail; then we consider Kuznetsov's 'K3 category of the cubic fourfold'; finally we consider the 'Z -manifold', an example of a rigid Calabi-Yau threefold.
Further interesting examples can be found in [Sch93,Sch96].

The mirror quartic
As we have already mentioned, when r = 1 our results amount to a proof of homological mirror symmetry for Greene-Plesser mirrors. There are 27 Greene-Plesser mirror pairs in dimension 2 [KS98], and 800 in dimension 3 [KS00]. We remark that our results imply both 'directions' of homological mirror symmetry: we prove both D π F(X) D b Coh(X) and D π F(X) D b Coh(X), for each pair of Greene-Plesser mirrors.
One interesting case is whenX is a Calabi-Yau hypersurface in projective space. It has been proved in [Sei14c] and [She15b] that D π F(X) D b Coh(X) for the appropriate mirror X , and we will not discuss this case further here. However our main result also applies to prove homological mirror symmetry in the other direction in these cases: i.e., we also prove that D π F(X) D b Coh(X), which is new. We feel it is illustrative to explain the case whenX is a quartic hypersurface in projective 3-space, and X is the mirror quartic.
The toric data in this case are as follows: r = 1, |I| = 4, and M := {m ∈ Z 4 : i m i ≡ 0 (mod 4)}. The simplex ∆ is illustrated in Figure 1: it is the convex hull of the vectors 4e i . The set Ξ 0 is also illustrated: it consists of all lattice points p = (p 1 , p 2 , p 3 , p 4 ) with p i ∈ Z ≥0 such that i p i = 4 and at least two of the p i are 0. In other words, Ξ 0 consists of all lattice points that lie at the vertices or on the edges of ∆. We have where val(b p ) = λ p . We have G ∼ = Z/4 in this case, and Γ is trivial. So is a quartic K3 surface in projective 3-space. By varying b we get a 22-dimensional family of hypersurfaces; however the algebraic torus G 3 m acting on P 3 preserves this family, so up to isomorphism we get a 19-dimensional family of K3 surfaces.
To describe the mirror, we consider the lattice M = M/e I equipped with the complete fan Σ whose rays are spanned by the vectors 4e i . It is illustrated in Figure 1: since we quotient by e I = i e i , the central point is now regarded as the origin. The corresponding singular toric variety Y is the quotient The hypersurface X ⊂ Y is the quotient of the Fermat hypersurface i z 4 i = 0 ⊂ CP 3 by H . Now X is not smooth: it has six A 3 singularities where it hits the pairwise intersections of the components of the toric boundary divisor of Y . We can resolve them by partially resolving the ambient toric variety. We do this by refining the fan Σ to Σ λ , which has rays spanned by vectors in the set Ξ 0 . We define X to be the proper transform of X in the corresponding partial toric resolution Y λ of Y . We  observe that the singularities of Y λ lie over the toric fixed points of Y , which X avoids: so X avoids the singularities and in fact is smooth (it is obtained from X by resolving the six A 3 singularities). We also observe that, while the toric variety Y λ depends on λ, the variety X does not.
We denote the intersections of X with the components of the toric boundary divisor by D p ⊂ X , for p ∈ Ξ 0 . We choose a Kähler form ω λ on X whose cohomology class is Poincaré dual to p λ p · [D p ]. Note that we have a 22-dimensional space of choices of λ; however the classes Poincaré dual to [D p ] only span a 19-dimensional space in H 2 (X), so up to symplectomorphism we get a 19-dimensional family of symplectic K3 surfaces.
Because we are in the Greene-Plesser case r = 1 the embeddedness condition holds, so we can apply Theorem C. It says that there exists Remark 1.21 Bayer and Bridgeland have computed the derived autoequivalence group of a K3 surface of Picard rank 1 [BB17], for example the general quartic surfaceX b . In work-in-preparation [SS], the authors combine Bayer-Bridgeland's result with (1-28) to derive consequences for the symplectic mapping class group of (X, ω λ ), for a generic Kähler class [ω λ ] (the genericity requirement on [ω λ ] ensures that the mirror has Picard rank 1).

The cubic fourfold
It is recognized that there is an intimate relationship between cubic fourfolds and K3 surfaces: see [Has16] and references therein. In particular, Hassett [Has00] explained that certain cubic fourfolds have an 'associated K3 surface' in a certain Hodge-theoretic sense. The moduli space C of cubic fourfolds is 20-dimensional, and Hassett showed that there exist certain irreducible divisors C d ⊂ C with an associated K3 surface that is polarized of degree d . It is conjectured (although not explicitly by Hassett) that a cubic fourfold is rational if and only if it has an associated K3 surface in this sense.
Relatedly, Kuznetsov [Kuz10] explained that any cubic fourfoldŽ has an associated category AŽ , which is the semi-orthogonal complement of the full exceptional collection O, O(1), O(2) ⊂ D b Coh(Ž). He observed that this category 'looks like' the derived category of a K3 surface, and conjectured that the cubic fourfold is rational if and only if it is equivalent to the derived category of an actual K3 surface (in which case the category is called 'geometric'). Addington and Thomas [AT14] showed that this holds for the general member of one of Hassett's divisors C d , establishing compatibility of these two rationality conjectures.
In this section we explain how Kuznetsov's category AŽ fits into our results.
To describe the mirror, we consider the elliptic curve E = C/ 1, e 2πi/6 , with the order-3 symmetry generated by z → ζ · z where ζ := e 2πi/3 . The quotient E/(Z/3) is isomorphic toX 1 ∼ =X 2 : it is a sphere with three orbifold points of order 3. We take the quotient of E × E by the anti-diagonal action of Z/3: i.e., (z 1 , z 2 ) → (ζ · z 1 , ζ −1 · z 2 ). This gives a surface X which has 9 A 2 singularities: resolving them we get a K3 surface X equipped with a divisor D which has 24 irreducible components. We consider a Kähler form ω λ on X with cohomology class Poincaré dual to p λ p · [D p ]. The classes Poincaré dual to [D p ] span a 20-dimensional space in H 2 (X), so up to symplectomorphism we get a 20-dimensional family of symplectic K3 surfaces.
These toric data do not satisfy the embeddedness condition (for example, e {1,4,5} ∈ V ∩ M does not lie in the span of e {1,2,3} and e {4,5,6} ), so we need to admit certain immersed Lagrangian tori into our Fukaya category in order for Theorem D to apply. If we do that, we obtain the existence Remark 1.22 The category AŽ b is only 'geometric' on some 19-dimensional loci inside the 20dimensional moduli space of cubic fourfolds, by [AT14] -indeed, for the generic cubic fourfold it does not even contain any point-like objects, simply for K -theory reasons. Thus it is striking that, on the A-side, the Fukaya category is 'geometric' (in the sense of being the Fukaya category of an honest symplectic manifold) on the entire 20-dimensional moduli space. The absence of point-like objects is mirrored by the absence of special Lagrangian tori T ⊂ (X, ω) for generic ω : the homology class [T] of such a special Lagrangian torus would have to be non-zero (because special), isotropic (because a torus), and lie in the transcendental lattice T(X) ∼ = −A 2 which however admits no non-zero isotropic vectors. In particular, there does not exist an SYZ fibration on (X, ω), so this version of homological mirror symmetry can not be proved using family Floer theory on X (it might, however, be possible to prove it via family Floer theory on a larger space, compare [AAK16]).
Remark 1.23 The construction generalizes to arbitrary values of r. The varietyŽ b is a cubic (3r − 2)fold, and the mirror X is a crepant resolution of the quotient E r /S, where S := ker((Z/3) r + − → Z/3).

Remark 1.24
The reverse direction of homological mirror symmetry in this case, which would relate a component of the Fukaya category of the cubic (3r−2)-foldŽ to the derived category of the Calabi-Yau r-fold E r /S, ought to follow from the results of [She16].

The Z -manifold
The Z -manifold is an example of a rigid Calabi-Yau threefold, i.e., one which has h 1,2 = 0 and therefore admits no complex deformations. In particular, it cannot be mirror to another Calabi-Yau threefold: the mirror would necessarily have h 1,1 = 0 and therefore could not be Kähler. It was first considered in the context of mirror symmetry in [CHSW85]; see [FG11] for a detailed study of its properties. It was explained in [CDP93] that the generalized mirror ought to be the quotient of the cubic sevenfold by a Z/3 action, and we verify this on the level of homological mirror symmetry here.
The toric data in this case are r = 3, The simplex ∆ is the convex hull of the vectors 3e i ∈ Z 9 . We have and acts on P 8 by multiplying homogeneous coordinates z 1 , z 2 , z 3 by ζ 1 , z 4 , z 5 , z 6 by ζ 2 , and z 7 , z 8 , z 9 by ζ 3 . Thus is a quotient of a cubic sevenfold by Z/3.
The mirror X (which is the Z -manifold) can be described as a crepant resolution of the quotient where E is as in § 1.7.2 and Γ * ∼ = Z/3 acts diagonally. Therefore Theorem D gives a quasi-equivalence The embeddedness condition does not hold (for example, e {1,4,7} ∈ V ∩ M is not in the span of the e I j ), so we must include certain immersed Lagrangian tori in our definition of the Fukaya category to obtain this result.

Outline
The strategy of proof of Theorems C and D has much in common with Seidel's proof of homological mirror symmetry for the quartic surface [Sei14c], and even more in common with the first-named author's proof of homological mirror symmetry for Calabi-Yau hypersurfaces in projective space in [She15b].
In particular, we use Seidel's idea [Sei02] to consider the Fukaya category of X relative to the simple normal crossings divisor D ⊂ X which is the intersection of X with the toric boundary divisor of Y λ . The relative Fukaya category can be regarded as a deformation of the Fukaya category of the complement of D, which in good situations one hopes is versal in an appropriate sense. The first step in the proof is to prove an equivalence of categories between the Fukaya category of X \ D and the category of perfect complexes on the central fibre of the mirror degeneration. This is done using the observation that X \ D is a cover of products of the 'pairs of pants' considered in [She11]. The next step is to show that the the relative Fukaya category and the mirror category of coherent sheaves match up to first order. This is done by a similar method, using the computation of the first-order deformation classes of the relative Fukaya category of the pair of pants from [She15b]. At that point a versality result takes over, and says that the relative Fukaya category is equivalent to the category of coherent sheaves to all orders, up to a formal change of variables corresponding to the mirror map [She17].
In [Sei14c,She15b], the irreducible components of D were all ample, which meant that the relative Fukaya category was defined over a formal power series ring. This made the deformation theory particularly simple. For arbitrary generalized Greene-Plesser mirrors however, D will have non-ample components which are created by the crepant resolution. In this case the relative Fukaya category will be defined over a more complicated ring which is related to the Kähler cone of X . This requires one to revisit the definition of the relative Fukaya category, and makes its deformation theory more complicated. The construction of the relative Fukaya category was explained in [She17], where relevant versality results were also proved.
It was explained in [She17] that the relative Fukaya category of a Calabi-Yau submanifold of a toric variety is automatically versal (for Kähler forms which are restricted from the ambient toric variety), so long as one can rule out deformation directions corresponding to H 2 (X \ D). This was done in [Sei14c,She15b] using symmetry, namely the action of the symmetric group on the homogeneous coordinates of X . Unfortunately that symmetry does not exist for all generalized Greene-Plesser mirrors. However, in [She17] it was also explained that one could use a real structure to rule out deformations in the direction of H 2 (X \ D). In this paper we verify that this does the job for the generalized Greene-Plesser mirrors.
N.S. was partially supported by a Sloan Research Fellowship, and by the National Science Foundation through Grant number DMS-1310604 and under agreement number DMS-1128155. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation. I.S. was partially supported by a Fellowship from EPSRC.

The ambient relative Fukaya category
In this section we recall the definition of the ambient relative Fukaya category given in [She17], and explain what it looks like in the present setting. Recall that Y λ is a (possibly singular) toric variety, and we denote the toric boundary divisor by D Y ⊂ Y λ . We consider the complete intersection X ⊂ Y λ , and equip it with the divisor D := X ∩ D Y . Our assumptions ensure that X is smooth and D is a simple normal-crossings divisor, so (X, D) is a snc pair in the sense of [She17]. Even though (Y λ , D Y ) need not be a snc pair because Y λ need not be smooth, we are going to call (X, D) ⊂ (Y λ , D Y ) a sub-snc pair in accordance with [She17, § 3.6], because all of the relevant arguments go through when Y λ has singularities so long as X avoids them, which it does when the MPCS condition holds.

Grading datum
The hypersurface X ⊂ Y λ is cut out by a section of a certain vector bundle L. We define , where the map from Z is induced by the inclusion of a fibre, and the map to Z/2 corresponds to the first Stiefel-Whitney class of the tautological real line bundle.
In this case the line bundle ∧ top (TY λ ⊕L) is trivial over Y λ (which is why X ends up being Calabi-Yau). Restricting this trivialization to Y λ \ D Y induces a splitting of the grading datum. This determines an isomorphism of G with the grading datum Note that there is also a morphism of grading data induced by the trivialization.

Relative Kähler form
We recall the covering group G :=M/M of the branched cover Y λ →Ỹ from § 1.3. The covering group G acts on (Y λ , D Y ), preserving the sub-snc pair (X, D). We also observe that Y λ has a real structure, as it is a toric variety and therefore defined over R: so it admits an anti-holomorphic involution. This involution preserves X , because its defining equation is real. The covering group G, together with the anti-holomorphic involution, generate a signed group (Ḡ, σ) which acts on (Y λ , D Y ) preserving the sub-snc pair (X, D), as outlined in [She17,§ 5.4].
Recall that a relative Kähler form on the snc pair (X, D) is a Kähler form ω on X equipped with a Kähler potential h on X \ D having a prescribed form near D [She17, Definition 3.2]. We abuse notation by denoting a relative Kähler form by ω ≡ (ω, h). Recall that a relative Kähler form defines a cohomology class [ω] ∈ H 2 (X, X \ D; R), which is specified by the linking numbers p corresponding to the irreducible components D p of D. Explicitly, [ω] = p p · PD(D p ).
Proof First let us suppose that Y λ is smooth. Recall that the support function of the divisor p λ p · D Y p on Y λ is the piecewise-linear function which is linear on each cone of Σ λ and equal to −λ p at each p ∈ Ξ 0 . It is clear that this coincides with the function ψ λ already defined, and that ψ λ is strictly convex by our assumptions, so this divisor is ample by [

Coefficient ring
We recall the definition of the coefficient ring of the ambient relative Fukaya category (see [She17,§3.7] for further details).
Recall that we can identify H 2 (X, X \ D; R) ∼ = R P with the space of R-divisors supported on D. The function ι : P Ξ 0 determines a map ι * : Z P Z Ξ 0 (when Y λ is smooth we identify it as the map H 2 (X, X \ D) → H 2 (Y λ , Y λ \ D Y ) induced by the inclusion). Applying Hom(−, R) gives a map ι * : R Ξ 0 → R P (when Y λ is smooth we identify it as the restriction map H 2 (Y λ , Y λ \ D Y ; R) → H 2 (X, X \ D; R)). We let Nef (X, D) ⊂ R P correspond to the cone of effective nef divisors supported on D, and we suppose that N ⊂ Nef (X, D) is a convex sub-cone. We denote the pre-image of N under ι * by N amb ⊂ R Ξ 0 . We will assume that N is amb-nice in the sense of [She17, Definition 3.39], rational polyhedral, contained in the ample cone, and that N amb contains λ in its interior. Such a cone exists by [She17, Lemmas 3.30 and 3.44], because λ ∈ Amp(Y λ , D Y ) by construction.
We denote the dual cone to N amb by N ∨ amb ⊂ R Ξ 0 , and NE amb (N) := N ∨ amb ∩ Z Ξ 0 . We observe that the interior of N amb contains λ and in particular is non-empty, so N ∨ amb is strongly convex. We define a C-algebra and equip it with a G-grading by putting the generator r p in degree 0 ⊕ p, for all p ∈ Ξ 0 . It has a unique toric maximal ideal m ⊂ R amb (N), and we define R amb (N) to be the G-graded completion of R amb (N) with respect to the m-adic filtration. We will abbreviate R := R amb (N).

Ambient relative Fukaya category
The ambient relative Fukaya category F amb (X, D, N) is a G-graded R-linear A ∞ category. Its objects are compact exact Lagrangian submanifolds L ⊂ X \ D, equipped with an anchoring, Pin structure and orientation.

Remark 2.3
The various versions of the Fukaya category (absolute, relative, and ambient relative) can be defined without requiring the Lagrangian branes to be oriented. In particular, [She17] did not require the Lagrangian branes to be oriented: there is a forgetful functor from the version we introduce here to the version considered in [She17] which forgets the orientation of each object. We need to consider oriented Lagrangians here in order for the open-closed map OC to be defined (see § 2.5), since that is required for us to prove split-generation of the Fukaya category in Proposition 4.8 using Abouzaid's criterion ( [She17] did not consider the open-closed map).
The morphism spaces in the relative Fukaya category are free R-modules generated by intersection points, and its A ∞ structure maps count pseudoholomorphic discs u : D → X with boundary on the Lagrangian branes, with a weight r ι * [u] ∈ R. In order to arrange that [u] ∈ NE amb (N), we Remark 2.4 The definition of the relative Fukaya category depends on a choice of relative Kähler form ω on (X, D) (e.g., the objects are Lagrangian with respect to the chosen symplectic form, and exact with respect to the chosen primitive for it on X \ D). It should be independent of ω in some sense, which is why we do not include it in the notation. However we have not proved this independence. Nevertheless, a weak version of it is proved in [She17, § 4.5]: namely, that F(X \ D) and the first-order deformation classes of F amb (X, D, N) are independent of ω . This weak version is all that we will use in this paper (see Remark 3.11), so we hope the reader will accept this notational imprecision in the name of readability.
We define a C-algebra homomorphism We regard it as a Λ-point a(λ) of the scheme  → F(X, ω λ ) bc .
Recall that the 'q * ' means we turn the G-graded category into a Z-graded one via the morphism q : G → Z, and the subscript 'a(λ)' means we take the fibre of the family of categories over the corresponding Λ-point. In other words, we turn the R-linear category into a Λ-linear one by tensoring with Λ (regarded as an R-algebra via the homomorphism a(λ) * ).

Assumptions about the Fukaya category
In this section we explain which properties of (the various versions of) the Fukaya category we will use, because the constructions of these categories and the proofs of their basic properties have not yet been carried out in full generality. We will discuss cases in which these assumptions have been proved. We assume that the ambient relative Fukaya category is defined and satisfies [She17, Assumption 5.1] (more precisely, the analogue of that assumption in the ambient case): namely, it is a G-graded (possibly curved) deformation of F(X \ D) over R. We assume that its first-order deformation classes are as prescribed in [She17, Assumption 5.3].
We assume that the absolute Fukaya category F(X, ω) bc is defined and satisfies [She17, Assumption 5.4]: namely, there is an embedding of Λ-linear, Z-graded A ∞ categories as in (2-6) (here and in what follows, we abbreviate ω = ω λ ). We assume the existence of the open-closed map, a map of Λ-vector spaces (2-7) OC : . We assume that the map HH n (F(X, ω) bc ) → Λ given by X OC(−) defines a weak proper Calabi-Yau structure on F(X, ω) bc (see e.g. [GPS15, Definition 6.3]). We assume the existence of the coproduct, which is a morphism of F(X, ω) bc -bimodules (2-8) from the diagonal bimodule F ∆ to the tensor product of left-and right-Yoneda modules for any object K . We assume the existence of the length-zero part of the closed-open map, a unital graded Λ-algebra homomorphism (2-9) CO 0 : QH • (X) → Hom • F(X,ω) bc (K, K) for any object K , where QH • (X) is equipped with the quantum cup product. We assume that the Cardy relation is satisfied, which means that the diagram (2-10) H * (µ) / / Hom • F(X,ω) bc (K, K) commutes for any object K , up to the sign (−1) n(n+1)/2 (see [Abo10] for notations). We assume that the open-closed map respects pairings, in the sense that for all α, β ∈ HH • (F(X, ω) bc ). Here −, − Muk denotes the 'Mukai pairing' on Hochschild homology, as defined by Shklyarov [Shk12] (see also [Cos07]).
Remark 2.5 When (X, ω) is positively monotone (which is not true in our case), versions of the relative and absolute Fukaya categories F amb (X, D, N) and F(X, ω) that satisfy the analogues of all of the above assumptions are constructed using classical pseudoholomorphic curve theory in [She16] (up to minor changes in conventions), with the exception of the proof that OC respects pairings. The proof that OC respects pairings is almost identical to the proof of the Cardy relation, when one uses the explicit formula for the Mukai pairing derived in [She15a, Proposition 5.22] (compare [Sei12,§5b]). We remark that it was also explained in [She15a] how to incorporate homotopy units and (weak) bounding cochains supported on (direct sums of) Lagrangians, which introduced a subtlety regarding the unitality of CO 0 .
Remark 2.6 When X is Calabi-Yau, the constructions and proofs referenced in Remark 2.5 go through with minor alterations so long as one can upgrade each object L of F amb (X, D, N) to a strictly unobstructed object, which is a pair (L, J L ) where L is a Lagrangian brane and J L an ω -compatible almost-complex structure such that there are no non-constant J L -holomorphic spheres intersecting L, or non-constant J L -holomorphic discs with boundary on L, where J L should be adapted to the system of divisors E. The construction of the absolute and relative Fukaya categories whose objects are such pairs (L, J L ) is discussed for example in [Sei14b,Sei14a]. The incorporation of bounding cochains is straightforward, following [She16]. We remark that the subtlety regarding unitality of CO 0 referenced in Remark 2.5 does not arise in the context of the present paper, because we need only consider bounding cochains (rather than weak bounding cochains), so we do not need homotopy units, which were the origin of the subtlety (see [She16, Remark 5.7]).
Remark 2.7 When dim C (X) ≤ 2 the condition that (L, J L ) should be strictly unobstructed is generic in J L , so any Lagrangian brane can be upgraded to a strictly unobstructed object in this case. It follows by Remark 2.6 that all of the above assumptions hold in this case. When dim C (X) ≥ 3, there is no reason to expect that the Lagrangians we consider in this paper can be upgraded to strictly unobstructed objects. In this case, virtual techniques may be required [FOOO10, AFO + ] to justify our assumptions, or recourse to the substitute mentioned in Remark 1.17.

Branched cover and the corresponding map of grading data
Recall that the morphism of fans Σ λ →Σ determines a toric morphism Y λ →Ỹ with covering group G =M/M , which induces a branched covering of sub-snc pairs in the sense of [She17, § 4.9]. We will denote the toric boundary divisor ofỸ byDỸ , and its intersection withX byD . We have for any p ∈ P.

The immersed Lagrangian sphere in the pants
Let us assume for the moment that r = 1. Then the hypersurfaceX \D ⊂Ỹ \DỸ is an (|I| − 2)dimensional pair of pants. In [She11], an exact immersed Lagrangian sphere L X \D was constructed, equipped with an anchoring and Pin structure, and the endomorphism algebraÃ I 0 := hom • F amb (X \D ) (L, L) was explicitly computed (up to A ∞ quasi-isomorphism). We briefly recall the result.
The grading datum associated toX \D isG = Z ⊕ Z I /(2(1 − |I|), e I ). We haveÃ I 0 ∼ = C[θ i ] i∈I on the cochain level, where θ i has degree (−1, e i ). The variables θ i are in odd degree, so this is an exterior algebra rather than a polynomial algebra. The algebra structure µ 2 is the exterior product, and the higher A ∞ products µ ≥3 define a Maurer-Cartan element in CC • (C[θ 1 , . . .]).
We have the Kontsevich formality quasi-isomorphism of L ∞ algebras [Kon03]: . .], where the variable z i has degree (2, −e i ), and the variables θ i are graded as before. The variables z i are even, so commute, and θ i are odd, so anti-commute. The right-hand side is a formal L ∞ algebra, i.e., it has L ∞ products s = 0 for s = 2, and 2 is the Schouten bracket. We would like to use this to compute the Hochschild cohomology ofÃ I 0 , following [She16, § 6.4].
Lemma 3.3 The pushforward of the Maurer-Cartan element µ ≥3 by Φ HKR is where we recall W 0 = −z e I .

Proof
The formula for the pushforward of a Maurer-Cartan element by an L ∞ morphism is It is computed in [She11] that the leading term is Φ 1 HKR µ ≥3 = W 0 , so it suffices to prove that the remaining terms in (3-12) vanish.
We need to compute HH • (Ã I 0 ), so we turn to that task now. Note that W 0 does not have an isolated singularity at 0, so the cohomology of K(dW 0 ) is not concentrated in degree 0.
Let U := C I , and H := U/e I . For any K ⊂ I we denote where I is the ideal generated by zK · ∧ top (H * K ) for all K ⊂ I (here, 'K ' denotes the complement of K ).
We now define an injectiveG-graded algebra map Lemma 3.5 The map f induces an isomorphism ofG-graded C-algebras Proof Suppose we have an element of the kernel of ι dW 0 : (3-20) It follows that K a K (z) · θ K ∈ im(f ): so ker(ι dW 0 ) ⊂ im(f ).
As W 0 is not a zero-divisor, it follows that f induces an isomorphism We now have The image of ι dW 0 is generated by the classes . Therefore the right-hand side of (3-26) spans zK · ∧ top (H * K ), completing the proof.
Now we consider the case r > 1. We consider the product exact Lagrangian immersion L := j L j j (X j \D j ) =X \D . Its endomorphism algebra is quasi-isomorphic to

Signed group action
We briefly recall the notion of a signed group action on an snc pair from [She17, Definition 5.8]. A signed group is a group with a homomorphism to Z/2, so that the group can be decomposed into 'odd' and 'even' elements. An action of a signed group on an snc pair (X, D) is an action of the group on X , preserving D as a set, such that even elements act by holomorphically and odd elements antiholomorphically. A signed group action on an snc pair (X, D), together with a morphism of grading data G(X \ D) → Z/4 that is preserved by the action, induces a signed group action on the relative Fukaya category by [She17,Lemma 5.12].
In our case, complex conjugation τ :X →X defines a signed action of Z/2 on (X ,D ). Any holomorphic volume form onX with poles alongD induces a map of grading data, v :G → Z (and hence a map to Z/4, by post-composing with Z → Z/4). Explicitly, if the volume form has a pole of order v i alongD i , and we denote v := i v i e i , then we have (3-29) v, e I j = |I j | − 1 for all j, and the morphism is defined by If we choose a real holomorphic volume form, i.e., one such that τ * Ω = Ω, then τ preserves the map of grading data v (see [She17,Example 5.11]).
Thus, τ together with v determine a signed action of Z/2 on F amb (X \D ). Furthermore, it was observed in [She11] that we have an isomorphism of branes L ∼ = τ L. As a result, τ induces an action of Z/2 on the vector spaceÃ 0 = hom • F amb (X \D ) (L, L). The non-trivial element of Z/2 acts on the endomorphism algebra of L by sending (it was erroneously claimed in [She11, Corollary 3.13] that the action sent θ K → −θ K ; the correct calculation appears in the post-publication update to the arXiv version of [She11]).
It is immediate that (3-32) defines a signed action of Z/2 on the endomorphism algebra of L, on the level of cohomology (and this is how one establishes that the endomorphism algebra is supercommutative). We would like to lift it to an action on the cochain level, but this may run into issues with equivariant transversality. To avoid them, we define a full subcategoryÃ 0 ⊂ F amb (X \D ), closed under shifts, which has two underlying unanchored Lagrangian branes: L and τ L. The advantage of this 'doubled' category is that Z/2 acts freely on the underlying set of unanchored Lagrangian branes, bypassing issues with equivariant transversality: so we have a signed action of Z/2 onÃ 0 up to shifts, by [She17,Lemma 5.12].
Because L ∼ = τ L, the inclusion of the full subcategory whose objects are L and its shifts is a quasiequivalence. In particular we have an isomorphism HH • (Ã 0 ) ∼ = J from the previous section. The signed action of Z/2 onÃ 0 induces an action on HH • (Ã 0 ) = J (see [She17,§ A.4

]).
Lemma 3.6 Let z a · h represent an element of J , where h ∈ ∧ |h| H . The non-trivial element of Z/2 sends Proof The element z a · h is represented by a sum of Hochschild cochains of the form where a + j∈K e j = j e i j (as can be seen from (3-18) and the explicit formula for the HKR isomorphism [She15b, Definition 2.89]). By (3-32), the non-trivial element of Z/2 sends this Hochschild cochain to a Hochschild cochain of the form The isomorphism CC • (F op amb ) → CC • (F amb ) then sends this to a Hochschild cochain of the form . The variables θ i j are all odd, so in fact vanishes.
Now we consider the branched cover of snc pairs φ from (3-1). By [She17,Lemma 4.17] combined with Lemma 2.1 we can equip (X, D) with a (Ḡ, σ)-invariant relative Kähler form ω so that φ becomes a branched cover of relative Kähler manifolds.
It follows that there is an embedding [She15b,Proposition 4.23], using the facts that p is the morphism of ambient grading data induced by the branched cover φ by Lemma 3.2, that this morphism is injective, and that the covering group G of φ is abelian. We denote the image of this embedding by A 0 . Proof Recall that the generators θ K ofÃ I j 0 correspond to self-intersections of L j for all K ⊂ I j except K = ∅, I j (which correspond to the generators of the cohomology of the underlying sphere). Therefore the generators θ K of the product L = L 1 × . . . × L r correspond to self-intersections for all K ⊂ I except K = j∈J I j where J ⊂ {1, . . . , r}.
The self-intersection θ K inX lifts to an intersection between two lifts of L in X . The two lifts of L coincide (i.e., θ K is a self -intersection) if and only if the degree of θ This isomorphism is Z/2-equivariant (for this it suffices that the morphism G → Z/4 factors through G, which is true by construction). Thus we have (3-44)

Deformation classes
We recall the graded vector spaces sh • amb (X ,D ) and sh • amb (X, D) defined in [She17, § § 4.3 and 4.9]. The basis elements of sh • amb (X ,D ) are denotedỹ u , where u ∈ H 2 (X ,D ) is a class that can be represented by a disc meetingD at a single point, where it meets each component ofD non-negatively. We denote the elements dual to the divisorsD i byỹ i :=ỹ ei . We denote the basis elements of sh • amb (X, D) similarly by y u and y i .
We recall the maps defined in [She17,§ 4.4]. The idea is that this is a version of the 'closed-open map': co(y u counts pseudoholomorphic discs with a single internal marked point at which the curve is required to have orders of tangency with the components ofD prescribed by u. We observe that co : The first-order deformation classes ofÃ are computed in [She15b, Proposition 6.2] up to sign: the result is that co(ỹ i ) is equal to ±z i ∈ J I j . It follows that the first-order deformation classes ofÃ 0 ⊂ F amb (X \D ) are co(ỹ i ) = ±z i ∈ J , by [She17,Proposition 4.25]. It follows that (3-48) co ỹ p = ±z p for all basis elementsỹ p of sh • amb (X ,D ), by [She17,Lemma 4.13], and in particular for all p ∈ Ξ 0 . We also consider the map from [She17, Definition 4.21]. We observe that it actually lands in sh • amb (X, D) G , as is clear from the definition. It follows from Lemma 3.1 that y q for all p ∈ Ξ 0 . The sum on the right-hand side is over all components D q that are contained in the component of D Y p of D Y . We observe that the image of the right-hand side under co is precisely the pth deformation class of the corresponding subcategory A ⊂ F amb (X, D, N), by our assumptions in § 2.5 (specifically, our assumption that [She17, Assumption 5.3] holds).
Lemma 3.8 HH 2 A 0 , A 0 ⊗ m Ḡ is contained in the R 0 -submodule generated by the deformation classes r p · z p . Furthermore, the deformation classes are all non-zero.
Applying (3-53) and Lemma 3.6, we find that the non-trivial element of Z/2 sends (since |h| is even) j · e I j + n σ , b by (3-56) = 1 + n σ , b (because n σ , e I j = 1 = v − e I , e I j by (3-29)) Thus, in order for r a z b h to represent a Z/2-invariant class, |h| must be divisible by 4. We already showed |h| ≤ 2, so we must have |h| = 0.
It follows that b ∈ Ξ. If b / ∈ Ξ 0 , then there exists some k ∈ I j such that z b is divisible by i∈I j \k z i . One easily verifies that the latter monomial is a generator of the ideal I j by which we quotient to get J I j , so z b vanishes in this case. Thus, in order for r a z b h to be non-vanishing and Z/2-invariant, we must have b ∈ Ξ 0 and |h| = 0.
The degree of r a z b is then equal to the degree of r b z b (since both are equal to 2), so r a has the same degree as r b . It follows that r a is a multiple of r b , because the coefficient ring R is 'nice' in the sense of [She17, Definition 2.3], by [She17, Lemma 3.42], because we chose N to be amb-nice in § 2.4. Therefore r a z b h is a multiple of the first-order deformation class r b z b , as required.
Finally, it is easy to check from the definitions that z b = 0 in J for all b ∈ Ξ 0 , so the first-order deformation classes are non-zero.
Remark 3.9 Lemma 3.8 may appear mysterious at first. The geometric reason for it is explained in [She17, Corollary 6.8]. In particular, one of the important steps in the proof of Lemma 3.8 was to rule out deformation classes r a z b h with |h| = 2. This corresponds, in [She17, Corollary 6.8], to showing that H 2 (X \ D)Ḡ ∼ = 0. Indeed, in this case we have H 2 (X \ D)Ḡ ∼ = H 2 (X \D ) Z/2 , so we must show that the anti-holomorphic involution τ * acts with sign +1 on H 2 (X \D ) (because τ is defined to act on H • (X \D ) by −τ * , see [She17, Equation (6-4)]). This follows because τ * acts with sign (−1) k on H k (Ỹ \DỸ ) ∼ = H k ((C * ) |I|−r ), and the restriction map H 2 (Ỹ \DỸ ) → H 2 (X \D ) is surjective. Now let A ⊂ F amb (X, D, N) denote the full subcategory corresponding to A 0 ⊂ F(X \ D). The category A is aḠ-equivariant deformation of A 0 over R relative to the action ofḠ on R by (3-52) (see [She17,Lemma 5.12]). We recall some terminology from [She17, § 2]: the equivariant deformation is said to be R-complete if, for anyḠ-equivariant deformation B of A 0 over R such that HH 2 (A 0 , A 0 ⊗ m/m 2 )Ḡ is contained in the span of the first-order deformation classes of B, there exists an automorphism Ψ * : R → R and a (possibly curved) A ∞ isomorphism If furthermore the map Ψ * : m/m 2 → m/m 2 is uniquely determined, the deformation is said to be R-versal.
Corollary 3.10 A is an R-versalḠ-equivariant deformation of A 0 over R.
Remark 3.11 Although we used a specific relative Kähler form ω to verify Corollary 3.10, namely one such that the branched cover φ respects relative Kähler forms, the analogous result follows for arbitrary (Ḡ, σ)-equivariant relative Kähler forms by [She17, Remark 5.14].
We finish with the following: Lemma 3.12 If the no bc condition holds (Definition 1.12), then the A ∞ isomorphism (3-61) is necessarily non-curved.
Proof Suppose to the contrary that the curvature of (3-61) is non-zero. The curvature defines a degree-1 endomorphism of each object in A (where '1' means '(1, 0) ∈ G'). Such an endomorphism can be written as r a · α, where r a ∈ m and α is a lift of some endomorphism of an object inÃ 0 .
Suppose that α is a lift of the endomorphism θ K . If θ K is to lift to an endomorphism in A, i.e., a self -intersection point of some lift of L, then we must have e K ∈ M (as in the proof of Lemma 3.7). On the other hand, for r a θ K to have degree (1, 0), we must have (following the proof of Lemma 3.8 and skipping some steps): Therefore we have e K ∈ M and (3-63) holds: this contradicts the no bc condition, so the proof is complete.

Graded matrix factorizations 4.1 Matrix factorizations
We make the G-graded ring R into aG-graded ring by pushing the grading forward by p: so r a has degree 2|a| − 2|k(a)|, a ∈G. We introduce theG-graded ring S := R[z i ] i∈I with z i in degree (2, −e i ). We define the element Assuming all terms of W to have degree ≥ 2, a minimal model forB dg was constructed in [She15b, §7.2] using the homological perturbation lemma. We denote it byB. The underlying R-module is R[θ i , .
. .] i∈I with θ i in degree (−1, e i ) (as in § 3.2). The A ∞ products have the form µ * = µ 2 ext +μ * , where µ 2 ext denotes the exterior product among the θ i , andμ * is everything else. The leading term in the HKR map (3-10) sends HKR μ * = W (4-4) by [She15b,Proposition 7.1] (the result there was stated in the case that W has degree ≥ 3, but the proof works also if W has quadratic terms).

Signed group action
Recall that on the A-side, the choice of a holomorphic volume form onX with poles alongD induced a Z/2-action onÃ 0 . We introduced the vector v ∈ Z I , where the ith entry v i is the order of pole of the volume form alongD i . This induces an involution on the coefficient ring R, defined in (3-52). We extend this to an involution : S → S by defining (z i ) := (−1) 1+vi z i . Proof The terms in W have the form r a z b , so can also be regarded as an element of (J ⊗ m) 2 . In Lemma 3.8 we considered an action of Z/2 on such elements: this action is the negative of the action of , because of the leading '1' in the sign † from Lemma 3.6. Since we verified in the proof of Lemma 3.8 that the action of Z/2 preserves the terms r a z b of degree 2, it follows that the action of reverses the sign of each term. Now recall that there is a canonical isomorphism of DG categories, MFG(S, W) ∼ = MFG(S, −W) op , given by dualization (see, e.g., [Dyc11,§ 4.3]). This is the analogue of the isomorphism F(X, ω) ∼ = F(X, −ω) op that goes into constructing the signed group action on the Fukaya category. On the level of objects, the isomorphism sends a matrix factorization K = (K, δ K ) of W to the dual matrix factorization . On the level of morphisms, it sends a morphism f ∈ Hom • S (K, L) to the morphism f ∨ ∈ Hom S (L ∨ , K ∨ ), where (4-7) f ∨ (α)(k) := (−1) |f |·|α| · α(f (k)).
The matrix factorization O 0 := (K, δ K ) has underlying S-module K := S[ϕ 1 , . . .] where the ϕ i have degree 1 ⊕ −e i and anticommute, and differential (4-8) . .] ∼ = K ∨ in the standard way, where the θ i have degree (−1, e i ) and anticommute: explicitly, we map (4-9) The dual differential is easily computed to be (4-10) (4-11) We now have the standard isomorphism of a Koszul complex with its dual: where ϕ top := ϕ 1 ϕ 2 . . . ϕ |I| . One easily verifies that this map commutes with the differentials (the sign (−1) k is needed so that the differential on K is the original δ K : without it, the map would commute with the differential −δ K on K ). We observe that this map has degree r − |I|, so this isomorphism is not an isomorphism in MFG(S, W) because it is not graded (recall that shifting in MFG(S, W) changes the sign of the differential). Nevertheless it defines a graded isomorphism of endomorphism DG algebras (4-13) hom • MFG(S,W) (K, K) ∼ = hom • MFG(S,W) (K, K) op , which is what we will need.

Versality
We now mirror the construction of A in the matrix factorization world. We define a subcategorỹ B dg ⊂ A ∞ (MFG(S, W)) which has objects O 0 and O ∨ 0 and all of their shifts, and equip it with a signed Z/2-action up to shifts by dualization. We construct a minimal modelB forB dg as above: we may do so in such a way that it also has an induced Z/2-action. LetB 0 be its order-0 A ∞ algebra: then it follows from the preceding computations that we have a Z/2-equivariant isomorphism of categories Let us denote the corresponding minimal model for a subcategory of MF(C[z i ] i∈I j , −z e I j ) byB It was shown in [She11,She15b] that there is an A ∞ isomorphismB Ã I j 0 . We can take the tensor product of these isomorphisms, by [Dyc11, § 6] and [Amo17], to obtain a Z/2-equivariant A ∞ isomorphismB 0 Ã 0 .
We now define B := p * B .
As a corollary, there is a non-curved A ∞ embedding If the no bc condition holds, then we can remove the 'bc' from (4-19).
Proof The A ∞ isomorphismB 0 Ã 0 induces an A ∞ isomorphism B 0 A 0 between the orderzero categories, so we may assume without loss of generality that B 0 = A 0 (see [She15b, Proof of Corollary 2.105]). We then observe that B and A are now (Ḡ, σ)-equivariant deformations of A 0 over R; and they have the same deformation classes r p z p up to sign, as we calculated in § 3.4 (on the A-side) and (4-4) (on the B-side). The existence of Ψ * and F then follows by Corollary 3.10. The fact that the first-order deformation classes coincide up to sign allows us to conclude (4-18).
To prove the corollary, we first observe that B is non-curved by definition, so we can equip each object with the zero bounding cochain. By [She17, Lemma 2.16], there is a non-curved A ∞ embedding (4-19) which sends each object of B to the corresponding object of Ψ * A equipped with a bounding cochain given by the curvature F 0 . If the no bc condition holds, then the A ∞ isomorphism F is already non-curved by Lemma 3.12, so the 'bc' can be removed from (4-19).

Graded matrix factorizations
We recall that the category of graded matrix factorizations [Orl09] can be formulated in terms of the grading datum G MF(d) := Z ⊕ Z/(2, −d) (see [She15b,§ 7.5]). Namely, we equip the polynomial ring with a G MF(d) -grading by putting z i in degree (0, q i ), then (4-21) is the unique morphism of grading data. However we want to consider the category of Γ-equivariant graded matrix factorizations. To that end we introduce a new grading datum , 0), and the sign map G ∆ → Z/2 sends (k, u) → [k]. We equip S Λ with a G ∆ -grading by putting z i in degree (0, e i ). There is a morphism of grading data t : t(k, m) := (k, q, m ), which recovers the G MF(d) -grading of the polynomial ring from the G ∆ -grading.
A Γ * -grading determines a Γ-action, whose invariant part is the part of degree 0 ∈ Γ * . In this case it is a simple matter to verify that (4-25) , where s : Z → G ∆ is the unique morphism of grading data. This justifies the following definition of the category of Γ-equivariant graded matrix factorizations: We define a morphism of grading data r :G → G ∆ (4-27) r(k, m) := (k + 2|m|, −m). Observe that r * S is a G ∆ -graded algebra, and one easily verifies that R is in degree 0, and z i is in degree (0, e i ). It follows that for any Λ-point b of Spec(R), we have fully faithful embeddings There is a commutative square of grading data: Proof The maps in the square send (recall that p is determined in Lemma 3.1). The commutativity follows because (2 n σ , m , −m) = 0 in G ∆ .
By the existence of the commutative square (4-29) and [She15b, Lemma 2.29], we have an isomorphism of categories where the subscript H denotes equivariance with respect to a certain action of the dual group H of the group In this case, we have G/Z ∼ = M/ e I j ,G/Z ∼ = Z I / e I j , and G ∆ /Z ∼ = Z I /M , so one easily verifies that (4-32) is 0: thus we may remove the H from (4-31).

W b has an isolated singularity
Let b ∈ A Ξ 0 have coefficients (b p ) p∈Ξ 0 , with val(b p ) = λ p . Let W b be as in (1-13). The aim of this section is to prove the following Proposition, which is based on the relationship between the tropical A-discriminant and the secondary fan (compare [GKZ94,DFS07]), although we will not use that language.
Proposition 4.4 If the MPCP condition holds, then W b has an isolated singularity at the origin.
Remark 4.5 We will apply this result (in the proofs of Propositions 4.7 and 4.8) with b = Ψ −1 (a(λ)). Note that the mirror map Ψ is at this stage undetermined; we only know that Ψ * (r p ) = ±r p + m 2 , which implies that val(b p ) = val(a p ) = λ p , but we do not know the precise coefficients b p . So it is a crucial feature of Proposition 4.4 that it needs only to make an assumption on the valuations of the coefficients of b, rather than requiring precise knowledge of the coefficients themselves.
We need some preliminary discussion before giving the proof of Proposition 4.4.
We have a decomposition of A I into toric orbits (G m ) K indexed by subsets K ⊂ I . In order to prove that W b has an isolated singularity at the origin, it suffices to prove that the vanishing locus of W b | (Gm) K is smooth for all K . We start with the case K = I .
Let B ⊂ Z I denote the set of monomials appearing in W b (their convex hull is the Newton polytope ∆). The valuations of the corresponding coefficients define a 'weight vector' for these vectors (see [MS15,Definition 2.3.8]), which is equal to 0 at e I j for 1 ≤ j ≤ r, and equal to λ p at p for p ∈ Ξ 0 . This weight vector induces a regular subdivision T λ of ∆. If T λ is a unimodular triangulation, then the vanishing locus of W b | (Gm) I is smooth by [MS15, Theorem 4.5.1]; in fact the proof goes through verbatim without the assumption of unimodularity when the field has characteristic zero, so it suffices for us to prove that T λ is a triangulation.
We consider the projection π : R I → M R from § 1.3, which sends all e I j to the origin. We set ∆ := π(∆) (this clashes with the notation from § 1.4, but no confusion should result) and B := π(B), and define a weight vector for B which is equal to 0 at the origin and λ p at π(p) for p ∈ Ξ 0 . We denote the induced regular subdivision of ∆ by T λ : by definition it coincides with the fanΣ λ , and therefore is a triangulation becauseΣ λ is simplicial by our assumption that the MPCP condition holds.
Proof of Proposition 4.4 The simplices σ cover ∆, so the simplices σ = π −1 (σ) ∩ ∆ cover ∆; it follows that T λ is a triangulation as required. Therefore the vanishing locus of W b | (Gm) K is smooth for K = I . It follows also that the restriction of T λ to any coordinate hyperplane is a triangulation, and hence that the analogous result holds for any K .
Proof of Lemma 4.6 The first claim follows immediately from the fact that the weight vector for B is pulled back from that for B. For the second claim, it is immediate that σ ⊂ π −1 (σ). What remains to prove is the reverse inclusion, so let x ∈ π −1 (σ) ∩ ∆; we will show that x lies in the convex hull of C.
Applying pr j to (4-36), we obtain pr j (x) = c∈C α π(c) · pr j (c) + β j · e I j . (4-38) Now any element of Ξ 0 must project to an element of Z I j with at least two vanishing coordinates, by definition of Ξ 0 , so the same is true of C ⊂ Ξ 0 . Furthermore, becauseΣ λ is assumed to be a refinement ofΣ := jΣ j , the projection of σ to Z I j /e I j lies inside a cone ofΣ j . It follows that pr j (C ) lies inside a coordinate hyperplane of Z I j . Examining (4-38), and observing that x ∈ ∆ ⊂ (R ≥0 ) I , it follows that β j ≥ 0.
Applying n σ , − to (4-36), we find that c∈C α π(c) + r j=1 β j = n σ , x = 1. (4-39) We now have two cases: if 0 ∈ C, then e I j ∈ C for all j, and (4-36) expresses the fact that x lies in the convex hull of C (since we have proved that the coefficients are non-negative and sum to 1). If 0 / ∈ C, then C = C so we have c∈C α π(c) = c∈C α c = 1, from which it follows by (4-39) that r j=1 β j = 0. Since we showed that β j ≥ 0, we conclude that β j = 0 for all j, so (4-36) again expresses the fact that x lies in the convex hull of C.
The third claim is equivalent to the claim that the set C is linearly independent. Suppose to the contrary that it is linearly dependent. We claim that this implies that C is linearly dependent. Indeed, if 0 / ∈ C, then C = C so there is nothing to prove. If 0 ∈ C, then (4-36) holds with x replaced by 0. The previous argument applies to show that β j = 0 for all j, and hence that C is linearly dependent. Now, linear dependence of C implies linear dependence of π(C ) = C , which contradicts our assumption that T λ is a triangulation. Therefore C must be linearly independent, so σ is a simplex as required. GrMF Γ (S Λ , W b(λ) ) F(X, ω λ ) bc .

Split-generation
We will denote C := q * B b(λ) , and regard it as a full subcategory C ⊂ GrMF Γ (S Λ , W b(λ) ) which is identified with a full subcategory C ⊂ F(X, ω λ ) bc in accordance with (4-40). In this section we prove: Proposition 4.7 If the MPCP condition holds, then C split-generates GrMF Γ (S Λ , W b(λ) ).
Proposition 4.8 If the MPCS condition holds, then C split-generates D π F(X, ω λ ) bc .
These two Propositions (together with the observation that the 'bc' can be removed everywhere from Lemma 4.2 onwards, if the no bc condition holds) complete the proof of Theorems C and D.
We start by recalling some background. Let D be a triangulated category (e.g., the cohomology category of a triangulated A ∞ category). Let E ⊂ D be a full subcategory; recall that the right orthogonal complement of E is the full subcategory of D consisting of all objects L such that Hom(E[i], L) ∼ = 0 for all objects E of E and all i ∈ Z. If the right orthogonal complement of E vanishes, we say that E weakly generates the category. Now let D be a triangulated category which admits arbitrary direct sums. Recall that an object K of such a category is called compact if Hom(K, −) commutes with direct sums, and denote by D c ⊂ D the full subcategory of compact objects. The following result is due to [TT90,Nee92] (a proof can also be found in [SP17, Proposition 13.34.6]). given by the shift functors (more precisely, the rightwards shift maps s −s(g),−s(g+h) r , see [She17, Appendix A.2]). Observe that because Z I /M is finite, ind lands in s * MF , which we recall is the category of finite-rank matrix factorizations. We leave the verification of the adjunctions ind res ind to the reader (it is a version of the standard fact that restriction and induction form a Frobenius pair of functors). Proof It suffices to show that ind(O 0 ) weakly generates s * MF ∞ , by Proposition 4.9. Suppose that Q is in the right orthogonal complement to ind(O 0 ); it follows by adjointness that res(Q) is in the right orthogonal complement to O 0 , and therefore res(Q) ∼ = 0 by Proposition 4.10 (since W b has an isolated singularity at the origin by Proposition 4.4). If we choose s(0) = 0, then Q is the direct summand of (4-44) Proof of Proposition 4.7 We observe that ind(O 0 ) is a direct sum of objects of C, by definition. It follows by Corollary 4.11 that C split-generates s * MF , which coincides with GrMF Γ (S Λ , W b(λ) ) by (4-26).
Proof of Proposition 4.8 This can be proved using the 'automatic split-generation criteria' of [PS15] or [Gan16]; we adopt the latter. By Proposition 4.7, D π (C) is quasi-equivalent to GrMF Γ (S Λ , W b(λ) ). By [Orl09], this is an admissible subcategory of the stacky bounded derived category D b Coh(Ž b ). The latter category is smooth and proper by [BLS16, Theorem 6.6], because the stackŽ b is smooth and proper by Proposition 4.4. It follows that D π (C) is smooth and proper, by [LS14,Theorem 3.24] (see also [Orl16,Theorem 3.25]). Therefore the Mukai pairing on HH • (C) is non-degenerate by [Shk12, Theorem 1.4].
Since the open-closed map OC : HH • (C) → QH •+n (X; Λ) respects pairings, and the pairing on HH • (C) is non-degenerate, it follows that OC is injective. In particular the map OC : HH −n (C) → QH 0 (X; Λ) is non-zero, since it is injective and the domain is non-zero, so it hits the unit. It follows that C split-generates by Abouzaid's criterion [Abo10], all of whose ingredients are contained in § 2.5.