Mirror symmetry for perverse schobers from birational geometry
W. Donovan T. Kuwagaki


Mirror symmetry for perverse schobers]Mirror symmetry for perverse schobers from birational geometry 

W. Donovan, Yau Mathematical Sciences Center, Tsinghua University, Haidian District, Beijing 100084, China.
donovan@mail.tsinghua.edu.cn

T. Kuwagaki, Kavli IPMU (WPI), UTIAS, University of Tokyo, Kashiwa, Chiba 277-8583, Japan.
tatsuki.kuwagaki@ipmu.jp


Perverse schobers are categorical analogs of perverse sheaves. Examples arise from varieties admitting flops, determined by diagrams of derived categories of coherent sheaves associated to the flop: in this paper we construct mirror partners to such schobers, determined by diagrams of Fukaya categories with stops, for examples in dimensions 2 and 3. Interpreting these schobers as supported on loci in mirror moduli spaces, we prove homological mirror symmetry equivalences between them. Our construction uses the coherent-constructible correspondence and a recent result of  <cit.> to relate the schobers to certain categories of constructible sheaves. As an application, we obtain new mirror symmetry proofs for singular varieties associated to our examples, by evaluating the categorified cohomology operators of  <cit.> on our mirror schobers.









[
    T. Kuwagaki
    December 30, 2023
=====================


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§ INTRODUCTION

Mirror symmetry is a collection of mysterious conjectural relationships between complex and symplectic geometry, inspired by the physics of superstring . For certain pairs of a complex geometry X and a symplectic geometry Y, key predictions are:


  * that a stringy Kähler moduli space _Käh for X is isomorphic to a complex structure moduli space _CS for Y, and 

  * an equivalence between a derived category of coherent sheaves on X and a Fukaya category for Y, called homological mirror symmetry.

Unifying these, mirror symmetry predicts:



  * an equivalence between a locally constant family of derived categories on _Käh, and a corresponding family of Fukaya categories on _CS.


We interpret a `locally constant family of categories' here as an appropriate categorical analog of a locally constant sheaf. Such sheaves arise as solution sheaves for ordinary differential equations: when these equations have singularities, it is natural to study them using certain generalized objects called perverse sheaves. 

 have now suggested categorical analogs of perverse sheaves <cit.>, named perverse schobers, or simply schobers. As perverse sheaves may be thought of as singular versions of locally constant sheaves, schobers give a notion of a locally constant families of categories with singular behaviour.

In the mirror symmetry situation, the locally constant family of categories on _Käh is expected to have singular behaviour at certain boundary points. We focus here on boundary points known as conifold points. Recent research suggests that one can extend the family as a schober over such points by using categories from birational geometry. By mirror symmetry, this schober should then have a mirror partner. Namely, we have: 
 For the locally constant family of Fukaya categories on _ CS,



  *  does this family extend to a perverse schober?


  *  does this schober satisfy a mirror symmetry equivalence?



In this paper, we give affirmative answers to these questions for some examples, showing that appropriate extensions may be elegantly formulated using Fukaya categories with stops. We apply this to give a new proof of homological mirror symmetry for singularities associated to these examples, by evaluating a categorified cohomology operator on the resulting schober of Fukaya categories.





 §.§ Background
 We give background on perverse schobers and mirror symmetry, before explaining our results in Section <ref>. 




  §.§.§ Stringy Kähler moduli heuristic
 In this paper, we take X to be a resolution of a surface or 3-fold quadric cone. As a heuristic for these examples, we take _Käh to be a punctured disk Δ - p, and p to be a conifold point. This should be thought of as a local slice of the full stringy Kähler moduli: for further physical discussion in the 3-fold case, see Aspinwall <cit.>. We therefore construct schobers on Δ, singular at p, to answer Question (<ref>): this will mean constructing a spherical pair, as follows.





  §.§.§ Spherical pairs
  give different categorifications of a perverse sheaf P on Δ, singular at p, for different . Take K a path which passes through 0 as in Figure <ref>. Then the sheaf of local cohomology of P with support in K is concentrated in some fixed degree (this important property is known as purity), and is constructible. Furthermore, because P is singular only at p, this sheaf has only 3 distinct stalks (P_± and P_0). These stalks, along with natural maps between them, turn out to determine P.

This motivates the definition of a spherical pair, illustrated in Figure <ref>, as the data of triangulated categories _± and _0, along with embeddings satisfying certain conditions, and further conditions on orthogonals to these embeddings: for details, see Section <ref>.

Examples of spherical pairs arise from birational geometry. Letting X_± be the two sides of a 3-fold flop, or more generally certain flops of families of curves, Bodzenta and Bondal <cit.> construct a spherical pair given by the following data. 

    [scale=1.7]
     (Bminus) at (-1.3,1) D(X_-);
     (Bzero) at (0,1) _0;
     (Bplus) at (+1.3,1) D(X_+);
    [right hook->] (Bminus) to node[above]  (Bzero); 
    [left hook->] (Bplus) to node[above]  (Bzero);

Here _0 is an appropriate quotient of the derived category D(X_B) of the fibre product of X_± over their common contraction. In this paper we give an alternative construction using a subcategory of D(X_B), in Theorem <ref> below.



  §.§.§ Flobers
 Bondal, Kapranov, and Schechtman give related constructions for certain webs of flops, under the name of flobers, which may be viewed as categorified perverse sheaves on ℂ^n, singular along a real hyperplane arrangement <cit.>. The prototype of a flober (on ℂ, singular at 0) is determined by data as follows.

    ∗[scale=1.8,baseline=([yshift=-.5ex]current bounding box.center)]
     (Bminus) at (-1.3,1) D(X_-);
     (Bzero) at (0,1) D(X_B);
     (Bplus) at (+1.3,1) D(X_+);
    [right hook->] (Bminus) to node[above]  (Bzero); 
    [left hook->] (Bplus) to node[above]  (Bzero);

A flober is defined by strictly weaker conditions than a spherical pair: in this case, these conditions amount to the usual Fourier–Mukai functors between D(X_-) and D(X_+) being equivalences.









  §.§.§ Mirror symmetry equivalences
 Our version of homological mirror symmetry combines work of the second author <cit.> and  <cit.>. For a large class of toric stacks X, this gives an equivalence between the bounded derived category of coherent sheaves D(X) and a certain Fukaya category. Namely, we take a torus T obtained from the (dual of the) toric data for X, and consider the wrapped Fukaya category _Λ^∞(T) with stop Λ^∞, a locus in contact infinity of the cotangent bundle T which Lagrangians in the category must avoid, also determined by the toric data of X. Further details are given in Section <ref>.




 §.§ Results
 Take X_0 to be one of the following singularities, along with two (stacky) resolutions X_± as described. 

 Take X_0 the quotient of ℂ^2 by ℤ/2ℤ acting by ± 1. Then let X_- be the associated Deligne–Mumford quotient stack, and X_+ be the minimal resolution of the singularity X_0.


 Take X_0 the conifold singularity { xy-zw = 0 }, and X_± two small resolutions related by an Atiyah flop.


We denote the fibre product X_B of X_± over the singularity X_0, and associated morphisms, as follows: see Section <ref> for an explicit construction of X_B.

    [xscale=1,scale=1.25]
    
     (Bminus) at (0,1) X_B;
     (Bplus) at (1,0) X_+;
    
     (Cminus) at (-1,0) X_-;
     (Cplus) at (0,-1) X_0;
    
    [->] (Bminus) to node[above right] _+  (Bplus);
    [->] (Cminus) to (Cplus);
    
    [<-] (Cminus) to node[above left] _-  (Bminus);
    [<-] (Cplus) to (Bplus);




  §.§.§ Spherical pairs
 Following our heuristic above, we take _CS and _Käh to be punctured disks Δ - p. The following is our main theorem. It constructs perverse schobers on partial compactifications _CS and _Käh, both identified with the disk Δ. We give these schobers in the form of spherical pairs, and show they are equivalent, answering Questions (<ref>) and (<ref>) for our examples.


[Theorems <ref>, <ref>]
For Examples <ref> and <ref>, we have



  * schobers given by data below, and


  * an equivalence of these schobers via homological mirror symmetry, explained in Section <ref>.




Symplectic side:
Let T be a torus and Λ_±^∞ be loci in contact infinity of T, all determined by the toric data of X_± in Section <ref>. Then take data as follows, with embeddings given in the course of the proof.

    A[scale=2.3,baseline=([yshift=-.5ex]current bounding box.center)]
     (Bminus) at (-1.5,1) _Λ_-^∞(T);
     (Bzero) at (0,1) _Λ_-^∞∪Λ_+^∞(T);
     (Bplus) at (+1.5,1) _Λ_+^∞(T);
    [right hook->] (Bminus) to node[above]  (Bzero);
    [left hook->] (Bplus) to node[above]  (Bzero);

Complex side:
Let   be the subcategory of D(X_B) generated by the images of the embeddings of D(X_±) in (<ref>). Then take data as follows.

    B[scale=1.7,baseline=([yshift=-.5ex]current bounding box.center)]
     (Bminus) at (-1.3,1) D(X_-);
     (Bzero) at (0,1) ;
     (Bplus) at (+1.3,1) D(X_+);
    [right hook->] (Bminus) to node[above]  (Bzero); 
    [left hook->] (Bplus) to node[above]  (Bzero);



The mirror operation to the 3-fold flop in Example <ref> is given by Fan, Hong, Lau, and Yau <cit.>. It would be interesting to relate this to the locally constant family of Fukaya categories on the punctured disk Δ - p which arises from Theorem <ref>.


It would be interesting to try to use the schobers appearing in Theorem <ref> to categorify perverse sheaves extending the A-model and B-model local systems of cohomology on _CS and _Käh, in our examples and more generally. In particular, this could lead to an a priori reason why the derived equivalences associated to a flop may be organized into a categorified perverse sheaf, as asked by  <cit.>.





  §.§.§ Symplectic flobers
 Some of the techniques used to prove Theorem <ref> then yield the following, giving a counterpart to the flober (<ref>) on the symplectic side.

[Theorem <ref>] For Examples <ref> and <ref>, the flober (<ref>) is equivalent to a flober as follows, where Λ_B^∞ is a locus in contact infinity of T, determined by the toric data of X_B.

    [scale=2.2]
     (Bminus) at (-1.5,1) _Λ_-^∞(T);
     (Bzero) at (0,1) _Λ_B^∞(T);
     (Bplus) at (+1.5,1) _Λ_+^∞(T);
    [right hook->] (Bminus) to node[above]  (Bzero); 
    [left hook->] (Bplus) to node[above]  (Bzero);





  §.§.§ Application
 Perverse sheaves admit certain cohomology operators which yield vectors spaces: these are used, for instance, to define intersection cohomology. By analogy,  <cit.> define certain categorified cohomology operators on perverse schobers which yield categories. In particular, they define the 2nd cohomology with compact support ℍ^2_c to be a homotopy push-out of the diagram defining the flober.  They prove that ℍ^2_c for (<ref>) is D(X_0), with X_0 the singular base given above. We prove a symplectic analog of this result, as follow.

[Proposition <ref>]
The 2nd cohomology with compact support ℍ^2_c for the symplectic flober in Theorem <ref> is given by the category

    _Λ_-^∞∩Λ_+^∞(Ω_T).



We immediately obtain the following statement of homological mirror symmetry for the singular space X_0.

[Corollary <ref>] The equivalence of Theorem <ref> induces an equivalence

    D(X_0) ≅_Λ_-^∞∩Λ_+^∞(Ω_T).



An equivalence of the categories in Corollary <ref> follows from work of the second author in <cit.>, as the methods there apply in singular cases. The corollary provides a new alternatively proof of this, which uses the results of <cit.> only for smooth cases, along with the categorified cohomology operator ℍ^2_c.

 also define 1st cohomology ℍ^1 of a flober <cit.>: for the flober (<ref>) this is by definition a quotient of D(X_B) by the triangulated subcategory generated by the embeddings of D(X_±). In this language, the spherical pairs of Theorem <ref> are special in the sense that their ℍ^1 is zero, whereas the flober (<ref>) and the flober in Theorem <ref> have non-zero ℍ^1.




 §.§ Method of proof
 Let X be one of the toric varieties or stacks appearing above, and let T be the corresponding dual torus: namely, if X is described by a toric fan Σ in N_ where N=(M,) for a lattice M, then we take T=M_/M. Write _Λ(T) for a category of wrapped constructible sheaves on T, with microsupport Λ in T, where Λ is the FLTZ skeleton for the toric data. Furthermore let Λ^∞ be the associated locus at infinity in T, where full details are given in Section <ref>. Then the homological mirror symmetry equivalences used above are obtained by composition as follows.

    _Λ^∞(T)  (1)≅_Λ(T)^ (2)≅ D(X)^ (3)≅ D(X)



  *  This is proved in recent work of  <cit.>, refining a result of Nadler–Zaslow <cit.>.

  *  This is a version of the coherent-constructible correspondence, as proved by the second author in <cit.>.

  *   Here we take the derived dual (,_X) to give a covariant composition.


We prove Theorem <ref> by showing that the diagrams (<ref>) and (<ref>) appearing there are equivalent to the following diagram (where, for convenience, we drop `' from the notation).

    C[scale=2.2,baseline=([yshift=-.5ex]current bounding box.center)]
     (Bminus) at (-1.5,1) _Λ_-(T);
     (Bzero) at (0,1) _Λ_- ∪Λ_+(T);
     (Bplus) at (+1.5,1) _Λ_+(T);
    [right hook->] (Bminus) to node[above]  (Bzero);
    [left hook->] (Bplus) to node[above]  (Bzero);

For the symplectic side diagram (<ref>), this equivalence is formal, following from (<ref>) and by construction. For the complex side (<ref>), we proceed as follows. We first apply the equivalence (<ref>) to the spaces which appear in the flober (<ref>). From these we deduce that there is an embedding 
    κ↪_Λ_- ∪Λ_+(T).

We prove that κ is an equivalence by constructing and comparing certain semiorthogonal decompositions of these categories, as follows. To illustrate our proof, and for interest, these decompositions are presented in Section <ref> for the surface case.


  * A decomposition of _Λ_- ∪Λ_+(T) with a component _Λ_±(T) arises from an explicit presentation of the skeleton Λ_- ∪Λ_+.

  * A decomposition of  with a component D(X_±) is standard in the  case. For the surface case, we construct one by studying derived categories of appropriate GIT quotients.

We thus establish that κ is an equivalence. It follows from the decompositions of  that the diagram (<ref>) gives a spherical pair, and we thence deduce that (<ref>) gives a spherical pair. That the diagram (<ref>) gives a spherical pair then follows by composing with the equivalence (<ref>).

Theorem <ref> is proved by a similar, but simpler, argument using a diagram of wrapped constructible categories as follows.

    [scale=2.2,baseline=([yshift=-.5ex]current bounding box.center)]
     (Bminus) at (-1.5,1) _Λ_-(T);
     (Bzero) at (0,1) _Λ_B(T);
     (Bplus) at (+1.5,1) _Λ_+(T);
    [right hook->] (Bminus) to node[above]  (Bzero); 
    [left hook->] (Bplus) to node[above]  (Bzero);


To prove Proposition <ref> we adapt a method of  to show that ℍ^2_c applied to the the diagram above gives _Λ_0(T), using explicit presentation of skeleta. The result follows again by composing with the equivalence (<ref>).

 In the course of the proof of Proposition <ref> we obtain a new proof of an instance of the coherent constructible correspondence, namely

    D(X_0) ≅_Λ_0(T).

At the end of paper, we present a conjectural picture about how this method could be extended to prove more general such results.




 §.§ Related work
 Nadler has also discussed mirror equivalences of schobers on the disk <cit.>. In this case a different skeleton K is used so that the schober takes the form of a spherical functor. He proves a homological mirror symmetry statement relating a certain Landau–Ginzburg A-model to the B-model for the higher-dimensional pair of pants <cit.> by deducing it from a mirror equivalence of such schobers <cit.>. The A-model schober in this case is over a disk in the space of values of the (complex) superpotential. 


Harder and Katzarkov have used schobers to given a new proof of a mirror equivalence for the projective space ^3 <cit.>.

The first author previously constructed schobers on the complex side of mirror symmetry, associated to wall crossings in GIT <cit.>. In subsequent work he applied these to study schobers associated to standard flops, which led to a discussion of the Questions (<ref>) and (<ref>) <cit.>. In forthcoming work <cit.>, the first author
 and M. Wemyss give a mathematical treatment of the stringy Kähler moduli for general 3-fold flops of irreducible curves.


In <ref> we explain some details of our surface example informally. In <ref> we give preliminaries. In <ref> we construct the data of spherical pairs and flobers, and in <ref> we prove that they satisfy the required properties, and give homological mirror symmetry equivalences between them, proving Theorems <ref> and <ref>. Finally, in <ref> we prove Proposition <ref>, deduce Corollary <ref>, and finish with some remarks and conjectures on further applications of flobers to the coherent-constructible correspondence.


 Both authors are grateful for conversations with A. Bondal and M. Kapranov at Kavli IPMU. The first author is grateful for discussions with T. Logvinenko, and thanks D. Auroux, T. Coates, D. Nadler for helpful conversations at an early stage of the project. The authors also thank the organizers of the conference `Categorical and Analytic Invariants in Algebraic Geometry V' at Osaka in 2018, where this collaboration started.

The two authors acknowledge the support of WPI Initiative, MEXT, Japan, and of JSPS KAKENHI Grants JP16K17561 and JP18K13405 respectively. The first author is supported by the Yau MSC, Tsinghua University, and the Thousand Talents Plan.


 We denote categories as follows: for details, see Section <ref>.


     constructible sheaves 

     quasi-constructible sheaves 

     wrapped constructible sheaves <cit.> 

D     bounded derived category of coherent sheaves





§ SURFACE EXAMPLE



In this section we present semiorthogonal decompositions for the surface case, to illustrate our proof, and for interest. 



 §.§ Setting
 We take the singularity X_0 = ^2/_2, and take (stacky) resolutions X_± given by the corresponding Deligne–Mumford stack X_- and the minimal resolution X_+, denoted as below. The fibre product X_B is a further  Deligne–Mumford stack, given explicitly in Section <ref>.

    [xscale=1,scale=1.25]
    
     (Bminus) at (0,1) X_B;
     (Bplus) at (1,0) X_+;
     (Dplus) at (2.5,0) ^2/_2;
     (DplusLabel) at (2.5,0.05) ^2/_2;
    
     (Cminus) at (-1,0) X_-;
     (Dminus) at (-2.5,0) [^2/_2];
    
     (Cplus) at (0,-1) X_0;
    
    [->] (Bminus) to  (Bplus);  
    [->] (Cminus) to node[below left] f_- (Cplus);
    
    [<-] (Cminus) to (Bminus); 
    [<-] (Cplus) to node[below right] f_+ (Bplus);
    
    [-,transform canvas=yshift=+] (Dminus) to (Cminus);
    [-,transform canvas=yshift=-] (Dminus) to (Cminus);
    
    [-,transform canvas=yshift=+] (Dplus) to (Bplus);
    [-,transform canvas=yshift=-] (Dplus) to (Bplus);



The fibres of f_- and f_+ over the singularity 0 are the stacky point [0/_2], and a projective line ^1. Let _- and _+ be the structure sheaves of these fibres, tensored by the non-trivial irreducible representation of _2, and the twisting sheaf (-1), respectively.

-0.48


Now take skeleta Λ_± determined by the toric data of X_±, as illustrated in Figure <ref>, which we explain now. The construction of these skeleta is discussed in Section <ref>. The squares in the figure show a fundamental domain for the quotient T=^2/^2. The `hairs' denote cotangent directions in Ω_T, and thereby indicate conical loci in Ω_T corresponding to the skeleton Λ. We let _- and _+ be microlocal skyscraper sheaves <cit.> in _Λ_- ∪Λ_+(T) corresponding to the cotangent directions marked.

[Propositions <ref>, <ref>]
The category _0 appearing in Theorem <ref> has semiorthogonal decompositions as follows.

    D(X_-), _-  = _0 =  D(X_+), _+

The category _Λ_- ∪Λ_+(T) has decompositions as follows.

    _Λ_-(T), _-  = _Λ_- ∪Λ_+(T) = _Λ_+(T), _+



In the course of the proof we see that these categories, and the components in the above decompositions, all correspond under the coherent-constructible correspondence.

The embeddings of the D(X_±) are the pullbacks p_± *, and it follows that the equivalences between D(X_-) and D(X_+) required in the definition of a spherical pair follow from simple cases of results of Bridgeland–King–Reid <cit.>.

 We get decompositions of Fukaya categories

    ⟨_-, _Λ_-^∞(T) ⟩ = _Λ_-^∞∪Λ_+^∞(T) = ⟨_+, _Λ_+^∞(T) ⟩

for certain objects _± of _Λ_-^∞∪Λ_+^∞(T), by applying the equivalence of . (That the categories  appear in the decompositions here on the right, whereas the categories  appear on the left in the proposition above, is due to the fact that this equivalence is contravariant.)




§ PRELIMINARIES








 §.§ Perverse schobers



We work with spherical pairs, as follows.
Recall that a semi-orthogonal decomposition
= ⟨, ⟩,
is determined by embeddings
 
    γ⟶    and    δ⟶,
 
and induces projection functors given by adjoints as follows.
 
    γ^⟶    and    δ^⟶.

We may then make the following definition.

<cit.>
A spherical pair  is  a triangulated category _0 with admissible subcategories _± and semi-orthogonal decompositions

    ⟨_-, _- ⟩ = _0 = ⟨_+, _+ ⟩,

such that compositions of the embeddings and projections above
 
    _- ⟷_+     and    _- ⟷_+,

are equivalences.


As indicated in Section <ref>, this data should be thought of as a categorification of vector space data determining a perverse sheaf. Further discussion is given by  in <cit.> and <cit.>.


 An isomorphism between spherical pairs  and ' consists of equivalences _∙≃'_∙ and _∙≃'_∙ intertwining the embeddings.


A weaker notion is discussed by  <cit.> omitting the condition on orthogonals _∙.

<cit.>
A flober or weak spherical pair  is a triangulated category  _0 with admissible subcategories _± with embeddings δ_± such that the 
compositions of δ_± and δ_±^ as follows 
 
    _- ⟷_+
 are equivalences.


A spherical pair yields a flober in the obvious way.

 Flobers arise from 3-fold flops and more general flops of curves, see <cit.> for discussion.

 make following definition.

<cit.>
For a weak spherical pair , the homology with compact support ℍ^2_c(Δ,) is defined as the homotopy push-out in the Morita model of the following.

    [xscale=1,scale=1.25]
    
     (Bminus) at (0,1) _0;
     (Bplus) at (1,0) _+;
    
     (Cminus) at (-1,0) _-;
    
    [->] (Bminus) to node[above right] δ_+^  (Bplus);
    [<-] (Cminus) to node[above left] δ_-^  (Bminus);



We will often say, for brevity, that a spherical pair  is determined by data written as follows, sometimes omitting the adjoint functors.

    [scale=2]
     (Bminus) at (-1,1) _-;
     (Bzero) at (0,1) _0;
     (Bplus) at (+1,1) _+;
    [right hook->,transform canvas=yshift=+] (Bminus) to node[above] δ_- (Bzero);
    [left hook->,transform canvas=yshift=+] (Bplus) to node[above] δ_+ (Bzero);
    [<-,transform canvas=yshift=-] (Bminus) to node[below] δ_-^ (Bzero);
    [<-,transform canvas=yshift=-] (Bplus) to node[below] δ_+^ (Bzero);

As we will study anti-equivalences, it will also be convenient to say that a spherical pair is determined by the opposite of categorical data written as follows, where we reuse the notation δ_± for the induced functors below.

    [scale=2]
     (Bminus) at (-1,1) ^_-;
     (Bzero) at (0,1) ^_0;
     (Bplus) at (+1,1) ^_+;
    [right hook->,transform canvas=yshift=+] (Bminus) to node[above] δ_- (Bzero);
    [left hook->,transform canvas=yshift=+] (Bplus) to node[above] δ_+ (Bzero);
    [<-,transform canvas=yshift=-] (Bminus) to node[below] δ_-^ (Bzero);
    [<-,transform canvas=yshift=-] (Bplus) to node[below] δ_+^ (Bzero);

In particular, given such data we have semi-orthogonal decompositions

    ⟨^_-, ^_- ⟩ = ^_0 = ⟨^_+, ^_+ ⟩,

and ℍ^2_c(Δ,) is given by a push-out of the following.

    [xscale=1,scale=1.25]
    
     (Bminus) at (0,1) ^_0;
     (Bplus) at (1,0) ^_+;
    
     (Cminus) at (-1,0) ^_-;
    
    [->] (Bminus) to node[above right] δ_+^  (Bplus);
    [<-] (Cminus) to node[above left] δ_-^  (Bminus);






 §.§ Fukaya category and microlocal sheaf theory


In this subsection, we give a brief introduction to a relationship between the Fukaya category and microlocal sheaf theory.
Let Z be a real analytic manifold and k_Z be the constant sheaf valued in a field k. We denote the bounded derived category of k_Z-modules by (k_Z). We define microsupport (also known as `singular support'), one of the most important notion in microlocal sheaf theory. 


The microsupport () of ∈(k_Z) is a subset of the cotangent bundle Z defined by its complement as follows: (x, ξ)∈Z (where x∈ Z and ξ∈Zx) is not contained in () if there exists an open neighbourhood U of (x, ξ) such that

    (_ zψ(z)≥ψ(y) )_y ≃ 0

for any point y∈ Z and any smooth function ψ with (dψ)⊂ U.


We view Z as a symplectic manifold with its standard exact symplectic structure. The microsupport () is a Lagrangian subset of Z if  is constructible:

A sheaf  is constructible (respectively quasi-constructible) if there exists a real analytic Whitney stratification  of Z such that the restriction of  to each stratum S∈ is a locally constant sheaf of finite rank (respectively locally constant sheaf possibly of infinite rank). For the definition of such a stratification, we refer to <cit.>.


An object ∈(k_Z) is quasi-constructible if and only if () is Lagrangian.

We write (Z) (respectively (Z)) for the (dg-)category of bounded complexes of constructible sheaves (respectively  unbounded complexes of quasi-constructible sheaves) over Z. For a subset Λ⊂Z, the full subcategory of (Z) (respectively (Z)) spanned by objects with microsupport living inside Λ is denoted by _Λ(Z) (respectively _Λ(Z)).



Before stating a relationship with Fukaya category, we introduce one more category.

Let Λ⊂Z be a subset. The full subcategory _Λ(Z) of _Λ(Z) is defined by the following: ∈_Λ(Z) if and only if 

    __Λ(Z)(, -)_Λ(Z)→Vect
 commutes with any direct sums. We call an object of _Λ(Z) a wrapped constructible sheaf.


A relationship between microlocal sheaf theory and the Fukaya category was first clearly stated by Nadler–Zaslow by using an `infinitesimally wrapped Fukaya category'. Our setting is a more recent variant of Nadler–Zaslow established by  <cit.>: Let Z be a real analytic manifold and Ω_Z be the cotangent bundle of Z. Let {x_i} be local coordinates of Z and {ξ_i} be the corresponding cotangent coordinates. Then Ω_Z has an exact symplectic structure locally written as ∑_idξ_i∧ dx_i. Let us fix g a Riemannian metric over Z. Then we define the cosphere bundle by 

    S^*Z_a:= (x, ξ)∈Ω_Z g_x(ξ, ξ)=a.

The symplectic structure induces a contact structure on S^*Z_a. Since the S^*Z_a for various a are contactoisomorphic to each other, we consider a→∞ virtually and call this abstract contact manifold contact infinity and denote it by Z. Let Λ^∞ be a subanalytic Legendrian of Z. Then one can define an A_∞-category _Λ^∞(Z), the wrapped Fukaya category of Z with the stop Λ^∞. Roughly, this category consists of 


  * Objects: (possibly noncompact) Lagrangian submanifold whose noncompact ends live away from Λ^∞.

  * Morphisms: wrapped Floer cohomology.


Given a Legendrian Λ^∞ as above, we obtain a locus in Z as follows.
 Let Λ⊂Z be given by Λ=(_>0·Λ^∞) ∪ Z where _>0 acts by scaling the cotangent fibers, and Z is the zero section in Z.

We can also go the other way around: namely, for a given conic (i.e. invariant under _>0) Lagrangian Λ in Z Z we have the following.
 We obtain a Legendrian Λ^∞⊂Z as follows. First, for any a we obtain a Legendrian in S^*Z_a by Λ_a=S^*Z_a∩Λ. The conicness implies that Λ_a is preserved under the isomorphism S^*Z_a≅ S^*Z_b for any a, b. Hence we get a Legendrian Λ^∞ in the abstract contact manifold Z.


We may now state the following theorem.


There exists an equivalence

    _Λ(Z)^≃_Λ^∞(Z)

of A_∞-categories, where  denotes the opposite category.


 We may remove the  in the above, but at the expense of negating the Liouville form, or performing a similar operation on Λ^∞ <cit.>.

It is expected that this theorem generalizes to the case of Weinstein manifolds instead of Z. This is known as Kontsevich's conjecture <cit.>. Progress towards a proof has been made by many people.

A point for us is that a Landau–Ginzburg model gives an isotropic subset Λ^∞, hence a partially wrapped Fukaya category. The A-brane category associated to a Landau–Ginzburg model is sometimes described as a Fukaya–Seidel category generated by Lefschetz thimbles (or vanishing cycles). The Fukaya–Seidel category and the partially wrapped Fukaya category associated to a single Landau–Ginzburg model are sometimes different.

We take W(z)=z on ^*. Then the Fukaya–Seidel category is empty but the partially wrapped Fukaya category is equivalent to the derived category of coherent sheaves over ^1.

Hence, to study mirror symmetry, the partially wrapped Fukaya category is appropriate.

The Lagrangian skeletons for mirror Landau–Ginzburg models of toric varieties can be combinatorially defined. Such skeletons were first proposed by Fang–Liu–Treumann–Zaslow. 

 We use standard notation in the toric setting, as follows.


M     a rank n free abelian group 

N     the dual of M 

Σ     a rational finite fan in N_:=N⊗_ 

X_Σ     the toric variety of Σ 

T     real torus M_/M 

     projection M_→ T



For a subset σ⊂ N_, we set

    σ^⊥:= m∈ M_ n(m)=0  for any n∈σ.



We then have the following.



    Λ_Σ:=⋃_σ∈Σ(σ^⊥)× (-σ)⊂ T× N_≅T.





The following result is due to Gammage–Shende <cit.>, Zhou <cit.>, and Ganatra–Pardon–Shende <cit.>.

The FLTZ skeleton at infinity Λ_Σ^∞ is a Weinstein skeleton of a generic fiber of the Hori–Vafa mirror potential. If moreover X_Σ is Fano, then _Λ_Σ^∞(T), and hence _Λ_Σ(T), is equivalent to the Fukaya–Seidel category of W.

We will not define `Weinstein skeleton' here. The main point for us is that _Λ_Σ^∞(T) in general can be considered as a generalization of a Fukaya–Seidel category.


Since mirror symmetry predicts an equivalence between the B-model on X_Σ and the mirror Landau–Ginzburg A-model, we can interpret homological mirror symmetry as an equivalence between the derived category of coherent sheaves over X_Σ and _(T). This is the content of the next subsection.




 §.§ Coherent-constructible correspondence

In this subsection, we review the result in <cit.> and <cit.> for the smooth variety case. We now take the field k to be . Let Λ_Σ and X_Σ be as in the previous section. For σ∈Σ, we have the corresponding affine coordinate i_σ U_σ⊂ X_Σ. Letting  denote the unbounded derived category of quasicoherent sheaves, there exists a unique functor

    κ_Σ(X_Σ)→_(X_Σ)

which maps Θ'(σ):=i_σ*_U_σ to Θ(σ):=_!_Int(σ^∨) where Int is the interior and σ^∨ is the polar dual of σ.


The restriction of κ_Σ to D(X_Σ) gives an equivalence

    D(X_Σ)_(T).


As discussed in the last part of the previous section, this equivalence is an instance of homological mirror symmetry.

According to <cit.>, we redefine the equivalence functor as

    K_Σ:=κ_Σ(-⊗ω_Σ^-1)

where ω_Σ is the canonical sheaf. Since ω_Σ is invertible, we have:

The functor K_Σ gives an equivalence

    D(X_Σ)_(T).


The reason why we use this modified functor is to get a commutativity with push-forwards on the B-side, as appeared in the proof of Theorem <ref>. 

As a corollary of the above theorem and Theorem <ref>, we get a homological mirror symmetry between coherent sheaves and a Fukaya category:


    D(X_Σ)_(T)^op



Next we would like to discuss the functoriality. Let Σ' be a refinement of Σ. We denote the corresponding map by X_Σ'→ X_Σ. Then there exists an inclusion relation Λ_Σ⊂Λ_Σ', which induces an inclusion

    ^*_(T)↪_Λ_Σ'(T).





    κ_Σ'∘^*≃^*∘κ_Σ





For a general inclusion of Lagrangians Λ⊂Λ', the inclusion 
    _Λ(T)↪_Λ'(T)
 induces _Λ(T)↪_Λ'(T) but does not induce _Λ(T)↪_Λ'(T). However, in our case, we have:

 If X_Σ is smooth, the functor (<ref>) above induces a functor

    ^*_(T)↪_Λ_Σ'(T).


By Proposition <ref> and Theorem <ref>, this is equivalent to saying that the pullback ^* (X_Σ ) →(X_Σ') restricts to a functor between bounded derived categories, which is true under the assumption.


We also note the following for later use.


For a general inclusion of Lagrangians Λ⊂Λ', a left adjoint ι^ of the natural embedding ι: _Λ(T)↪_Λ'(T) restricts to an essentially surjective functor 
    ι^w: _Λ'(T)→_Λ(T).
 

This is by general nonsense, see Nadler <cit.>.





 §.§ Toric stacks

Here we recall a definition of toric stacks, following Gerashenko–Satriano <cit.>. Let L and N be free abelian groups and 
    f L→ N
 be a morphism. We can associate a morphism between algebraic groups 
    f⊗_^*  L⊗_^*→ N⊗_^*.
 Let Σ be a fan in L and X_Σ be the associated toric variety. Then the toric stack associated to the data (Σ,f) is defined by the quotient stack

    [X_Σ/(f⊗_^*)]

where (f⊗_^*) acts on X_Σ via the action of L⊗_^* on X_Σ. 



§ SCHOBER CONSTRUCTIONS



We give the required description of the geometry of our examples, before constructing schobers on the A-side and B-side of mirror symmetry.



 §.§ Surface example


Consider the A_1 quotient singularity X_0 = ℂ^2/ ℤ_2 where ℤ_2 = {± 1 }. This has resolutions a Deligne–Mumford quotient stack, and a minimal resolution, which we denote as follows.

    X_- = [ ℂ^2/ ℤ_2 ]      X_+ = ℂ^2/ ℤ_2

The minimal resolution may be realised as
the total space of a line bundle.

    X_+ =Tot_ℙ^1(-2)

The fibre product of X_± over X_0 is a further Deligne–Mumford stack, as follows.

    X_B = [Tot_ℙ^1(-1)/ ℤ_2].

Here ℤ_2 acts as ± 1 on the fibres of the bundle. The associated morphisms 
    _± X_B → X_±
 are described as follows.



  * The map _- is the blowup of [ 0 / ℤ_2 ] ⊂ X_-, in other words the blowup of 0∈ℂ^2, noting that this is equivariant with respect to the .

  * The map _+ is given by the root stack construction along ℙ^1 (for a general discussion, see <cit.>), namely a family version of the following morphism.

    [ℂ/ℤ_2 ] →ℂ z ↦ z^2





  §.§.§ Toric description
 Let N be a rank 2 lattice with basis e_1, e_2, and let M be its dual. The faces of the cone (e_1, e_1+2e_2)⊂ N_ give a fan Σ_0 representing the singularity X_0. Then the small resolution X_+ is represented by a refinement Σ_+ of Σ_0 given by the faces of the following cones.

    (e_1, e_1+e_2), (e_1+e_2, e_1+2e_2) ⊂ N_


black
0.5pt

    [xscale=1,scale=1]
    [fill,color=] (0,0) – (1,0) – (1,2) – cycle;
    [fill,color=] (0,0) circle ();
     in 0,...,2[fill,color=] (1,) circle (); [semithick,-stealth] (0,0) – (1,0);
    [semithick,-stealth] (0,0) – (1,2);
     at (0.2,1.3) Σ_0;
    
    4cm[fill,color=] (0,0) – (1,0) – (1,2) – cycle;
    [fill,color=] (0,0) circle ();
     in 0,...,2[fill,color=] (1,) circle (); [semithick,-stealth] (0,0) – (1,0);
    [semithick,-stealth] (0,0) – (1,1);
    [semithick,-stealth] (0,0) – (1,2);
     at (0.2,1.3) Σ_+;


To obtain toric data for the stack X_-, in the sense of Section <ref>, take a further rank 2 lattice L with basis g_1, g_2 and consider the standard fan representing ^2, generated by faces of (g_1, g_2)⊂ L_, and denote it Σ_-. The lattice map

    f    L→ N 
        g_1 ↦ e_1 
        g_2 ↦ e_1+2e_2

then induces a map between the corresponding algebraic tori

    T_L → T_N
     (a,b) ↦ (ab, b^2).

The kernel of this map is _2=± 1. It follows that the toric stack corresponding to the data of 
Σ_L and f is the toric stack [^2/_2]=X_-, the stacky resolution of the A_1-singularity, see Figure <ref>. The variety X_+ is realized as a toric stack by taking the data Σ_+ and _N.


For the toric data corresponding to the fibre product X_B, take the fan Σ_B given by faces of the following cones

    (e_1, e_1+e_2), (e_1+e_2, e_2) ⊂ L_.

Note that this fan refines Σ_-, and corresponds to Tot_ℙ^1(-1). Then we take data Σ_B and f to obtain X_B, see Figure <ref>. Note this is not a toric Deligne–Mumford stack in the sense of Borisov–Chen–Smith <cit.>.





Now the projections _± from X_B are induced by appropriate commutative squares of lattices, as shown.

    [xscale=1,scale=1.5,baseline=([yshift=-.5ex]current bounding box.center)]
    
     (A) at (0,0) X_B;
     (Aminus) at (0,-1) X_-;
     (Aplus) at (0,1) X_+;
    
    [->] (A) to node[left] _-  (Aminus);
    [->] (A) to node[left] _+  (Aplus);
    
    2cm (C) at (0,0) L;
     (Cminus) at (0,-1) L;
     (Cplus) at (0,1) N;
    
    [-,transform canvas=xshift=+] (C) to (Cminus);
    [-,transform canvas=xshift=-] (C) to (Cminus);
    [->] (C) to node[left] f  (Cplus);
    
    1cm (D) at (0,0) N;
     (Dminus) at (0,-1) N;
     (Dplus) at (0,1) N;
    
    [-,transform canvas=xshift=+] (D) to (Dminus);
    [-,transform canvas=xshift=-] (D) to (Dminus);
    [-,transform canvas=xshift=+] (D) to (Dplus);
    [-,transform canvas=xshift=-] (D) to (Dplus);
    
    [->] (Cminus) to node[above] f  (Dminus);
    [->] (C) to node[above] f  (D);
    [->] (Cplus) to node[above] _N  (Dplus);






 §.§ Threefold example
 


We recall the geometry of the Atiyah flop local model. Namely, take X_0 to be the conifold singularity (xy-zw=0) in ℂ^4. This has crepant resolutions X_± with exceptional curves E_±≅ℙ^1 such that

    X_±≅Tot_E_±(-1)^⊕ 2,

related by an Atiyah flop as follows.
 
    [xscale=0.9,scale=1.25,baseline=([yshift=-.5ex]current bounding box.center)]
    
     (Bplus) at (1,0) X_+;
     (Cminus) at (-1,0) X_-;
     (Cplus) at (0,-1) X_0;
    
    [<->,dashed] (Cminus) to (Bplus);
    [->] (Cminus) to (Cplus);
    [<-] (Cplus) to (Bplus);

Let X_B be the fibre product of this diagram. This is isomorphic to the blowup of X_± along E_±. Write E for the common exceptional divisor ℙ^1 ×ℙ^1.






  §.§.§ Toric data
 
Let N be a rank 3 lattice with basis e_i, and dual M. The set of faces of the cone (e_1, e_2, e_1+e_3, e_2+e_3)⊂ N_ gives a fan Σ_0 representing the conifold. The small resolutions X_± come from

    Σ_+   :=Σ_0∪{(e_2, e_1+e_3)}
    Σ_-   :=Σ_0∪{(e_1, e_2+e_3)}

and the blow-up X_B from

    Σ_B:=Σ_+∪Σ_-∪{_≥ 0· (e_1+e_2+e_3)}.

This data is sketched in Figure <ref>.
Note that the fan Σ_B refines Σ_± giving the morphisms 
    _± X_B → X_±.















 §.§ B-side constructions
 We construct schobers involving the categories D(X_±) by an appropriate analysis of semiorthogonal decompositions. For the 3-fold case which we study first, our schober should be equivalent to the one constructed, in a more general setting, by Bodzenta–Bondal in <cit.>. We here use a different method to theirs, which also applies to the stacky surface case.



  §.§.§ Threefold case
 First note we have _±^*  D(X_±) → D(X_B) with left adjoints _± !, where we take p_! = p_* (ω_p[dim p] ⊗). By Bondal–Orlov <cit.>, _± *_∓^* are equivalences, so by taking left adjoints _∓ !_±^* are equivalences.
Let  be full triangulated subcategory of D(X_B) generated by the images of _+^* and _-^*.

Then we have _±^*  D(X_±) → with left adjoints _± ! obtained by restriction, because  is a full subcategory of D(X_B), and we deduce that the compositions D(X_-) ↔ D(X_+) in the following diagram are equivalences _∓ !_±^*.

    [scale=2,baseline=([yshift=-.5ex]current bounding box.center)]
     (Bminus) at (-1,1) D(X_-);
     (Bzero) at (0,1) ;
     (Bplus) at (+1,1) D(X_+);
    [right hook->,transform canvas=yshift=+] (Bminus) to node[above] _-^* (Bzero);
    [left hook->,transform canvas=yshift=+] (Bplus) to node[above] _+^* (Bzero);
    [<-,transform canvas=yshift=-] (Bminus) to node[below] _-! (Bzero);
    [<-,transform canvas=yshift=-] (Bplus) to node[below] _+! (Bzero);


To show that this gives (the opposite of) a spherical pair, and for use in our equivalence proof in Section <ref>, we construct semi-orthogonal decompositions of . Set notation for the blowup of X_± in E_± as follows.

    [xscale=1.25,scale=0.75,baseline=([yshift=-.5ex]current bounding box.center)]
    
     (Bminus) at (-1,1) X_±;
     (Bplus) at (+1,1) X_B;
    
     (Cminus) at (-1,-1) E_±;
     (Cplus) at (+1,-1) E;
    
    [<-] (Bminus) to node[above] _±  (Bplus);
    [<-] (Cminus) to node[below] _±  (Cplus);
    
    [right hook->] (Cminus) to node[left] j_±  (Bminus);
    [right hook->] (Cplus) to node[right] i  (Bplus);


 The data of (<ref>) yields the opposite of a spherical pair. In particular, we have semi-orthogonal decompositions

    = _±^* D(X_±), i_* _±^* _E_± (-1) .



Recall that by a result of Orlov <cit.>, see for instance <cit.>, we have

    D(X_B) = _+^* D(X_+), i_* _+^* D(E_+) .

To obtain a semi-orthogonal decomposition of ⊂ D(X_B), we therefore calculate the image of _-^* D(X_-) ⊂ under the projection to D(E_+): the projection functor is _+ * i^!. Recalling that i^! = ω_i [dim i] ⊗ i^*, it thence suffices to calculate the image of the following functor on D(X_-).

    _+*( ω_i ⊗ i^* _-^* () )

Now ω_i = _E|X, and it is straightforward to show that

    _E|X = _+^* _E_+ (-1) ⊗_-^* _E_- (-1),

see <cit.> for instance. The functor (<ref>) is then given as follows.

    _+*( ω_i ⊗ i^* _-^* () )    ≅_+*( ω_i ⊗_-^* j_-^* () ) commutativity of (<ref>)
       ≅_+*( _+^* _E_+ (-1) ⊗_-^* _E_- (-1) ⊗_-^* j_-^* () )
       ≅_E_+ (-1) ⊗_+*( _-^* _E_- (-1) ⊗_-^* j_-^* () )
       ≅_E_+ (-1) ⊗_+*_-^* (  _E_- (-1) ⊗ j_-^* () )

Here the third line is obtained by the projection formula. Now, writing r_± E_±→, we have _+*_-^* ≅ r_+^* r_-* by flat base change. The functor (<ref>) is therefore isomorphic to the composition of 

    r_-*(  _E_- (-1) ⊗ j_-^* () )     then    _E_+ (-1) ⊗ r_+^* ().

The first functor is essentially surjective onto D(): to see this, apply it to s_-^* _E_- (+1) where s_- is the bundle projection of X_-, and note that j_-^* s_-^* ≅𝕀. The image of (<ref>) is thence the image of the second functor, namely the subcategory of D(E_+) generated by _E_+ (-1).
We deduce a semi-orthogonal decomposition as in the statement, with the other sign following by symmetry.


Finally we show the schober conditions. The required equivalences between D(X_-) and D(X_+) are explained above (<ref>). For the equivalences between the orthogonals,
these are generated by single objects, and therefore by symmetry it suffice to prove the following lemma.


Writing _± = i_* _±^* _E_± (-1), we have the following.

 The image of _- under the projection to D(E_+) is _+[-2]. 



The projection to D(E_+) is _+ * i^!. Noting that i is the embedding of a divisor we have a triangle of functors

    𝕀→ i^! i_* → (⊗_E|X) [-1]  →.

Applying this to _-^* _E_- (-1) and using the expression (<ref>) for _E|X gives

    _-^* _E_- (-1) → i^! i_* _-^* _E_- (-1) →_+^* _E_+ (-1) ⊗_-^* _E_- (-2) [-1] →.

Applying _+ * to the lefthand term gives zero, where we use flat base change _+*_-^* ≅ r_+^* r_-*. Applying to the righthand term gives _+[-2] because r_-*_E_- (-2) =  _[-1], and the claim follows. 





  §.§.§ Surface case
 

We apply the same approach to the surface case, constructing the required semiorthogonal decompositions using variation of GIT. For the surfaces X_±, we have fibre product diagrams as in (<ref>) as follows.

    [xscale=1.25,scale=0.75,baseline=([yshift=-.5ex]current bounding box.center)]
    
     (Bminus) at (-1,1) X_±;
     (Bplus) at (+1,1) X_B;
    
     (Cminus) at (-1,-1) E_±;
     (Cplus) at (+1,-1) E;
    
    [<-] (Bminus) to node[above] _±  (Bplus);
    [<-] (Cminus) to node[below] _±  (Cplus);
    
    [right hook->] (Cminus) to node[left] j_±  (Bminus);
    [right hook->] (Cplus) to node[right] i  (Bplus);

Here we take 

    E_- = [0 / ℤ_2]      and     E_+ = ℙ^1,

with j_- the obvious embedding, and j_+ the embedding of the zero section of the bundle X_+. The common fibre product E is isomorphic to [ℙ^1 / ℤ_2] with trivial ℤ_2-action.

We have  a full subcategory of D(X_B) defined as in the 3-fold case.

 For the surfaces X_±, the data shown in (<ref>) yields the opposite of a spherical pair. In particular, have semi-orthogonal decompositions

    = _±^* D(X_±), i_* _±^* _E_± (-1) .

Here _E_- (-1) is used to denote the sheaf on the stack E_- = [0 / ℤ_2] corresponding to the non-trivial irreducible representation of ℤ_2.

 There is existing literature on semiorthogonal decompositions for `stacky blowups': this term is convenient to describe both blowups of stacks and root stack construction (see for instance <cit.>), as both arise naturally in birational geometry of Deligne–Mumford stacks. Derived categories of root stacks have been considered in <cit.>, and these results generalized in <cit.> and <cit.>. Kawamata has also constructed semiorthogonal decompositions in this situation, for discussion see in particular <cit.> with proofs found in <cit.>. Although the results below are likely covered by this work, we found it convenient to apply the variation of GIT method herein.


We use a general construction of Coates, Iritani, Jiang, and Segal <cit.> to realize the blowup p_- as a variation of GIT. We then use standard technology which relates derived categories under variation of GIT <cit.> to obtain a semi-orthogonal decomposition. The semi-orthogonal decomposition for p_+ follows by an appropriate adaptation of this argument.

Case (-): First recall that _- is the blowup in E_- = [ 0 / ℤ_2 ] ⊂ X_-. We choose a bundle  on X_- with a section σ cutting out E_-. Following <cit.>, we consider the total space Tot (⊕) with a ℂ^*-action of weight -1 on the fibres of , and weight +1 on the fibres of . Writing (v,z) for fibre coordinates, and π for the projection to X_-, we take the substack

    [M / ℂ^*] := { v z = π^*σ}⊂Tot (⊕).

The stack [M / ℂ^*] has GIT quotients

    X_-   and  Bl_E_- X_- ≅ X_B,

and fixed locus M^ℂ^*≅ E_-. Furthermore, we obtain a semiorthogonal decomposition as follows.

    D(X_B) = _-^* D(X_-), i_* _-^* D(E_-)


We explain some of the details, so that we can adapt them to the case of p_+ below. By general theory, the GIT quotients are derived equivalent to certain `windows' in D  [M / ℂ^*] of the form

    _d = {∈ D  [M / ℂ^*]  | ^∙ Li_Z^*  have weights in [0,d)}.

Namely, the restriction functors 

    _rk→ D(X_B)     _1 → D(X_-)

are equivalences. Furthermore, there is a semiorthogonal decomposition of with D(X_B) with components D(X_-) and D(E_-), the latter appearing with multiplicity rk -1 = 1. In particular, <cit.> is used to show that D(X_-) embeds via _-^*, and the embedding of the orthogonal follows from <cit.>.

We then obtain the required semiorthogonal decomposition of  with component D(X_-) from (<ref>) by following the argument for the 3-fold case. Note in particular that the expression (<ref>) for _E|X holds verbatim.

Case (+): We argue similarly, considering E_+ = ℙ^1 ⊂ X_+, and taking a line bundle  on X_+ with a section σ cutting out E_+. For this case we take Tot (⊕) with a ℂ^*-action of weight -2 on the fibres of , denote fibre coordinates as above, and consider the substack

    [N/^*] := { v  z^2 = π^*σ}⊂Tot (⊕).

By the arguments of <cit.>, we see that [N/^*] has a GIT quotient X_+. We claim furthermore that it has X_B as a GIT quotient.

For this, first note that X_+ may be described as a ℂ^*-quotient of ℂ^3 with weights (1     1     -2 ), coordinates (x_1, x_2, y), and semistables {(x_1, x_2) ≠ 0}. Under this description [N/^*] is the locus {(x_1, x_2) ≠ 0,   vz^2=y} in the stack [ℂ^5 / ℂ^*2] given as follows.
 -2 
14pt D>p<@
    DDDDDx_1   x_2   y   v   z   
    (DDDDD)
    1   1   -2   -2   0   
    
    0   0   0   -2   1

By projecting away fromy, this is isomorphic to the left-hand stack[ℂ^4 / ℂ^*2]below. By row and column operations, this is isomorphic to the right-hand one, which can be seen to have a GIT quotientX_B= [Tot_ℙ^1(-1)/ ℤ_2]. By inspection, the variation of GIT for[N/^*]induces the mapp_+ X_B → X_+.

    DDDD   x_1   x_2   v   z   
    (DDDD)   1     1     -2     0   
        0     0     -2     1   
        ⟶      DDDD   x_1   x_2   z   v   
    (DDDD)   1     1     -1     0    
        0     0     1     -2


We then again use the argument of <cit.> to obtain a semi-orthogonal decomposition ofwith componentD(X_+). This proceeds as above, except that_2 ·rktakes the place of_rk. In particular, the multiplicity of componentsD(E_+)in the semiorthogonal decomposition is now given by2 ·rk -1 = 2· 1 - 1 =1,as required.


Finally we show the schober conditions. The required equivalences betweenD(X_-)andD(X_+)are an instance of the main theorem of Bridgeland–King–Reid <cit.>. The proof of the equivalences between orthogonals goes through as for the3-fold case, using the expression (<ref>) for_E|Xas before, which applies verbatim.


 §.§ A-side constructions
 We  setΛ':=Λ_Σ'andΛ_±:=Λ_Σ_±. We also write :=Λ_+∪Λ_-.

 The skeleton  is not represented as Λ_Σ by some Σ.

Using thatΛ_±⊂, there exist natural embeddings:

    _Λ_±(T)↪_(T)

We want to define such a functor on the level of wrapped constructible sheaf categories. For this, recall the following.


  *  Using smoothness of X_± and X_B, Proposition <ref> gives functors

    _Λ_±(T)↪_Λ_B(T).

  *  Using the inclusion ⊂Λ_B, Proposition <ref> gives a functor

    ι^w _Λ_B(T) →_(T).
 Composing (<ref>) and (<ref>), we write 

    ^*_±_Λ_±(T)→_(T).

Using Proposition <ref>, we also have functors
 
    _± !_(T) →_Λ_±(T).
We use the notations ^*_±, _± ! because these functors will turn out to be mirror to ^*_±, _± ! between D(X_±) and _0.
 The functor ^*_± is isomorphic to the restriction of functor (<ref>) to _Λ_±(T). In particular, it is an embedding.By functoriality, composing (<ref>) with 
the embedding 
    ι_(T)↪_Λ_B(T)
 gives the embeddings ι_±_Λ_±(T)↪_Λ_B(T). Writing ι^ for a left adjoint, we have ι^ι≃𝕀. Therefore (<ref>) is isomorphic to the composition of ι_± with ι^. Restricting the latter composition to _Λ_±(T) gives ^*_± by definition. Embeddings _±^* give a spherical pair,  which is mirror to the B-side spherical pair from Section <ref>.
We will prove this claim by using mirror symmetry in the next section.   
   
   
   

§ MIRROR EQUIVALENCES

In this section we prove homological mirror symmetry statements for the A-side and B-side schobers of the previous section, obtaining proofs of Theorems <ref> and <ref>.



 §.§ Threefold proof

Theorem <ref> gives equivalences

    κ_Σ_B D(X_B)_Λ_B(T)     κ_Σ_± D(X_±)_Λ_±(T)

which we write asκ_Bandκ_±for brevity.




  §.§.§ Mirror symmetry at the conifold point

Let us consider the composition

    κ_↪ D(X_B)_Λ_B(T)_(T).
κ_ is fully faithful.

Let us consider the precomposition of the inclusion D(X_±)↪ with κ_ and denote it κ_±. Then by the functoriality in Proposition <ref>, this is the same as the composition of κ_± and the functor
_±^*_Λ_±(T)↪_(T) constructed in Section <ref>. Hence κ_± is fully faithful. Since D(X_±) jointly generate  by definition, the functor κ_ is also fully faithful.


Take a point(x,ξ)∈Ω_Twhich lies in, but not inΛ_. Let_x, ξbe a microlocal skyscraper sheaf.


Skyscraper sheaves are exceptional objects. However microlocal skyscraper sheaves are not exceptional in general. 

However, in our case,
_x,ξ is exceptional.

Let us consider the locally closed subset D in M_, as shown in Figure <ref>, consisting of those m∈ M_ such that the following hold.

    m, e_1+e_3≤ -1    m, e_2≥ 0    m, e_1 >-1    m, -e_2+e_3 >-1

The 1-cell Δ of D cut out by

    m, e_1+e_3 = -1    m, e_2 = 0

has conormal -Λ_, and it follows that the sheaf _!_D gives _x,ξ by non-characteristic deformation as in <cit.>. Since (_!_D, _!_D) is the microstalk of _!_D, which is rank 1 and degree 0 from the picture below, the claim follows.

    _(T)≃_Λ_(T), _x,ξ.

Since _x,ξ is exceptional, we have a semi-orthogonal decomposition

    _(T)≃, _x,ξ

where  is the left orthogonal of _x, ξ. By the definition of microlocal skyscraper, ⊂_Λ_-(T).

Since we now have an inclusion _Λ_(T)⊂_(T), we also have j_Λ_(T)↪. So it suffices to show that this latter inclusion is essentially surjective.

By the definition of semi-orthogonal decomposition, we have a left adjoint i^_(T)→ of i↪_(T). On the other hand, we also have a composition of the inclusion j_Λ_(T)↪ and 
    ι^w_(T)→_Λ_(T)
 which is the left adjoint of ι_Λ_(T)↪_(T). Then we have, for any ∈_(T) and ∈,

    _(j∘ι^w(), )   ≃__Λ_(ι^w(), )
       ≃__(, )
       ≃__(, i())
       ≃_(i^L(),).

Hence we have i^≃ j∘ι^w. Since i^ is essentially surjective, we can conclude that j is also essentially surjective. This completes the proof.

We also have a semi-orthogonal decomposition of the form

    _(T)≃_Λ_+(T), _x,ξ

where (x, ξ)∈⋃Λ_±Λ_-. In this case, F_x, ξ is π_!_D' where D' is the image of D under the reflection e_1↔ e_2.


We also have a semiorthogonal decomposition≃ D(X_), D()from Proposition <ref>. Note that the functorκ_Brestricts to the equivalenceκ_-. 


κ_ is an equivalence.

We have already shown that κ_ is fully faithful, and that its restricts on D(X_) to an equivalence D(X_)→_Λ_(T). Using the semiorthogonal decompositions above we deduce that the restriction of κ_ to the orthogonal D() of D(X_) gives a functor  

    κ_ D() →_x,ξ.

This is fully faithful, therefore via the equivalence _x,ξ≃ D() it is isomorphic to a homological shift [s]. Hence κ_ is essentially surjective, and so the same holds for κ_, completing the proof.


  §.§.§ Relating mirror schobers
 We use Proposition <ref> above to conclude equivalences of schobers.

We now have an opposite spherical pair

    [scale=2]
     (Bminus) at (-1.3,1) D(X_-);
     (Bzero) at (0,1) ;
     (Bplus) at (+1.3,1) D(X_+);
    [right hook->,transform canvas=yshift=+] (Bminus) to node[above] _-^* (Bzero);
    [left hook->,transform canvas=yshift=+] (Bplus) to node[above] _+^* (Bzero);
    [<-,transform canvas=yshift=-] (Bminus) to node[below] _-! (Bzero);
    [<-,transform canvas=yshift=-] (Bplus) to node[below] _+! (Bzero);

Let us set_0':=_0⊗ω_X_B⊂ D(X_B).
By tensoring with the canonical sheaves on each category of the above schober, we get an equivalent opposite spherical pair.

    [scale=2]
     (Bminus) at (-1.3,1) D(X_-);
     (Bzero) at (0,1) ';
     (Bplus) at (+1.3,1) D(X_+);
    [right hook->,transform canvas=yshift=+] (Bminus) to node[above] _-^! (Bzero);
    [left hook->,transform canvas=yshift=+] (Bplus) to node[above] _+^! (Bzero);
    [<-,transform canvas=yshift=-] (Bminus) to node[below] _-* (Bzero);
    [<-,transform canvas=yshift=-] (Bplus) to node[below] _+* (Bzero);

To show this we use the isomorphism

    p_±^! ≅ω_X_B⊗ p_±^*(ω_X_±^-1⊗ -),

which holds because the relative dimension ofp_±is zero, and its left adjoint.

 The opposite spherical pair

    [scale=2]
     (Bminus) at (-1.3,1) D(X_-);
     (Bzero) at (0,1) ';
     (Bplus) at (+1.3,1) D(X_+);
    [right hook->,transform canvas=yshift=+] (Bminus) to node[above] _-^! (Bzero);
    [left hook->,transform canvas=yshift=+] (Bplus) to node[above] _+^! (Bzero);
    [<-,transform canvas=yshift=-] (Bminus) to node[below] _-* (Bzero);
    [<-,transform canvas=yshift=-] (Bplus) to node[below] _+* (Bzero);

is equivalent via the mirror symmetry functor K_Σ_± to an opposite spherical pair as follows.

    [scale=2.2]
     (Bminus) at (-1.5,1) _Λ_-(T);
     (Bzero) at (0,1) _(T);
     (Bplus) at (+1.5,1) _Λ_+(T);
    [right hook->,transform canvas=yshift=+] (Bminus) to node[above] _-^* (Bzero);
    [left hook->,transform canvas=yshift=+] (Bplus) to node[above] _+^* (Bzero);
    [<-,transform canvas=yshift=-] (Bminus) to node[below] _-! (Bzero);
    [<-,transform canvas=yshift=-] (Bplus) to node[below] _-! (Bzero);
 
By the argument above, in particular using Theorem <ref>, Proposition <ref>, and Proposition <ref>, we have commutative squares as follows

    [xscale=1.5,scale=2]
    
     (Bminus) at (-1,1) D(X_-);
     (Bzero) at (0,1) _0;
     (Bplus) at (+1,1) D(X_+);
    
     (Aminus) at (-1,0) _Λ_-(T);
     (Azero) at (0,0) _(T);
     (Aplus) at (+1,0) _Λ_+(T);
    
    [->] (Bminus) to node[above,sloped] ∼ (Aminus);
    [->] (Bzero) to node[above,sloped] ∼ (Azero);
    [->] (Bplus) to node[above,sloped] ∼ (Aplus);
    
    [->] (Bminus) to node[above] _-^* (Bzero);
    [->] (Bplus) to node[above] _+^* (Bzero);
    
    [->] (Aminus) to node[above] _-^* (Azero);
    [->] (Aplus) to node[above] _+^* (Azero);

where the vertical arrows are the functors κ.

Taking left adjoints, we furthermore obtain the following diagram.

    [xscale=1.5,scale=2]
    
     (Bminus) at (-1,1) D(X_-);
     (Bzero) at (0,1) _0;
     (Bplus) at (+1,1) D(X_+);
    
     (Aminus) at (-1,0) _Λ_-(T);
     (Azero) at (0,0) _(T);
     (Aplus) at (+1,0) _Λ_+(T);
    
    [<-] (Bminus) to node[above,sloped] ∼ (Aminus);
    [<-] (Bzero) to node[above,sloped] ∼ (Azero);
    [<-] (Bplus) to node[above,sloped] ∼ (Aplus);
    
    [<-] (Bminus) to node[above] _-! (Bzero);
    [<-] (Bplus) to node[above] _+! (Bzero);
    
    [<-] (Aminus) to node[above] _-! (Azero);
    [<-] (Aplus) to node[above] _+! (Azero);

Since the upper lines of the diagrams give the data of a weak spherical pair and satisfy the condition to be a spherical pair, we deduce that the lower lines give the data of an equivalent spherical pair. Note that the associated semi-orthogonal decompositions are given in Proposition <ref> and Remark after the proposition.

Composing the tensor of the inverse of canonical sheaves with the above schober equivalence, we have

    [xscale=1.5,scale=2]
    
     (Bminus) at (-1,1) D(X_-);
     (Bzero) at (0,1) _0';
     (Bplus) at (+1,1) D(X_+);
    
     (Aminus) at (-1,0) _Λ_-(T);
     (Azero) at (0,0) _(T);
     (Aplus) at (+1,0) _Λ_+(T);
    
    [->] (Bminus) to node[above,sloped] ∼ (Aminus);
    [->] (Bzero) to node[above,sloped] ∼ (Azero);
    [->] (Bplus) to node[above,sloped] ∼ (Aplus);
    
    [->] (Bminus) to node[above] _-^! (Bzero);
    [->] (Bplus) to node[above] _+^! (Bzero);
    
    [->] (Aminus) to node[above] _-^* (Azero);
    [->] (Aplus) to node[above] _+^* (Azero);

where the vertical arrows are κ∘ (-⊗ω^-1)=K.


Let:=(-, ω)be the Grothendieck duality. Then

    ∘ p^!∘   ≃∘(p^*(-, ω)⊗ω⊗ p^*ω^-1)
       ≃∘(p^*(-)^∨⊗ p^*ω⊗ω⊗ p^*ω^-1)
       ≃∘(p^*(-)^∨⊗ω)
       ≃(p^*(-)^∨⊗ω, ω)
       ≃(p^*(-)^∨)^∨
       ≃ p^*.

Therefore applyingto the opposite spherical pair given by the top line in the diagram below

    [scale=2, xscale=1.2]
    
     (Bminus) at (-1,1) D(X_-);
     (Bzero) at (0,1) _0';
     (Bplus) at (+1,1) D(X_+);
    
     (Aminus) at (-1,0) D(X_-);
     (Azero) at (0,0) _0';
     (Aplus) at (+1,0) D(X_+);
    
    [->] (Bminus) to node[left]  (Aminus);
    [->] (Bzero) to node[right]  (Azero);
    [->] (Bplus) to node[right]  (Aplus);
    
    [->] (Bminus) to node[above] _-^! (Bzero);
    [->] (Bplus) to node[above] _+^! (Bzero);
    
    [->] (Aminus) to node[above] p_-^* (Azero);
    [->] (Aplus) to node[above] p_+^* (Azero);

we get an anti-equivalent spherical pair. By the construction,_0'is the image of the functorsp_±^*and hence_0'=_0. Consequently, we have a spherical pair as follows.

    [scale=2]
     (Bminus) at (-1.3,1) D(X_-);
     (Bzero) at (0,1) ;
     (Bplus) at (+1.3,1) D(X_+);
    [right hook->,transform canvas=yshift=+] (Bminus) to node[above] _-^* (Bzero);
    [left hook->,transform canvas=yshift=+] (Bplus) to node[above] _+^* (Bzero);
    [<-,transform canvas=yshift=-] (Bminus) to node[below] _-* (Bzero);
    [<-,transform canvas=yshift=-] (Bplus) to node[below] _+* (Bzero);

Combining this with Theorem <ref> we obtain the following.

 In the 3-fold setting of Section <ref>, there exists a spherical pair as follows

    [scale=2,xscale=1.3]
     (Bminus) at (-1.5,1) _Λ_-^∞(T);
     (Bzero) at (0,1) _^∞(T);
     (Bplus) at (+1.5,1) _Λ_+^∞(T);
    [right hook->,transform canvas=yshift=+] (Bminus) to node[above] _-^* (Bzero);
    [left hook->,transform canvas=yshift=+] (Bplus) to node[above] _+^* (Bzero);
    [<-,transform canvas=yshift=-] (Bminus) to node[below] _-* (Bzero);
    [<-,transform canvas=yshift=-] (Bplus) to node[below] _-* (Bzero);

which is equivalent via mirror symmetry to the spherical pair as follows.

    [scale=2]
     (Bminus) at (-1.3,1) D(X_-);
     (Bzero) at (0,1) ;
     (Bplus) at (+1.3,1) D(X_+);
    [right hook->,transform canvas=yshift=+] (Bminus) to node[above] _-^* (Bzero);
    [left hook->,transform canvas=yshift=+] (Bplus) to node[above] _+^* (Bzero);
    [<-,transform canvas=yshift=-] (Bminus) to node[below] _-* (Bzero);
    [<-,transform canvas=yshift=-] (Bplus) to node[below] _+* (Bzero);

Note that ^∞=()^∞ by the definition.
We take the functors between Fukaya categories as the compositions of the functor in Theorem <ref> with the functors between wrapped constructible sheaves. The commutativity and equivalences follow from this description.
The functors q_±* between Fukaya categories are isomorphic to stop removal functors as given in <cit.>. The functors q^*_± are a bit more subtle, as we used the smoothness of mirrors to construct them. A Fukaya-categorical description may be as follows: the image of a Lagrangian submanifold is its result under the Reeb flow until it stops. However, making this claim precise is a problem.


 §.§ Surface proof


We return to the setting of Section <ref>, in particular we use notation as follows.

    X_- = [ ℂ^2/ ℤ_2 ]      X_+ = ℂ^2/ ℤ_2


For a toric Deligne–Mumford stack, we have to generalize the definition ofΛ_Σ. We only describe the result:

    Λ_-:=Λ_Σ_0∪ (Λ_Σ_0+12· [e_1^∨])

where+1/2· [e_1^∨]means the translation by the class1/2· [e_1^∨]∈ T=M_/M.

As before, let us take a point(x,ξ)∈Twhich lies in, but not inΛ_. Let_x, ξbe a microlocal skyscraper sheaf.

_x, ξ is exceptional.
 Again, we have an explicit description.

    0

We take the constant sheaf on the shaded triangle, where the blue side is included, and other sides are not.





By the same argument as in the conifold case: 


    _(T)≃_Λ_(T), _x,ξ.

The skeleton Λ_+ is the usual FLTZ skeleton, as shown in Figure <ref>. Again, we have a microlocal skyscraper on a point in Λ_+ as follows:

    4

We can prove a semi-orthogonal decomposition for this case: 
    _(T)≃_Λ_+(T), _x',ξ'.




In the same way as in the conifold case, we can get a mirror equivalence between schobers, as follows.

 In the surface setting of Section <ref>, the analog of Theorem <ref> holds.


 §.§ Proof for flober


The coherent-constructible correspondence and an argument similar to the one presented in the last two subsections also gives a mirror equivalence of flobers. Let us only describe the result. 
We have an opposite flober

    [scale=2]
     (Bminus) at (-1.3,1) D(X_-);
     (Bzero) at (0,1) D(X_B);
     (Bplus) at (+1.3,1) D(X_+);
    [right hook->,transform canvas=yshift=+] (Bminus) to node[above] _-^! (Bzero);
    [left hook->,transform canvas=yshift=+] (Bplus) to node[above] _+^! (Bzero);
    [<-,transform canvas=yshift=-] (Bminus) to node[below] _-* (Bzero);
    [<-,transform canvas=yshift=-] (Bplus) to node[below] _+* (Bzero);

which is anti-equivalent to a flober as follows.

    [scale=2]
     (Bminus) at (-1.3,1) D(X_-);
     (Bzero) at (0,1) D(X_B);
     (Bplus) at (+1.3,1) D(X_+);
    [right hook->,transform canvas=yshift=+] (Bminus) to node[above] _-^* (Bzero);
    [left hook->,transform canvas=yshift=+] (Bplus) to node[above] _+^* (Bzero);
    [<-,transform canvas=yshift=-] (Bminus) to node[below] _-* (Bzero);
    [<-,transform canvas=yshift=-] (Bplus) to node[below] _+* (Bzero);

We denote the latter one by_B.

On the other hand, the coherent-constructible correspondenceKtakes the former opposite flober to another flober

    [scale=2]
     (Bminus) at (-1.5,1) _Λ_-(T);
     (Bzero) at (0,1) _Λ_B(T);
     (Bplus) at (+1.5,1) _Λ_+(T);
    [right hook->,transform canvas=yshift=+] (Bminus) to node[above] _-^* (Bzero);
    [left hook->,transform canvas=yshift=+] (Bplus) to node[above] _+^* (Bzero);
    [<-,transform canvas=yshift=-] (Bminus) to node[below] _-! (Bzero);
    [<-,transform canvas=yshift=-] (Bplus) to node[below] _-! (Bzero);

and then Theorem <ref> takes this to an anti-equivalent flober as follows.

    [scale=2,xscale=1.5]
     (Bminus) at (-1.5,1) _Λ_-^∞(T);
     (Bzero) at (0,1) _Λ_B^∞(T);
     (Bplus) at (+1.5,1) _Λ_+^∞(T);
    [right hook->,transform canvas=yshift=+] (Bminus) to node[above] _-^* (Bzero);
    [left hook->,transform canvas=yshift=+] (Bplus) to node[above] _+^* (Bzero);
    [<-,transform canvas=yshift=-] (Bminus) to node[below] _-* (Bzero);
    [<-,transform canvas=yshift=-] (Bplus) to node[below] _-* (Bzero);

We denote this by_A.


The two flobers _A and _B are equivalent via mirror symmetry.


§ APPLICATIONS TO SINGULARITIES


We first explain work of  calculating cohomology of the flober_B, before giving an analogous calculation for the flober_A, proving Proposition <ref> and Corollary <ref>.



 §.§ B-side calculation


Bondal–Kapranov–Schechtman showed the following:
<cit.>
The flober _B has the 2nd compact cohomology

    ^2_c(Δ, _B)≃ D(X_0),

meaning that the diagram

    [xscale=1,scale=1.5,baseline=([yshift=-.5ex]current bounding box.center)]
    
     (Bminus) at (0,1) D(X_B);
     (Bplus) at (1,0) D(X_+);
    
     (Cminus) at (-1,0) D(X_-);
     (Cplus) at (0,-1) D(X_0);
    
    [->] (Bminus) to node[above right] _+*  (Bplus);
    [->] (Cminus) to node[below left] f_-*  (Cplus);
    
    [<-] (Cminus) to node[above left] _-*  (Bminus);
    [<-] (Cplus) to node[below right] f_+*  (Bplus);

is a push-out in the Morita model category of dg-categories, where the f_± are the resolutions X_±→ X_0. 


Let us recall their logic. First we extend the diagram to the following, defining_+(respectively_-) to be the kernel ofp_+*(respectivelyf_-*).

    [xscale=1,scale=1.5]
    
     (Aminus) at (-1,2) _+;
     (Aplus) at (-2,1) _-;
     (Aphantom) at (2,1) _-;
    
     (Bminus) at (0,1) D(X_B);
     (Bplus) at (1,0) D(X_+);
    
     (Cminus) at (-1,0) D(X_-);
     (Cplus) at (0,-1) D(X_0);
    
    [->] (Aminus) to node[above left] h  (Aplus);
    [->] (Aminus) to (Bminus);
    [->] (Aplus) to (Cminus);
    
    [->] (Bminus) to node[above right] _+*  (Bplus);
    [->] (Cminus) to node[below left] f_-*  (Cplus);
    
    [<-] (Cminus) to node[above left] _-*  (Bminus);
    [<-] (Cplus) to node[below right] f_+*  (Bplus);
<cit.>
In this situation, we have D(X_B)/_+≃ D(X_+) and D(X_-)/_-≃ D(X_0).


We moreover have the following general proposition, which is a slight modification of <cit.>.

Let u_1→_1 be a fully-faithful dg-functor between Karoubian pre-triangulated dg-categories and v_1→_2 be a dg-functor. Let _2 be the thick triangulated hull of objects of v(_1). Then the push-out of 

    [xscale=1,scale=1.5]
    
     (Bminus) at (0,1) _1;
     (Bzero) at (1,0) _1/_1;
    
     (Aminus) at (-1,0) _2;
    
    [->] (Bminus) to node[above left] v (Aminus);
    
    [->] (Bminus) to (Bzero);

is Morita-equivalent to _2/_2. 

Let _2' be the full sub dg-category of _2 spanned by objects of v(_1). Then <cit.> says the desired push-out is quasi-equivalent to _2/_2'. Since _2/_2' is Morita-equivalent to _2/_2 (for example, one can deduce it from <cit.>), we complete the proof.




The situation of the above proposition is: 

    [xscale=1,scale=1.5]
    
     (Bminus) at (-1,2) _1;
     (Bzero) at (0,1) _1;
     (Bplus) at (1,0) _1/_1;
    
     (Aminus) at (-2,1) _2;
     (Aphantom) at (2,1) _2;
     (Azero) at (-1,0) _2;
     (Aplus) at (0,-1) _2/_2;
    
    [->] (Bminus) to node[above,sloped]  (Aminus);
    [->] (Bzero) to node[above left] v  (Azero);
    [->] (Bplus) to node[above,sloped]  (Aplus);
    
    [->] (Bminus) to node[below left] u  (Bzero);
    [->] (Bzero) to node[above]  (Bplus);
    
    [->] (Aminus) to node[above]  (Azero);
    [->] (Azero) to node[above]  (Aplus);

To apply this proposition to our situation, the following is sufficient.
<cit.>
The functor h_+→_- is a split generation. 

This completes the logic to proveD^b(X_0)≃push-out of (<ref>).



 §.§ A-side calculation

Let us now turn to the A-side. We would like to prove the following:

The A-model flober _A has 2nd compact cohomology

    ^2_c(Δ, _A)≃_⋂Λ_±^∞(Ω_T).


We would first like to set up a somewhat general situation. LetZbe a real analytic manifold. LetΛbe a conic Lagrangian inΩ_ZandΛ'be a closed conic Lagrangian subset ofΛ. Then there exists a canonical inclusion 
    ι_Λ'(Z)→_Λ(Z)
 and a left adjointι^_Λ(Z)→_Λ'(Z).

Let S_Λ(ΛΛ') be the set of microlocal skyscraper sheaves in _Λ(Z) over ΛΛ' and S_Λ(Λ') be the set of microlocal skyscraper sheaves in _Λ(Z) over Λ'. 


For brevity, we set 
    S:=S_Λ(ΛΛ')∪ S_Λ(Λ')     and     R:=S_Λ(ΛΛ').

Thenι^mapsRto0andS Rto a set of generators of_Λ'(Z). Then_Λ(Z)= Sby Nadler's generation result <cit.>. By Thomason's localization theorem <cit.>, we have

    R^ =  R∩_Λ(Z)

where R^is the smallest thick subcategory containingRand 

    (_Λ(Z)/ R)^c≃_Λ(Z)/ R∩_Λ(Z).

where(-)^cis the full subcategory spanned by compact objects (more preciselyℵ_0-compact). Combining these, we have  the following.

    (_Λ(Z)/ R)^c≃_Λ(Z)/ R^


To prove the next lemma, we recall some facts. 

Let  be a triangulated category and  be a thick subcategory. The Bousfield localization functor for the pair (, ) is a right adjoint of the quotient functor →/.
<cit.>
The Bousfield localization functor is fully faithful.


Now we would like to prove the following:

    _Λ(Z)/ R≃_Λ'(Z).

Again, general nonsense tells us that there exists a right adjoint _1^ of the quotient _1_Λ(Z)→_Λ(Z)/ R (cf. <cit.>). Hence this is a Bousfield localization and hence fully faithful. For an object ∈_Λ(Z)/ R and an r∈ R, we have 

    __Λ(Z)(r, _1^())   ≃__Λ(Z)/ R(_1(r), )≃ 0.

Hence (_1^())⊂Λ' by the definition of microlocal skyscraper sheaves. So we have a fully faithful functor _1^_Λ(Z)/ R→_Λ'(Z) such that _1^=ι∘_1^. We would like to see that this is essentially surjective. 

Let us take ∈_Λ'(Z). It is enough to prove _1^∘_1∘ι ()≃ for our purpose. Letting _2 be the functor _Λ(Z)/ R→_Λ'(Z) induced by ι^, which satisfies _2∘_1= ι^.

First, let us see π_2 is the left adjoint of π_1∘ι. Let us consider an object π_1()∈_Λ(Z)/  R which is represented by ∈_Λ(Z). We have

    __Λ'(Z)(π_2∘π_1(), )    ≃__Λ'(Z)(ι^(),)
       ≃__Λ(Z)(, ι())
       ≃__Λ(Z)/ R(π_1(), π_1∘ι()).

For the last equality, we used the fact that R is in the left orthogonal of _Λ'(Z). Next, we can see π_1∘ι is fully faithful because of the following.

    __Λ(Z)/ R(π_1∘ι('), π_1∘ι())   ≃__Λ'(Z)(π_2∘π_1∘ι('), )
       ≃__Λ'(Z)(ι^∘ι('), )
       ≃__Λ'(Z)(', )

Finally, we have

    __Λ'(Z)(', _1^∘_1∘ι ())   ≃__Λ(Z)(ι('), _1^∘_1∘ι())
       ≃__Λ(Z)/ R(_1∘ι ('), _1∘ι())
       ≃__Λ'(Z)(',).

Yoneda then completes the proof.

The functor _2 is an equivalence.

From the proof above, we also see that _1∘ι_Λ'(Z)→_Λ(Z)/ R is an equivalence. Since π_2 is the left adjoint of π_1∘ι, we actually have 
    π_2=π_1^.

The functor _2 induces an equivalence as follows.

    _Λ'(Z)≃_Λ(Z)/ R^




Let us go back to our situation. We now have a diagram.

    [xscale=1,scale=1.75]
    
     (Aminus) at (-1,2) ⟨ S_Λ_B(Λ_BΛ_+)⟩^;
     (Aplus) at (-2,1) ⟨ S_Λ_-(Λ_-⋂Λ_±)⟩^;
     (Aphantom) at (2,1) ⟨Λ_- (⋂Λ_±)⟩^;
    
     (Bminus) at (0,1) _Λ_B(T);
     (Bplus) at (1,0) _Λ_+(T);
    
     (Cminus) at (-1,0) _Λ_-(T);
     (Cplus) at (0,-1) _⋂Λ_±(T);
    
    [->] (Aminus) to  node[above left] h (Aplus);
    [->] (Aminus) to (Bminus);
    [->] (Aplus) to (Cminus);
    
    [->] (Bminus) to node[above right] _+!  (Bplus);
    [->] (Cminus) to node[pos=,below left,transform canvas=xshift=8pt] f_-! (Cplus);
    
    [<-] (Cminus) to node[above left] _-!  (Bminus);
    [<-] (Cplus) to node[pos=1-,below right] f_+! (Bplus);

The two sequences from upper left to lower right are Verdier–Drinfeld quotients by Corollary 6.5. The functorhis the restriction of_-!to the uppermost category. The functors f_±!are the restrictions of the left adjoints of the inclusions_⋂Λ_±(T)⊂_Λ_±(T).


The category ⟨Λ_-⋂Λ_±⟩^ is split-generated by the image of h.

Note that _-! takes a microlocal skyscraper sheaf in _Λ_B(T) over a point in Λ_- to a microlocal skyscraper sheaf in _Λ_-(T) over the same point in Λ_- <cit.>. Note also that _-! takes a microlocal skyscraper sheaf in _Λ_B(T) over a point in Λ_BΛ_- to zero <cit.>. These imply the well-definedness of h. Since Λ_BΛ_+⊃Λ_-⋂Λ_±, these also imply the surjectivity of h on the split generators.





Repeating the logic presented in the beginning of the section, we can conclude

    [xscale=1,scale=1.5]
    
     (Bminus) at (0,1) _Λ_B(T);
     (Bplus) at (1,0) _Λ_+(T);
    
     (Cminus) at (-1,0) _Λ_-(T);
     (Cplus) at (0,-1) _⋂Λ_±(T);
    
    [->] (Bminus) to node[above right] _+!  (Bplus);
    [->] (Cminus) to node[pos=,below left,transform canvas=xshift=8pt] f_-! (Cplus);
    
    [<-] (Cminus) to node[above left] _-!  (Bminus);
    [<-] (Cplus) to node[pos=1-,below right] f_+! (Bplus);

is a homotopy push-out in the Morita model. Noting that(⋂Λ_±)^∞=⋂Λ_±^∞by the definition of taking infinity, Theorem <ref> takes this diagram to another push-out diagram.

    [xscale=1,scale=1.5]
    
     (Bminus) at (0,1) _Λ_B^∞(Ω_T);
     (Bplus) at (1,0) _Λ^∞_+(Ω_T);
    
     (Cminus) at (-1,0) _Λ^∞_-(Ω_T);
     (Cplus) at (0,-1) _⋂Λ_±^∞(Ω_T);
    
    [->] (Bminus) to node[above right] _+*  (Bplus);
    [->] (Cminus) to node[pos=,below left,transform canvas=xshift=8pt] f_-* (Cplus);
    
    [<-] (Cminus) to node[above left] _-*  (Bminus);
    [<-] (Cplus) to node[pos=1-,below right] f_+* (Bplus);

Here f_±*is defined as the composition of the functor in Theorem <ref> and f_±!. This implies Proposition <ref>.

On the other hand, Theorem <ref> implies that^2_c(Δ, _A)≃^2_c(Δ, _B). Combining with Proposition <ref>, we have homological mirror symmetry for singular varieties

    D(X_0)≃_⋂Λ_±^∞(Ω_T).

In this section, we used flobers to study the compact cohomology rather than spherical pairs. If one instead uses the spherical pairs from Theorem <ref>, then one will arrive at the same conclusion as presented here, i.e. the flobers and spherical pairs have the same 2nd compact cohomology.

The method presented here might be generalized to another general proof of the coherent-constructible correspondence for singular toric varieties, as proved by the second author in <cit.>.

Let X be a singular toric variety. Let S_X be the category consisting of 

  *  Object: smooth toric Deligne–Mumford stack refining X.

  *  Morphism: morphism along refinement.

There exists a functor _X from S_X to the category of dg-categories defined by X'↦ D(X') and morphisms are mapped to push-forwards along them. 

The universality morphism gives an equivalence

    lim_⟶
     X'∈_X D(X')≃ D(X).

One can define the A-model counterpart and conjecture

The universality morphism gives an equivalence

    lim_⟶
     X'∈_X_Λ_X'(T)≃_Λ_X(T).

These two conjectures would allow us to conclude an equivalence between D(X)≃_Λ_X(T) from the smooth coherent-constructible correspondence.

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