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PERVERSE SCHOBERS AND ORLOV EQUIVALENCES

NAOKI KOSEKI AND GENKI OUCHI

Abstract. A perverse schober is a categorification of a perverse sheafproposed by Kapranov–Schechtman. In this paper, we construct exam-ples of perverse schobers on the Riemann sphere, which categorify theintersection complexes of natural local systems arising from the mirrorsymmetry for Calabi-Yau hypersurfaces. Orlov equivalence plays a keyrole for the construction.

Contents

1. Introduction 12. Quick Review on dg categories 43. Local systems of categories and perverse schobers 74. Derived factorization category 105. Construction of perverse schobers 166. Decategorification 217. Mirror symmetry for elliptic curves 248. Orlov equivalences via VGIT 26References 30

1. Introduction

1.1. Motivation and Results. A Perverse schober is a conjectural cat-egorification of a perverse sheaf introduced in a seminal paper [KS15] byKapranov–Schechtman. There is a well-known way to define the notionof local systems of categories, which are the simplest examples of perverseschobers. However, it is not clear how to define perverse sheaves of cat-egories in general. At this moment, a general definition is only availableon Riemann surfaces and on affine spaces stratified by hyperplane arrange-ments [DKSS21, KS15]. A key observation for the categorification in thecase of Riemann surfaces is the classical result of [Bei87, GGM85, GMV96]that describes the category of perverse sheaves on a disk in terms of quiverrepresentations. Quiver representations consist of linear algebraic data, andhence we can categorify them by replacing vector spaces and linear mapswith categories and functors, respectively. More precisely, we will use thenotion of spherical functors between dg-categories developed in [AL17]. Alot of interesting classes of perverse schobers have been constructed via bi-rational geometry [BKS18, Don19a, Don19b, SdB19, SdB20] and symplecticgeometry [DK21, KS15, KSS20].

1

http://arxiv.org/abs/2201.05902v1

2 NAOKI KOSEKI AND GENKI OUCHI

In this paper, we construct new examples of perverse schobers on theRiemann sphere P1, which arise from mirror symmetry of Calabi-Yau hy-persurfaces. Let X ⊂ Pn+1 be a smooth hypersurface of degree n+2, whichis a Calabi-Yau variety of dimension n. Then there exists the mirror fam-ily of X over P1 \ {1,∞}, with the unique orbifold point at 0 ∈ P1. By avariation of complex structures, we obtain a natural homomorphism

π1(P1 \ {0, 1,∞}) → Auteq(Dπ Fuk(X∨)),

where X∨ denotes the mirror of X. Applying the conjectural mirror sym-metry equivalence Dπ Fuk(X∨) ≃ Db(X) (partly proved in [PZ98, Sei15,She15]), we get a homomorphism

(1.1) π1(P1 \ {0, 1,∞}) → Auteq(Db(X)),

which we think of as a local system of categories with the fiber Db(X).It is proved (cf. [BH06, CIR14, Hor99]) that the morphism (1.1) maps thesimple loops around the points∞, 1 to the autoequivalences (−)⊗OX(1) andSTOX , respectively, where we put OX(1) := OPn+1(1)|X , and STOX denotesthe spherical twist around the structure sheaf. Note that by taking thecohomology group ofX, and restricting it to the subring ΛH(X) ⊂ H∗(X,Q)generated by the hyperplane class, the homomorphism (1.1) induces a usuallocal system L with the fiber ΛH(X):

(1.2) L : π1(P1 \ {0, 1,∞}) → Aut(ΛH(X)).

There is a canonical way to extend L as a perverse sheaf on P1, called theintersection complex and denoted by IC(L).

The aim of this paper is to give a categorification of the intersectioncomplex IC(L):

Theorem 1.1 (Theorems 5.7 and 6.7). There exists a perverse schober P

on P1 which extends the local system (1.1) of categories. More precisely, theperverse schober P is a categorification of the intersection complex IC(L)associated to the local system (1.2).

A computation shows that the intersection complex IC(L) has a (anti-)symmetric pairing. In particular, it is Verdier self-dual. In Proposition 5.9,we will also prove that our perverse schober P has a categorification of thisproperty.

In the case of elliptic curves, we also construct a perverse schober PA onP1 from the A-side and prove:

Theorem 1.2 (Theorem 7.3). Let X be an elliptic curve, X∨ its mirror.Then the perverse schober PA on P1 has a generic fiber Dπ Fuk(X∨), andit is identified with the perverse schober P in Theorem 1.1 under the mirrorequivalence Db(X) ≃ Dπ Fuk(X∨).

1.2. Idea of proof. To construct the perverse schober as in Theorem 1.1,we need to find spherical functors which induce the autoequivalences (−)⊗OX(1),STOX , and STOX ◦(⊗OX(1)). For the first two autoequivalences, wecan find natural spherical functors using the derived categories of varieties.

For the last equivalence STOX ◦(⊗OX(1)), we use the following Orlovequivalence

PERVERSE SCHOBERS AND ORLOV EQUIVALENCES 3

Db(X) ≃ HMFgr(W ),

to find the natural spherical functor, where W is the homogeneous polyno-mial definingX ⊂ Pn+1. We find that the autoequivalence STOX ◦(⊗OX(1))has a natural presentation as the twist of a spherical functor between cate-gories of graded matrix factorizations.

Note that, for a given autoequivalence, there are many ways to express itas a twist of a spherical functor. We choose the natural expressions so thatwe recover the intersection complex after decategorification.

1.3. Related works. Various examples of perverse schobers, consisting ofderived categories of varieties, have been constructed using birational geom-etry [BKS18, Don19a, Don19b, SdB19, SdB20]. Our perverse schober hasa different origin from these examples, in the sense that it does not involvebirational geometric structure. Moreover, it is the first example involving aLandau-Ginzburg model, which defines the category of graded matrix fac-torizations.

Donovan–Kuwagaki [DK21] verified a mirror symmetry type statementfor some examples of perverse schobers coming from non-compact geometry,including the case of Atiyah flops. Theorem 1.2 is an analogue of their result,for the case of elliptic curves.

1.4. Open questions.

(1) It would be interesting to construct a perverse schober using Fukayacategories, which is mirror to our schober in higher dimension. Forexample, in the case of a quartic K3 surface, we need to under-stand mirror symmetry for the categories Db(C) and HMFgr(C3,W ),where C is a smooth projective curve of genus 3, and W is a homo-geneous polynomial of degree 4.

(2) In the case of a quartic K3 surface, we can think of the Riemannsphere P1 as a compactification of a certain quotient of the space ofBridgeland stability conditions. It would be interesting to generalizeour construction to arbitrary K3 surfaces.

1.5. Plan of the paper. The paper is organized as follows: In Section 2,we review the theory of dg-categories used in this paper. In particular, werecall the notion of spherical dg functors. In Section 3, we define the notionof perverse schobers on a Riemann surface following Kapranov–Schechtman[KS15]. In Section 4, we review the theory of derived factorization categories.In particular, we recall various constructions of dg-enhancements of thesecategories.

In Section 5, we construct our perverse schober. In Section 6, we provethat our perverse schober categorifies the intersection complex. In Section7, we discuss the mirror symmetry of perverse schobers for elliptic curves.Finally in Section 8, we review the proof of Orlov equivalences via a variationof GIT quotients, following [BFK19]. Using this, we construct an exampleof a spherical pair, which is another categorification of a perverse sheaf ona disk.


Acknowledgement. The authors would like to thank Professor ArendBayer, Dr. Yuki Hirano, and Dr. Kohei Kikuta for valuable discussions.The authors would also like to thank the participants of the seminar onperverse schobers, held at the University of Edinburgh on Fall 2021, wherethey learned a lot about perverse schobers. In particular, they would like tothank Professor Pavel Safronov for organizing the seminar.

N.K. was supported by ERC Consolidator grant WallCrossAG, no. 819864.G.O. is supported by JSPS KAKENHI Grant Number 19K14520.

Notation and Convention. Throughout the paper, we work over thecomplex number field C.

• For a variety X, Db(X) denotes the bounded derived category ofcoherent sheaves on X.

• For an integer d ∈ Z, χd : C∗ → C∗ denotes the character defined by

χd(t) := td.

2. Quick Review on dg categories

In this section, we briefly review the theory of dg categories used in thispaper. We refer [AL17, Kel94, Kel06, Toe07, Toe11] for more details.

2.1. Basic definitions. We denote by dgcatC the category of small dg cate-gories over C. For a dg category A ∈ dgcatC, we denote by [A] the homotopycategory of A.

Definition 2.1. Let A,B ∈ dgcatC be dg categories, and let F : A → B bea dg functor. We say that the functor F is quasi-equivalent if it satisfies thefollowing two conditions:

(1) For any objects a, a′ ∈ A, the morphism

HomA(a, a′) → HomB(F (a), F (a

′))

is a quasi-isomorphism.(2) The induced functor [F ] : [A] → [B] on the homotopy categories is

essentially surjective.

We denote by hodgcatC the localization of dgcatC by quasi-equivalences.A morphism in hodgcatC is called a quasi-functor.

2.1.1. Dg modules and derived categories. We denote by ModC the dg cat-egory of complexes of C-vector spaces. Let A ∈ dgcatC be a dg category.A right A-module is a dg functor A → ModC. We denote by ModA the dgcategory of right A-modules. We have the dg Yoneda embedding

(2.1) A → ModA, a 7→ HomA(−, a).

Definition 2.2. (1) An object C ∈ ModA is acyclic if for every a ∈ A,the complex C(a) ∈ ModC is acyclic.

(2) An object P ∈ ModA is projective if for every acyclic module C ∈ModA, we have HomModA

(P,C) = 0.

We denote by P(A) ⊂ ModA the dg subcategory consisting of projectivemodules. The derived category D(A) is the localization of the homotopycategory [ModA] by acyclic A-modules. The derived category D(A) has the


structure of a triangulated category, and we have a canonical equivalence[P(A)] ≃ D(A).

We denote by perf(A) ⊂ D(A) the full triangulated subcategory consist-ing of compact objects. A right A-module is called perfect if its class in thederived category D(A) is compact. We denote by Pperf(A) ⊂ P(A) the dgsubcategory of perfect projective modules. Note that we have an equivalence[Pperf(A)] ≃ perf(A).

The dg Yoneda embedding (2.1) induces the embedding

[A] → D(A).

Note that the homotopy category [A] is not triangulated in general. We de-fine the triangulated category tri(A) ⊂ D(A) to be the smallest triangulatedsubcategory containing [A]. Then we have the following inclusions:

[A] ⊂ tri(A) ⊂ perf(A) ⊂ D(A).

Definition 2.3. A dg category A ∈ dgcatC is pre-triangulated (resp. trian-gulated) if the inclusion [A] ⊂ tri(A) (resp. [A] ⊂ perf(A)) is an equivalence.

Remark 2.4. A pre-triangulated dg category A is triangulated if and onlyif its homotopy category [A] is idempotent complete (cf. [Kel06, Theorem3.8]).

2.1.2. Bimodules. Let A,B ∈ dgcatC be dg categories. An A-B-bimoduleis an Aop ⊗ B-module. We denote by AModB the dg category of A-B-bimodules, and by D(A-B) its derived category.

Definition 2.5. Let M be an A-B-bimodule. We say that M is A-perfect(resp. B-perfect) if M(b) ∈ Mod -Aop (resp. M(a) ∈ Mod -B) is perfect forall b ∈ B (resp. a ∈ A).

We denote by DA- perf(A-B) (resp. DB- perf(A-B)) the full triangulatedsubcategory of D(A-B) consisting of A-perfect (resp. B-perfect) bimodules.

Given a bimodule M ∈ AModB, we have the tensor product functor

(−)⊗A M : ModA → ModB, a 7→ (b 7→ b⊗A M(a)),

and its derived functor

(−)⊗L

A M : D(A) → D(B).

Similarly, we have the functors

M ⊗B (−) : ModBop → ModAop , M ⊗L

B (−) : D(Bop) → D(Aop).

We have the following characterizations of A-perfect and B-perfect bi-modules (see the first and second paragraphs in [AL17, p2590]):

• M is A-perfect if and only if the derived tensor product (−)⊗L

A Mrestricts to the functor perf(A) → perf(B).

• M is B-perfect if and only if the derived tensor product M ⊗L

B (−)restricts to the functor perf(Bop) → perf(Aop).


2.2. Dg enhancements. We first define the notion of dg enhancements:

Definition 2.6. Let D be a triangulated category.

(1) A dg enhancement of D is a pair (A, ǫ) consisting of a pre-triangulated

dg category A and an exact equivalence ǫ : [A]∼−→ D.

(2) A Morita enhancement of D is a pair (A, η) consisting of a dg cate-

gory A and an exact equivalence η : perf(A)∼−→ D.

Remark 2.7. Suppose that a dg category A is triangulated. If we have adg enhancement (A, ǫ) of a triangulated category D, it also gives a Moritaenhancement via

perf(A) ≃ [A]ǫ−→ D.

Example 2.8. Let A ∈ dgcatC be a dg category. The dg category Pperf(A)is triangulated and gives a Morita enhancement of perf(A).

For our purpose, we also need the notion of enhancements of exact func-tors of triangulated categories.

Theorem 2.9 ([Toe07]). Let A,B ∈ dgcatC be dg categories. There existsa dg category RHom(A,B) with the following properties:

(1) There exists a bijection

Isom ([RHom(A,B)]) = HomhodgcatC(A,B),

where the left hand side denotes the set of isomorphism classes ofobjects in [RHom(A,B)].

(2) There exists an equivalence

[RHom(Pperf(A),Pperf (B))] ≃ DB- perf(A-B)

such that for an object M ∈ DB- perf(A-B), the corresponding exactfunctor [M ] : perf(A) → perf(B) is ⊗L

AM .

Proof. The existence of the dg category RHom(A,B) is proved in [Toe07,Theorem 6.1]. The first assertion is [Toe07, Corollary 4.8]; the second asser-tion is proved in [Toe07, Theorem 7.2]. �

Definition 2.10. Let Φ: C → D be an exact functor of triangulated cate-gories. Suppose that the triangulated categories C,D have dg enhancementsA,B, respectively. Then a dg enhancement of the functor Φ is an objectF ∈ [RHom(A,B)] together with an isomorphism [F ] ≃ Φ: C → D.

2.3. Spherical functors. Let A,B ∈ dgcatC be dg categories, let S ∈D(A-B) be an A-perfect and B-perfect bimodule. Recall from Theorem2.9 (2) that S defines an isomorphism class of quasi-functors Pperf(A) →Pperf(B) whose underlying exact functor perf(A) → perf(B) is isomorphicto the derived tensor product (−)⊗L

A S. We denote as

(2.2) s := (−)⊗L

A S : D(A) → D(B).

By [AL17, Corollary 2.2], there exist B-perfect and A-perfect objects L,R ∈D(B-A) such that the functors

l := (−)⊗L

B L, r := (−)⊗L

B R : D(B) → D(A)

are the left, right adjoints of the functor (2.2), respectively.


Let us denote by SR the object R ⊗L

A S ∈ D(B-B). We define objectsSL ∈ D(B-B), RS,LS ∈ D(A-A) in a similar way. Then the correspondingderived tensor products are isomorphic to the functors sr, sl, rs, ls, respec-tively. By [AL17, Defenitions 2.3, 2.4], there exist morphisms

SR→ B, B → SL, A → RS, LS → A,

which induce adjoint (co)units on the underlying exact functors. Notethat we regard A,B as diagonal A-A-bimodule, B-B-bimodule, respectively.Namely, we define A ∈ AModA as

A : Aop ⊗A → ModC, (a, b) 7→ HomA(b, a),

and similarly for B ∈ B ModB. By taking (shifts of) cones, we obtain theexact triangles

SR→ B → T,

T ′ → B → SL,

F → A → RS,

LS → A → F ′.

We call T (resp. T ′, F, F ′) as twist (resp. dual twist, cotwist, dual cotwist)of S. We denote by t, t′, f, f ′ their underlying exact functors.

The following is the main result of [AL17]:

Definition-Theorem 2.11 ([AL17, Theorem 5.1]). Suppose that any twoof the following conditions hold:

(1) t is an autoequivalence of D(B).(2) f is an autoequivalence of D(A).(3) The composition lt[−1] → lsr → r is an isomorphism of functors.(4) The composition r → rsl→ fl[1] is an isomorphism of functors.

Then all four hold. If this is the case, we call the object S ∈ D(A-B)spherical.

The following result will be useful for our purpose:

Theorem 2.12 ([Bar20, Theorem B]). Let A,B, C ∈ dgcatC be small dg-categories. Suppose that we have spherical functors D(A) → D(C) andD(B) → D(C) whose twists are tA and tB, respectively.

Then there exists a dg category R ∈ dgcatC with a semi-orthogonal de-composition D(R) = 〈D(A),D(B)〉, and a spherical functor D(R) → D(C)whose twist is tA ◦ tB.

3. Local systems of categories and perverse schobers

In this section, we recall the notion of perverse schobers on Riemannsurfaces, which categorifies perverse sheaves. We refer [Don19a, Don19b,KS15] for the details.

3.1. Local systems of categories.

Definition 3.1. Let M be a manifold and let Z ⊂ M be a finite subset.We define a Z-coordinatized local system of categories to be an action of thefundamental groupoid π1(M,Z) i.e., it consists of the following datum.


• A category Dz for each z ∈ Z,• An equivalence ρg : Dz → Dz′ for each path g ∈ π1(M,Z),• A natural isomorphism θg,h : ρgρh → ρgh for each pair (g, h) of com-posable paths,

such that the diagrams

(3.1) ρgρhρkρgθh,k

//

θg,hρk

��

ρgρhk

θg,hk

��ρghρk

θgh,k// ρghk

commute for all composable paths g, h, k ∈ π1(M,Z).

3.2. Quiver description of perverse sheaves on a disk. To motivatethe definition of perverse schobers, we first recall the quiver description ofthe category of usual perverse sheaves on a disk.

Let ∆ be the unit disk, B = {b1, · · · , bn} ⊂ ∆ a finite subset. We denoteby Perv(∆, B) the category of perverse sheaves on ∆ singular at B (i.e.,whose restrictions to ∆ \B are local systems).

Definition 3.2. We define Pn to be the category of data (D,Di, ui, vi)ni=1

consisting of finite dimensional vector spaces D,Di and linear maps ui : D →Di, vi : Di → D such that the endomorphsms Ti := id−vi ◦ ui : D → D areisomorphisms for all i = 1, · · · , n.

Fix a point p ∈ ∂∆. A skeleton is a union of simple arcs joining p andbi for all bi ∈ B, coninciding near p. Let C be the set of isotopy classes ofskeletons. We will see that there are equivalences between the categoriesPerv(∆, B) and Pn parametrized by the set C, compatible with the naturalaction of the Artin braid group

Brn := 〈s1, · · · , sn−1 : sisi+1si = si+1sisi+1〉 .

For a generator si ∈ Brn, we define an autoequivalence fsi : Pn → Pn asfollows: For an object (D,Dj , uj , vj)

nj=1 ∈ Pn, we define

fsi(D,Dj , uj , vj) := (D′,D′j , u

′j , v

′j),

where

(3.2)

D′ = D, D′j = Dj , u′j = uj, v′j = vj (j 6= i, i+ 1),

D′i+1 = Di, D′

i = Di+1,

u′i = ui+1, v′i = vi+1, u′i+1 = uiTi+1, v′i+1 = T−1i+1vi.

For a general element σ ∈ Brn, we then define an equivalence fσ : Pn → Pnas the composition of fsi’s and their inverses.

Proposition 3.3 ([GMV96]). For each class K ∈ C, there is an equivalence

FK : Perv(∆, B) → Pn


of categories, such that, for every σ ∈ Brn, the following diagram commutes:

Pnfσ

// Pn

Perv(∆, B).

FK

ee❑❑❑❑❑❑❑❑❑❑ Fσ(K)

99ssssssssss

3.3. Perverse schobers on Riemann surfaces. First we define the no-tion of perverse schobers on a disk. We keep the notations in the previoussubsection.

Definition 3.4 ([KS15, Don19b]). (1) A coodinatized schober on (∆, B)is a datumS = (D,Di, Si)

ni=1 consisting of triangulated dg categories

D,Di and spherical functors Si : Di → D. For each i, we denote byRi the right adjoint of Si, and by Ti := Cone(SiRi → idD) thecorresponding twist equivalence.

(2) For a given coordinatized schober S and an element σ ∈ Brn,we define a new coordinatized schober fσ(S) as in (3.2), replacing(D,Di, ui, vi) with (D,Di, Ri, Si), and endomorphisms Ti = id−viuiwith the twist functors Ti.

(3) A perverse schober on (∆, B) is a collection (SK)K∈C of coordina-tized schobers SK such that for each σ ∈ Brn, we have a compatibleidentification

fσSK∼−→ Sσ(K).

We now define the notion of perverse schobers on Riemann surfaces. LetΣ be a Riemann surface, B ⊂ Σ a finite set of points. Take a disk ∆ ⊂ Σcontaining the set B. Denote by U the closure of Σ \∆.

We fix a point p ∈ ∂∆ and a finite set Y ⊂ Σ \B. We set Z := Y ∪ {p}.

Definition 3.5 ([KS15, Don19b]). A perverse schober on (Σ, B) is a datumconsisting of

• a perverse schober S∆ on (∆, B),• a Z-coordinatized local system L of categories on U ,

such that the induced {p}-coordinatized local systems S∆|∂∆ and L|∂∆ areisomorphic.

The following result is very useful for our purpose:

Proposition 3.6 ([Don19b, Proposition 4.10]). Fix a point p ∈ Σ \ B.Let SΣ\{p} be a schober on (Σ \ {p}, B), T : D → D be the correspondingmonodromy autoequivalence around the point p.

Given a presentation of the equivalence T as the twist of a spherical func-tor, we obtain a schober on (Σ, B ∪ {p}).

Remark 3.7. Note that there are many ways presenting an equivalence T asa spherical twist, and a schober on Σ in the above proposition depends onsuch a choice. We will find the most natural presentations in our setting.

Finally we recall the notion of Calabi-Yau perverse schobers:

Definition 3.8 ([KSS20, Definitions 3.1.8 and 3.1.14]). Let n ∈ Z>0 be apositive integer.


(1) A perverse schober on (∆, 0), represented by a spherical functorD1 → D, is n-Calabi-Yau (CY) if the following conditions hold:(a) The category D is CY of dimension n,(b) The shifted cotwist f [n+ 1] is the Serre functor of D1.

(2) A perverse schober S on (Σ, B) is n-Calabi-Yau if the followingconditions hold:(a) The restriction S|Σ\B is a local system of n-CY categories, and

the monodromy autoequivalences preserve the CY structure,(b) For each point b ∈ B, the restriction of S to a disk around b is

n-CY.

Remark 3.9. The notion of n-CY pervrese schobers categorifies perversesheaves with (anti-)symmetric structures with respect to the Verdier duality,see [KSS20, Proposition 1.3.22]. In particular, these perverse sheaves areVerdier self-dual.

3.4. Spherical pairs. Consider the special case where Σ = ∆ and B = {0}.In this case, we have another categorification of perverse sheaves on (∆, 0),called spherical pair:

Definition 3.10. A spherical pair is a pair of semi-orthogonal decomposi-tions

E0 = 〈E⊥− , E−〉 = 〈E⊥

+ , E+〉

of a triangulated category E0 such that the compositions of the inclusionsand the projections

E⊥± → E⊥

∓ , E± → E∓

are equivalences.

Remark 3.11. As proved in [KS15, Proposition 3.7], given a spherical pair,we obtain a spherical functor E− → E⊥

+ . Hence a spherical pair induces aperverse schober on (∆, 0).

As we will see in Section 8, the theory of variations of GIT quotients,developed by [BFK19, HL15], provides natural classes of spherical pairs.

4. Derived factorization category

4.1. Derived factorization category. In this subsection, we recall thenotion of derived factorization categories following [BFK14].

Let G be an affine algebraic group acting on a variety X, and χ : G→ C∗

be a character of G. Let W : X → C be a χ-semi-invariant regular function,i.e. W (g·x) = χ(g)W (x) for any g ∈ G and any x ∈ X. The data (X,χ,W )G

is called a gauged Landau-Ginzburg model. Denote the character invertiblesheaf of χ on X by OX(χ). First, we define the dg category factG(X,χ,W )of factorizations of (X,χ,W )G.

Definition 4.1. A factorization of (X,χ,W )G is a sequence

(4.1) E =(

E1ϕE1−−→ E0

ϕE0−−→ E1(χ))

of morphisms of G-equivariant coherent sheaves on X such that

ϕE0 ◦ ϕE1 =W · idE1 , ϕE1 (χ) ◦ ϕ

E0 =W · idE0 .


A factorization E =(

E1ϕE1−−→ E0

ϕE0−−→ E1(χ))

of (X,χ,W )G is called a

locally free factorization if E1 and E0 are G-equivariant locally free sheaveson X.

Definition 4.2. Let

E =(

E1ϕE1−−→ E0

ϕE0−−→ E1(χ))

, F =(

F1ϕF1−−→ F0

ϕF0−−→ F1(χ))

be factorizations of (X,χ,W )G. We define the Z-graded C-vector spaceHomfactG(X,χ,W )(E,F ) of morphisms from E to F as follows. For an integerl, we define

Hom2lfactG(X,χ,W )(E,F ) := Hom(E1, F1(χ

l))⊕Hom(E0, F0(χl)),

Hom2l+1factG(X,χ,W )(E,F ) := Hom(E1, F0(χ

l))⊕Hom(E0, F1(χl+1)).

Then we define

HomfactG(X,χ,W )(E,F ) :=⊕

n∈Z

HomnfactG(X,χ,W )(E,F ).

Moreover, we define the linear map

dE,F : HomfactG(X,χ,W )(E,F ) → HomfactG(X,χ,W )(E,F )

as follows. Take f = (f1, f0) ∈ HomnfactG(X,χ,W )(E,F ). When n = 2l for

some integer l, we put

dE,F (f) := (ϕF1 (χl) ◦ f1 − f0 ◦ ϕ

E1 , ϕ

F0 (χ

l) ◦ f0 − f1(χ) ◦ ϕE0 ).

When n = 2l + 1 for some integer l, we put

dE,F (f) := (ϕF0 (χl) ◦ f1 + f0 ◦ ϕ

E1 , ϕ

F1 (χ

l+1) ◦ f0 + f1(χ) ◦ ϕE0 ).

Then d2E,F = 0 holds and the degree of dE,F is one. We have a dg C-module

(HomfactG(X,χ,W )(E,F ), dE,F ).

Thus, we have the dg category of factorizations of (X,χ,W )G.

Definition 4.3. Denote the dg category of factorizations of (X,χ,W )G byfactG(X,χ,W ). Let vectG(X,χ,W ) be the dg subcategory of locally freefactorizations of (X,χ,W )G.

We obtain triangulated categories from the dg categories in Definition 4.3.

Proposition 4.4 ([BFK14, Proposition 3.7]). The dg categories factG(X,χ,W )and vectG(X,χ,W ) are triangulated dg categories.

Next, we define the derived factorization categories Dabs factG(X,χ,W )and Dabs[factG(X,χ,W )] of (X,χ,W )G.

Remark 4.5. The category Z0(factG(X,χ,W )) has the structure of an abeliancategory, where the category Z0(factG(X,χ,W )) is defined as follow. Theobjects of Z0(factG(X,χ,W )) are same as factG(X,χ,W ). For objectsE,F ∈ Z0(factG(X,χ,W )), we define HomZ0(A)(E,F ) := Kerd0E,F .

To introduce the notion of acyclic objects, we need the totalization of anobject in Z0(factG(X,χ,W )).


Definition 4.6. Let E• = (· · · → Eiδi−→ Ei+1 → · · ·) be a complex of

objects in the abelian category Z0(factG(X,χ,W )). For i ∈ Z, we put

Ei =(

Ei1ϕE

i

1−−→ Ei0ϕE

i

0−−→ Ei1(χ))

.

We define G-equivariant quasi-coherent sheaves T1, T0 on X as

T1 :=⊕

l∈Z

(

E−2l−10 (χl)⊕ E−2l−2

1 (χl+1))

,

T0 :=⊕

l∈Z

(

E−2l0 (χl)⊕ E−2l−1

1 (χl+1))

.

We define morphisms t1, t0 of G-equivariant quasi-coherent sheaves as

t1 :=∑

l∈Z

((

δ−2l−10 (χl)− ϕE

−2l−1

0 (χl))

⊕(

δ−2l−21 (χl+1) + ϕE

−2l−2

1 (χl+1)))

,

t0 :=∑

l∈Z

((

δ−2l0 (χl) + ϕE

−2l

0 (χl))

⊕(

δ−2l−11 (χl+1)− ϕE

−2l−1

1 (χl+1)))

.

The totalization Tot(E•) ∈ factG(X,χ,W ) of E• is an object defined as

Tot(E•) =(

T1t1−→ T0

t0−→ T1(χ))

.

Definition 4.7. A factorization E ∈ factG(X,χ,W ) is called acyclic if thereis an acyclic complex E• of objects in Z0(FactG(X,χ,W )) such that E isisomorphic to Tot(E•) in Z0(factG(X,χ,W )). Let acycG(X,χ,W ) be thefull dg subcategory of acyclic factorizations in factG(X,χ,W ).

Definition 4.8. We define the absolute derived categoryDabs[factG(X,χ,W )]of [factG(X,χ,W )] to be the Verdier quotient

Dabs[factG(X,χ,W )] := [factG(X,χ,W )]/[acycG(X,χ,W )].

Definition 4.9. We define the absolute derived categoryDabs factG(X,χ,W )of factG(X,χ,W ) to be the dg quotient

Dabs factG(X,χ,W ) := factG(X,χ,W )/ acycG(X,χ,W ).

The categories Dabs factG(X,χ,W ) and Dabs[factG(X,χ,W )] are also calledthe derived factorization categories of (X,χ,W )G.

Proposition 4.10 ([BFK14, Corollary 5.9, Proposition 5.11]). The dg cat-egory Dabs factG(X,χ,W ) is a dg enhancement of Dabs[factG(X,χ,W )].If X is smooth affine and G is reductive, the the canonical dg functorvectG(X,χ,W ) → Dabs factG(X,χ,W ) is a quasi-equivalence.

4.2. Derived functors. In this subsection, we recall the construction ofderived functors which we will use later. Let X and Y be smooth affinevarieties, and G a reductive affine algebraic group acting on X and Y . Letf : X → Y be a G-equivariant morphism. Take a character χ : G → C∗

of G. Let W : Y → C be a χ-semi-invariant regular function on Y . Then(Y, χ,W )G and (X,χ, f∗W )G are gauged Landau-Ginzburg models.


Definition 4.11. The dg functor f∗ : vectG(Y, χ,W ) → vectG(X,χ, f∗W )

is defined as follows. For an object E ∈ vectG(Y, χ,W ), we define an object

f∗(E) :=(

f∗E1f∗ϕE1−−−→ f∗E0

f∗ϕE0−−−→ f∗E1(χ))

∈ vectG(X,χ, f∗W ).

For a morphism p = (p1, p0) ∈ HomvectG(Y,χ,W )(E,F ), we define a morphism

f∗p := (f∗p1, f∗p0) ∈ HomvectG(X,χ,f∗W )(f

∗E, f∗F ).

By Proposition 4.10, we have the exact functor

Dabs[factG(Y, χ,W )] → Dabs[factG(X,χ, f∗W )]

and denote it by f∗.

Definition 4.12. Assume that f is a closed immersion. We define the dgfunctor f∗ : factG(X,χ, f

∗W ) → factG(Y, χ,W ) as follows. For an objectE ∈ factG(X,χ, f

∗W ), we define an object

f∗(E) :=(

f∗E1f∗ϕE1−−−→ f∗E0

f∗ϕE0−−−→ f∗E1(χ))

∈ factG(Y, χ,W ).

For a morphism p = (p1, p0) ∈ HomfactG(X,χ,f∗W )(E,F ), we define a mor-phism

f∗p := (f∗p1, f∗p0) ∈ HomfactG(Y,χ,W )(f∗E, f∗F ).

Since f is a closed immersion, f : factG(X,χ, f∗W ) → factG(Y, χ,W ) sends

acyclic factorizations to acyclic factorizations. Therefore, we obtain thedg functor f∗ : Dabs factG(X,χ, f

∗W ) → Dabs factG(Y, χ,W ) and the exactfunctor f∗ : D

abs[factG(X,χ, f∗W )] → Dabs[factG(Y, χ,W )].

Definition 4.13. Let (X,χ,W )G be a gauged Landau-Ginzburg model.Let L be a G-equivariant line bundle on X. We define the dg functor− ⊗ L : factG(X,χ,W )

∼−→ factG(X,χ,W ) as follows. For an object E ∈

factG(X,χ,W ), we define an object

E ⊗ L :=(

E1 ⊗ LϕE1 ⊗idL−−−−−→ E1 ⊗ L

ϕE0 (χ)⊗idL−−−−−−−→ E1 ⊗ L(χ)

)

.

For a morphism f = (f1, f0) ∈ HomfactG(X,χ,W )(E,F ), we define a morphism

f ⊗ L := (f1 ⊗ idL, f0 ⊗ idL) ∈ HomfactG(X,χ,W )(E ⊗ L,F ⊗ L).

Restricting this dg functor to the full dg subcategory vectG(X,χ,W ), we

obtain the dg functor −⊗L : vectG(X,χ,W )∼−→ vectG(X,χ,W ). Note that

it induces the autoequivalence

−⊗ L : Dabs[factG(X,χ,W )]∼−→ Dabs[factG(X,χ,W )].

More generally, for E ∈ factG(X,χ,W ), we have the derived tensor functor

−⊗L E : Dabs[factG(X,χ, 0)] → Dabs[factG(X,χ,W )]

by [Hir17, Proposition 4.23]. By [Hir17, Definition 3.14], there is the exactfunctor Υ : Db(CohG(X)) → Dabs[factG(X,χ, 0)] sending a G-equivariantcoherent sheaf A to the factorization (0 → A → 0). For simplicity, we alsodenote the composition Υ(−)⊗L E by −⊗L E as in [Hir17, Definition 4.24].


4.3. Koszul factorizations. Let (X,χ,W )G be a gauged Landau-Ginzburgmodel. Assume that X is smooth. Let E be a G-equivariant locally free sheafon X of finite rank. Let s : E → OX and t : OX → E(χ) be morphisms of G-equivariant coherent sheaves such that t◦s =W · idE and s(χ)◦t =W · idOX .Let Zs ⊂ X be the zero scheme of the section s ∈ H0(E∨)G. The section sis called regular if the codimension of Zs in X is equal to the rank of E .

Definition 4.14. We define theKoszul factorizationK(s, t) ∈ vectG(X,χ,W )of s and t as

K(s.t) :=(

K1ϕK1−−→ K0

ϕK0−−→ K1(χ))

,

where

K1 :=⊕

n≥0

(

2n+1∧

E)

(χn) , K0 :=⊕

n≥0

(

2n∧

E)

(χn)

and

ϕK0 , ϕK1 := ∨s+ t ∧ −.

We will use the following lemma later:

Lemma 4.15 ([BFK14, Proposition 3.20, Lemma 3.21]). The followingstatements hold:

(1) There is the natural isomorphism

K(s, t)∨ ≃ K(t∨, s∨).

(2) If s is regular, there are the natural isomorphisms

OZs ≃ K(s, t), OZs ⊗rk E∧

E [− rk E ] ≃ K(s, t)∨

in Dabs[factG(X,χ,W )]

4.4. Knorrer periodicity. Let X be a smooth quasi-projective variety.Consider the trivial action of C∗ on X. Let E be a G-equivariant locally freesheaf on X of finite rank. Take a G-invariant regular section s ∈ H0(E∨)G.Then we have the χ1-semi-invariant regular function Qs : V (E(χ1)) → C

induced by s. Let V (E(χ1)) := SpecX(Sym(E(χ)∨)) be the G-equivariantvector bundle on X. Take the restriction p : V (E(χ1))|Zs → Zs of the G-equivariant vector bundle q to Zs. Then there is the following commutativediagram:

V (E(χ1))|Zsi

//

p

��

V (E(χ1))

q

��

Zsj

// X.

Shipman [Shi12] proved the following theorem. See also [Isi13], [Hir17,Theorem 4.1].

Theorem 4.16 ([Shi12, Theorem 3.4]). We have the equivalence

(4.2) i∗ ◦ p∗ : Dabs[factC∗(Zs, χ1, 0)]

∼−→ Dabs[factC∗(V (E(χ1)), χ1, Qs)].


The equivalence in Theorem 4.16 is called the Knorrer periodicity. By[Hir17, Proposition 2.14], there is the canonical equivalence

(4.3) Db(Zs)∼−→ Dabs[factC∗(Zs, χ1, 0)].

We also denote by i∗ ◦p∗ the composition of the equivalences (4.2) and (4.3).

4.5. Graded matrix factorizations. Let Sn := C[x1, · · · , xn] be the poly-nomial ring. We regard Sn as a graded C-algebra, where the degree of xiis one for each i = 1, · · · , n. Let gr(Sn) be the category of finitely gener-ated graded Sn-modules whoese morphisms are degree preserving homomor-phisms. Let grfr(Sn) ⊂ gr(Sn) be the full subcategory of finitely generatedgraded free Sn-modules.

We recall the definition of graded matrix factorizations. Fix a positiveinteger d ∈ Z>0. Let W ∈ Sn be a homogeneous polynomial of degree d.

Definition 4.17. A graded matrix factorization of W is a sequence

(4.4) M =(

M1ϕM1−−→M0

ϕM0−−→M1(d))

,

where M1 and M0 are finitely generated free graded Sn-modules and ϕM1and ϕM0 are degree preserving homomorphisms such that

ϕM0 ◦ ϕM1 =W · idM1 , ϕF1 (d) ◦ ϕ

F0 =W · idM0 .

As similar to Definitions 4.2 and 4.3, we can define the dg categoryMFgr(W ) of graded matrix factorizations of W . Let HMFgr(W ) be the ho-motopy category [MFgr(W )]. For an integer l ∈ Z and a graded Sn-moduleP = ⊕i∈ZPi, we define the graded Sn-module P (l) = ⊕i∈ZP (l)i by P (l)i :=

Pi+l. Then we obtain the degree shift functors (l) : grfr(S)∼−→ grfr(S) and

(l) : MFgr(W )∼−→ MFgr(W ). We put τ := [(1)] : HMFgr(W )

∼−→ HMFgr(W ).

4.6. Matrix factorizations and Landau-Ginzburg models. Fix a pos-itive integer d and a homogeneous polynomialW ∈ Sn of degree d. Considerthe action of C∗ on the affine space Cn defined by

λ · (x1, · · · , xn) := (λx1, · · · , λxn)

for λ ∈ C∗ and (x1, · · · , xn) ∈ Cn. Let χd : C∗ → C∗ be the character ofC∗ defined by χd(t) := td. Then W : Cn → C is a χd-semi-invariant regularfunction. The data (Cn, χd,W )C

∗

is a gauged Landau-Ginzburg model.Denote the category of C∗-equivariant locally free coherent sheaves on

Cn by vectC∗(Cn). Taking global sections, we have the equivalence Γ :

vectC∗(Cn)∼−→ grfr(S). Note that, for E ∈ vectC∗(Cn), the Sn-module

Γ(E) has the structure of a graded Sn-module induced by the weight de-composition with respect to the induced action of C∗ on Γ(E). This in-

duces the equivalence Γ : vectC∗(Cn, χd,W )∼−→ MFgr(W ) of dg categories.

Hence by Proposition 4.10, we have the quasi-equivalence MFgr(W ) →Dabs factC∗(Cn, χd,W ). Taking homotopy categories, we obtain the equiva-lence

HMFgr(W )∼−→ Dabs[factC∗(Cn, χd,W )].

Remark 4.18. By [Orl09, Theorem 40] and [BDF+18, Lemma 4.8], the tri-angulated category HMFgr(W ) is idempotent complete.


4.7. Serre functor. Fix a positive integer d and a homogeneous polynomialW ∈ Sn of degree d. The Serre functor of HMFgr(W ) is described as follows.

Theorem 4.19 ([KST09, Theorem 3.8]). The Serre functor S of HMFgr(W )is given by S := τd−n[n− 2].

Using Theorem 4.19, we describe the Serre functor of Dabs[factG(Cn, χd,W )].

Note that the following diagram commutes:

(4.5) vectC∗(Cn, χd,W )Γ

//

−⊗OCn(χ1)��

MFgr(W )

(1)��

vectC∗(Cn, χd,W )Γ

// MFgr(W ).

Taking homotopy categories, we obtain the commutative diagram:

(4.6) Dabs[factC∗(Cn, χd,W )]∼

//

−⊗OCn(χ1)��

HMFgr(W )

τ

��

Dabs[factC∗(Cn, χd,W )]∼

// HMFgr(W ).

By Theorem 4.19 and (4.6), we have the following.

Theorem 4.20 ([KST09, Theorem 3.8]). The functor

−⊗OCn(χd−n)[n− 2] : Dabs[factC∗(Cn, χd,W )]∼−→ Dabs[factC∗(Cn, χd,W )]

is the Serre functor.

5. Construction of perverse schobers

Let n ≥ 1 be a positive integer, W ∈ C[x1, · · · , xn+2] be a homogeneouspolynomial of degree n + 2. We take the polynomial W general so thatthe hypersurface X := (W = 0) ⊂ Pn+1 is smooth, which is a projectiveCalabi–Yau variety of dimension n. We denote by OX(1) := OPn+1(1)|X .

In this section, we construct a perverse schober using the following Orlov’stheorem:

Theorem 5.1 ([Orl09]). There exists an equivalence

(5.1) ψ : Db(X)∼−→ HMFgr(W )

between triangulated categories. Moreover, the following diagram commutes:

(5.2) Db(X)ψ

//

STOX◦(−⊗OX(1))

��

HMFgr(W )

τ

��

Db(X)ψ

// HMFgr(W ).

Remark 5.2. More precisely, we have equivalences

ψw : Db(X) → HMFgr(W )

indexed by integers w ∈ Z, and we have the commutative diagram (5.2) fora particular choice of w ∈ Z. See Section 8 for more detail.


Take a disk ∆ ⊂ P1 \ {∞, 0} containing the point 1. We fix two distinctpoints x± ∈ ∆ \ {1}. First we construct a {x±}-coordinatized local systemof categories on P1 \ {0, 1,∞}.

Let a, b ∈ π1(P1 \{0, 1,∞}, x−) be simple loops around ∞, 1, respectively.

Let γ be a path from x− to x+ which is contained in ∆ and does not goaround the point 1 ∈ ∆. See Figure 1 below. Then the groupoid π1(P

1 \{0, 1,∞}, {x±}) is freely generated by the three paths a, b, γ.

We first prepare the following two lemmas:

Lemma 5.3. There is a {x−}-coodinatized local system K on P1 \ {0, 1,∞}with assignments

x− 7→ Db(X),

a 7→ (−)⊗OX(1), b 7→ STOX .

Proof. Consider a homomorphism

(5.3) G := π1(P1 \ {0, 1,∞}, {x−}) → AuteqDb(X),

which sends the generators a, b to the autoequivalences (−)⊗OX(1), STOX ,respectively.

By [BO20, Theorem 2.1 (1)] (see also Section 2.3 in the same paper), thereexists an obstruction class o ∈ H3(G,C∗) for lifting the homomorphism (5.3)to an action of G on the category Db(X). Since G = π1(P

1\{0, 1,∞}, {x−})is a free group, we have the vanishingH3(G,C∗) = 0, and hence the assertionholds. �

Remark 5.4. Although [BO20, Theorem 2.1] is stated only for finite groups,its proof works for infinite groups without any modifications.

Lemma 5.5 ([BO20, Lemma 2.2]). Let f, f ′ : B → C, g, g′ : A → B befunctors, and let α : f → f ′, β : g → g′ be natural transforms. Then thefollowing diagram commutes:

f ◦ gαg

//

fβ��

f ′ ◦ g

f ′β��

f ◦ g′αg′

// f ′ ◦ g′.

Proposition 5.6. There is an {x±}-coordinatized local system L on P1 \{0, 1,∞} with assignments

(5.4)x− 7→ Db(X), x+ 7→ HMFgr(W ),

a 7→ (−)⊗OX(1), b 7→ STOX , γ 7→ ψ.

Proof. Let K be the local system of categories constructed in Lemma 5.3.By definition, it consists of the data

µg′ : Db(X)

∼−→ Db(X), νg′,h′ : µg′µh′

∼−→ µg′h′


for all elements g′, h′ ∈ π1(P1 \ {0, 1,∞}, {x−}). Moreover, we have the

following commutative diagrams

(5.5) µg′µh′µk′µg′νh′,k′

//

µg′,h′µk′

��

µg′µh′k′

νg′,h′k′

��µg′h′µk′

νg′h′,k′// µg′h′k′

for all g′, h′, k′ ∈ π1(P1 \ {0, 1,∞}, {x−}).

We extend this local system K to an {x±}-coordinatized local system L.First observe that any morphism g in the groupoid π1(P

1 \ {0, 1,∞}, {x±})can be uniquely written as follows:

(5.6) g = γǫg′γδ, ǫ ∈ {0, 1}, δ ∈ {0,−1}, g′ ∈ π1(P1 \ {0, 1,∞}, {x−}).

To the morphism (5.6), we associate the functor

ρg := ψǫ ◦ µg′ ◦ ψδ .

For each pair g, h of composable morphisms, we associate a natural trans-form θg,h : ρgρh → ρgh as follows:

• When g = γ, h = γ−1, then we define θγ,γ−1 to be the adjoint counit

ηψ−1 : ψ ◦ ψ−1 → idHMFgr(W ) .

• When we have g = γǫg′γ−1, h = γh′γδ for some ǫ,−δ ∈ {0, 1},g′, h′ ∈ π1(P

1 \ {0, 1,∞}, {x−}), then we define θg,h as

θg,h := νg′,h′ηψ : ρgρh → ρgh,

where ηψ : ψ−1 ◦ ψ → idDb(X) is the adjoint counit.

• Otherwise, we put θg,h := νg′,h′ .

We need to check the commutativity of the diagram (3.1) for all compos-able paths g, h, k. We prove it only for the case when we have h = γh′ withh′ ∈ π1(P

1 \ {0, 1,∞}, {x−}). The other cases can be proved in a similarway. As in (5.6), we write g = γǫg′γ−1. k = k′γδ.

Consider the following diagram:

(ψǫµg′ψ−1)(ψµh′)(µk′ψ

δ)θg,h

//

ηψ**❚❚

❚❚❚❚

❚❚❚❚

❚❚❚❚

❚❚

θh,k=νh′,k′

��

(ψǫµg′h′)(µk′ψδ)

θgh,k=νg′h′,k′

��

(ψǫµg′)µh′(µk′ψδ)

νg′,k′

55❧❧❧❧❧❧❧❧❧❧❧❧❧❧

νh′,k′

��

(ψǫµg′)(µh′k′ψδ)

νg′,h′k′

))❘❘❘

❘❘❘❘

❘❘❘❘

❘❘❘

(ψǫµg′ψ−1)(ψµh′k′ψ

δ)θg,hk

//

ηψ44❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥

ψǫµg′h′k′ψδ .

The upper and the lower triangles are commutative by the definitions of θg,hand θg,hk; the left square is commutative by Lemma 5.5; the right square


is commutative by the commutativity of (5.5). Hence we conclude that thewhole diagram commutes as required. �

ψ

Db(X) HMFgr(W )

(−)⊗OX(1)

STOX

Figure 1. Local system of categories

Next we shall extend the local system L of categories to a perverse schoberon P1. Let us take a smooth hyperplane section C ∈ |OX(1)|, and denoteby i : C → X the natural inclusion. We also take a general linear sectionj : Cn+1 → Cn+2. Note that by using the equivalences (see (4.6))

HMFgr(W ) ≃ Dabs[

factC∗(Cn+2, χn+2,W )]

,

HMFgr(W |Cn+1) ≃ Dabs[

factC∗(Cn+1, χn+2,W |Cn+1)]

,

we have the natural push-forward functor j∗ : HMFgr(W |Cn+1) → HMFgr(W ).

Theorem 5.7. There exists a perverse schober P on (P1, {0, 1,∞}) withthe following properties:

(1) The restriction of the schober P to P1 \ {0, 1,∞} coincides with thelocal system L of categories constructed in Proposition 5.6,

(2) Restricting the schober P to small disks around ∞, 1, 0, we obtainthe spherical functors

(5.7)

i∗ : Db(C) → Db(X),

Db(pt) → Db(X), C 7→ OX ,

ψ−1 ◦ j∗ : HMFgr(W |Cn+1) → HMFgr(W )∼−→ Db(X),


respectively.

Remark 5.8. Our choices of spherical functors (5.7) are compatible with thegeneral result in Theorem 2.12. Indeed, by [Orl09], we have an SOD

HMFgr(W |Cn+1) = 〈Db(C),Db(pt)〉,

and the twist of the spherical functor ψ−1 ◦ j∗ is exactly the compositionSTOX ◦(− ⊗OX(1)).

Our point is that we have explicit descriptions of the source categoriesof spherical functors in terms of derived categories and matrix factorizationcategories, rather than abstract dg-categories.

Proof of Theorem 5.7. We apply Proposition 3.6 to the local system L threetimes. It is enough to show that the functors (5.7) are spherical and theirtwists coincide with the autoequivalences (−)⊗OX(1),STOX ,STOX ◦((−)⊗OX(1)), respectively.

We only prove the assertion for the functor

(5.8) ψ−1 ◦ j∗ : HMFgr(W |Cn+1) → Db(X),

namely, we show that it lifts to a spherical dg functor and the correspondingtwist autoequivalence of Db(X) is isomorphic to STOX ◦(− ⊗ OX(1)). Wetake dg enhancements of the categories HMFgr(W ) and HMFgr(W |Cn+1) tobeDabs factC∗(Cn+2, χn+2,W ) andDabs factC∗(Cn+1, χn+2,W |Cn+1), respec-tively. Note that they also give Morita enhancements, since the categoriesDb(X) and HMFgr(W |Cn+1) are idempotent complete. Since j : Cn+1 →Cn+2 is a closed embedding, we have the functor

j∗ : Dabs factC∗(Cn+1, χn+2,W |Cn+1) → Dabs factC∗(Cn+2, χn+2,W ),

which gives a dg enhancement of the functor (5.8).Next we construct dg lifts of the left and right adjoint functors

j∗ ◦ ψ, j! ◦ ψ : Db(X) → HMFgr(W |Cn+1).

For this we take dg enhancements of HMFgr(W ) and HMFgr(W |Cn+1) asvectC∗(Cn+2, χn+2,W ) and vectC∗(Cn+1, χn+2,W |Cn+1), respectively. Thenwe have the dg enhancement

j∗ : vectC∗(Cn+2, χn+2,W ) → vectC∗(Cn+1, χn+2,W |Cn+1).

Similarly, we have the enhancement of the functor j! = j∗(−)⊗O(χ1)[−1],where O(χ1) is the C∗-equivariant line bundle of weight 1. Note that thedg enhancements Dabs fact(−) and vect(−) are quasi-equivalent (see Propo-sition 4.10).

Let us consider the twist T of j∗, which fits into the exact triangle

(5.9) j∗ ◦ j! → id → T

of quasi-functors on Dabs factC∗(Cn+2, χn+2,W ). Applying the underlyingexact functors of (5.9) to an object E ∈ HMFgr(W ), we obtain a triangle

(5.10) E(χ1)|Cn+1 [−1] → E → T (E)

in HMFgr(W ). The exact triangle (5.10) is isomorphic to the triangle ob-tained by tensoring E to the sequence O(χ1)|Cn+1 [−1] → O → O(χ1). In


other words, we have isomorphisms

(5.11) [T ] ≃ (−⊗O(χ1)) ≃ τ

between functors. In particular, the functor [T ] is an autoequivalence ofHMFgr(W ). Similarly, we can check that the endofunctor [F ] of HMFgr(W |Cn+1)is an equivalence, where F is the cotwist of j∗ defined by the triangle

F → id → j! ◦ j∗.

Indeed, we can show that

(5.12) [F ] ≃ SHMFgr(W |Cn+1)[−n− 1],

where S(−) denotes the Serre functor.The above arguments show that the push-forward functor j∗ is a spherical

functor. Moreover, its twist is isomorphic to the degree shift functor by(5.11). Via the equivalence ψ : Db(X) ≃ HMFgr(W ), the degree shift functorτ corresponds to the autoequivalence STOX ◦(− ⊗ OX(1)) as required (seeTheorem 5.1). �

Proposition 5.9. The schober P constructed in Theorem 5.7 is n-Calabi-Yau.

Proof. The local system L clearly satisfies the Calabi-Yau property.By the isomorphism (5.12), the schober P restricted to a disk around 0 is

n-Calabi-Yau. Similarly, one can check that it is n-Calabi-Yau around thepoints ∞, 1. �

6. Decategorification

In this section, we prove that the decategorifications of our perverseschobers coincide with the intersection complexes.

6.1. Intersection complex. We first recall the description of intersectioncomplexes under the equivalence in Proposition 3.3:

Proposition 6.1 ([Don19a, Proposition 3.2]). Given a local system L on∆ \ {0} with fiber F and monodromy m, the intersection complex IC(L) ∈Perv(∆, 0) corresponds to the diagram

(6.1) F/Fmid−m

//F,

uoo

where Fm ⊂ F denotes the m-invariant part, and u : F → F/Fm denotesthe quotient map.

Remark 6.2. In [Don19b], Donovan uses a slightly different version of thequiver description of the category Perv(∆, 0). Namely, our category P1 isthe category of data (D,D1, u, v) such that id−v ◦u is an isomprphism. Onthe other hand, Donovan [Don19b] considers the condition that id+v ◦ u isan isomprphism. As a result, the map F/Fm → F in (6.1) is id−m in ourcase, while it is m− id in [Don19b].

The following lemma is useful for our purpose:


Lemma 6.3. Let X be a smooth projective variety, U ⊂ X be an opensubset, and L be a local system on U .

Given a perverse sheaf P on X, the following conditions are equivalent:

(1) We have an isomorphism P ≃ IC(L).(2) There exists an analytic open covering X = ∪iVi such that for every

i, we have an isomorphism P |Vi ≃ IC(L|Vi∩U ) ∈ Perv(Vi).

Proof. This is a well-known property, and follows from the characterizationof the intersection complexes given in [Max19, Definition 8.4.3 (2)], togetherwith the fact that the category of perverse sheaves forms a stack in analytictopology, see e.g., [Max19, Remark 8.2.11]. �

6.2. Decategorification of perverse schobers. In this subsection, weconsider the decategorification of our schober when n ≥ 2. Let H ∈|OPn+2(1)| be the hyperplane class. We also denote by H its restrictionsto X and C.

Definition 6.4. (1) We define the H-part ΛH(X), (resp.ΛH (C)) of thecohomology H∗(X,Q) (resp. H∗(C,Q)) to be the subring generatedby the hyperplane class H.

(2) We define the H-part ΛH(W |Cn+1) of the numerical Grothendieckgroup Knum (HMFgr(W |Cn+1))Q to be ΛH(C)⊕Q.

Remark 6.5. By [Orl09], we have an SOD

HMFgr(W |Cn+1) = 〈Db(C),Db(pt)〉.

Hence we have a natural embedding

ΛH(W |Cn+1) → Knum (HMFgr(W |Cn+1))Q .

The proof of the following lemma is straightforward:

Definition-Lemma 6.6. The following assertions hold:

(1) The local system L of categories in Proposition 5.6 induces a localsystem on P1 \ {0, 1,∞} whose fiber is ΛH(X). We denote it by L.

(2) The spherical functors (5.7) induce linear maps

(6.2)

i∗ : ΛH(C) → ΛH(X), i! : ΛH(X) → ΛH(C),

Q → ΛH(X), ΛH(X) → Q,

j∗ : ΛH(W |Cn+1) → ΛH(X), j! : ΛH(X) → ΛH(W |Cn+1),

which define perverse sheaves on (∆, 0).(3) The local system L in (1) and the data (6.2) define a perverse sheaf

P on P1.

We call L,P the decategorifications of L,P, respectively.

Theorem 6.7. Suppose that n ≥ 2. Let L,P be the decategorifications ofL,P as in Definition-Lemma 6.6.

Then we have an isomorphism P ≃ IC(L) ∈ Perv(P1). Moreover, theperverse sheaf P is Verdier self-dual.


Proof. By Lemma 6.3, it is enough to check the isomorphism of the perversesheaves IC(L) and P in the neighborhoods of the points 0, 1,∞. To showthis, we apply Proposition 6.1 to the data in (6.2).

We only prove the assertion for

(6.3) ΛH(W |Cn+1)j∗

//ΛH(X)

j!oo ,

namely, we prove that this is isomorphic to the diagram (6.1) with F :=ΛH(X) and m := STOX (− ⊗ OX(1)). We claim that Fm = 0 in this case,and hence the diagram (6.1) becomes

(6.4) ΛH(X)id−m

//ΛH(X).

idoo

To prove the claim, recall that we have the triangle

(6.5) RHom(OX ,−⊗OX (1))⊗OX → (−)⊗OX (1) → STOX (−⊗OX(1)).

Hence we have

m(~x) = ~x.eH − χ(~x.eH)(1, 0, · · · , 0)

for any ~x = (x0, · · · , xn) ∈ ΛH(X). Assume that ~x is fixed by m. Firstly,the second to the last components of ~x and m(~x) must be equal, which isonly possible when x0 = x1 = · · · = xn−1 = 0. Then the first componentof m(~x) is −χ(~x.eH) = −xn, while we have seen x0 = 0. Hence the m-fixedpart is trivial as claimed.

Next observe that we have the following morphism in the category P1 (cf.Definition 3.2):

ΛH(X)id−m

//

j!

��

ΛH(X)id

oo

id��

ΛH(W |Cn+1)j∗

//ΛH(X).

j!oo

Hence, to prove that the diagrams (6.3) and (6.4) are isomorphic, it isenough to show that the map j! : ΛH(X) → ΛH(C

n+1,W |Cn+1) is an iso-morphism. Since we assume that n ≥ 2, C ⊂ X is a connected smoothvariety, and the vector spaces ΛH(C

n+1,W |Cn+1) and ΛH(X) have the samedimension. Hence it is enough to show that j∗j

! = id−m is an isomor-phism. From the triangle (6.5), we can see that the map j∗j

! = id−m isrepresented by the following ((n + 1) × (n+ 1))-matrix with respect to thebasis 1,H,H2, · · · ,Hn ∈ ΛH(X):

B :=

χ(OX) ∗ · · · 0 −10

A ...0

,


where A is the lower-triangular (n× n)-matrix

A =

−1

−1/2 −1 0...

. . .. . .

−1/(n!) · · · −1/2 −1

.

Note that the first row of the matrix B is determined by the Todd classtdX = (1, 0, · · · , χ(OX)). We conclude that j!j∗ is an isomorphism, and inparticular so is j!.

The Verdier self-duality of P follows from Proposition 5.9, see also Remark3.9. �

Remark 6.8. When n = 1, X is an elliptic curve and C consists of 3 points.Then we would have dimΛH(W |C2) = 4 > 2 = dimΛH(X). Hence the abovecomputation shows that P and IC(L) are not isomorphic. This is relatedto the fact that OX(1) has degree three, and hence does not generate thePicard group when n = 1.

We will modify the construction in the next section to get the categorifi-cation of the intersection complex for elliptic curves.

Remark 6.9. Even when n ≥ 2, if we consider the full numerical K-groupsinstead of the H-parts, the result fails in general. Indeed, it may happenthat dimKnum(HMFgr(W |Cn+1))Q > dimKnum(X)Q.

7. Mirror symmetry for elliptic curves

In this section, we denote by X a smooth elliptic curve. We constructperverse schobers in both A- and B-models, which are identified via thehomological mirror symmetry for an elliptic curve.

A key observation is that, in the case of elliptic curves, our schober onlyconsists of spherical objects, while in higher dimension we have more generalspherical functors.

7.1. Perverse schober on the B-side. The following is a straightforwardmodification of the construction in Theorem 5.7:

Proposition 7.1. Fix a point p ∈ X. There exists a perverse schober PB

on (P1, {0, 1,∞}) satisfying the following properties:

(1) The schober PB restricted to the open subset P1\{0, 1,∞} is isomor-phic to a local system LB of categories with the fiber Db(X), definedby the assignments

a 7→ (−)⊗OX(p), b 7→ STOX ,

(2) Restricting the schober PB to small disks around ∞, 1, 0, we obtainthe spherical functors

Db(pt) → Db(X), C 7→ Op,

Db(pt) → Db(X), C 7→ OX ,

〈Db(pt),Db(pt)〉 → Db(X),

respectively.


(3) By taking the Grothendieck groups, we obtain a local system LB anda perverse sheave PB in the usual sense. We have PB ≃ IC(LB).

Proof. The proof is almost identical to that of Theorem 5.7. The first dif-ference is that we use (−) ⊗ OX(p) instead of the degree three line bundleOP2(1)|X . The second difference is that the skyscraper sheaf Op is sphericalfor an elliptic curve X, whose twist is isomorphic to (−)⊗OX(p).

Finally, to construct a spherical functor 〈Db(pt),Db(pt)〉 → Db(X) whosetwist is STOX ◦(− ⊗OX(p)), we use Theorem 2.12. �

7.2. Perverse schober on the A-side. Here, we construct the mirror per-verse schober using the homological mirror symmetry for an elliptic curve.

Let X = X/(Z ⊕ τZ) be an elliptic curve, where τ is an element of theupper half plane. Then its mirror is defined to be a pair X∨ := (T, ωC),where T = C/Z⊕2 is a torus, and ωC := τdx ∧ dy is a complexified Kahlerform. We denote by π : C → T the quotient map.

We briefly recall the definition of the Fukaya category Dπ Fuk(X∨), fol-lowing [PZ98]. An object of the Fukaya category Dπ Fuk(X∨) is isomorphicto a tuple (L,α,M), where

• L ⊂ T is a special Lagrangian submanifold,• α ∈ R is a real number such that the equation

L = π(

{z(t) ∈ C : z(t) = z0 + eiπαt, t ∈ R})

holds for some z0 ∈ C.• M is a C-local system on L whose monodromy has eigenvalues inthe unit circle.

Morphisms in the Fukaya category are defined to be the morphisms be-tween local systems restricted to the intersection of Lagrangian submani-folds. Finally, the A∞-structure is defined by using the complexified Kahlerform ωC and the moduli spaces of pseudo-holomorphic disks in X∨.

We denote by L0 = (L0, 1/2,CL0) (resp. L1 = (L1, 0,CL1

)) ∈ Dπ Fuk(X∨),where L0 (resp. L1) is the image of the imaginary axis (resp. the real axis)in C under the projection map π : C → T .

Theorem 7.2 ([PZ98]). There exists an equivalence

(7.1) Φmirror : Db(X)

∼−→ Dπ Fuk(X∨),

which sends Oe,OX to L0,L1, respectively, where e ∈ X is the origin.

In particular, the objects Li ∈ Dπ Fuk(X∨) (i = 0, 1) are spherical ob-jects. We denote by STLi ∈ Auteq(Dπ Fuk(X∨)) the corresponding spheri-cal twists, which are called the Dehn twists along Li.

Combining Proposition 7.1 and Theorem 7.2, we obtain the following:

Theorem 7.3. There exists a perverse schober PA on (P1, {0, 1,∞}) satis-fying the following properties:

(1) The schober PB restricted to the open subset P1\{0, 1,∞} is isomor-phic to a local system LA of categories with the fiber Dπ Fuk(X∨),defined by the assignments

(7.2) a 7→ STL0 , b 7→ STL1 ,


(2) Restricting the schober PA to small disks around ∞, 1, 0, we obtainthe spherical functors

Db(pt) → Dπ Fuk(X∨), C 7→ L0,

Db(pt) → Dπ Fuk(X∨), C 7→ L1,

〈Db(pt),Db(pt)〉 → Dπ Fuk(X∨),

respectively.

Under the mirror symmetry equivalence (7.1), the perverse schober PA isidentified with the schober PB with p = e ∈ X.

8. Orlov equivalences via VGIT

In this section, we review the proof of Orlov’s theorem (Theorem 5.1)via the magic window theorem and Knorror periodicity, following [BFK19,Section 7]. This approach naturally gives an example of spherical pairs inthe sense of Definition 3.10.

8.1. VGIT and window shift autoequivalences. Let us put G := C∗ ×C∗, Y := Cn+2 × C, and define an action of G on X as follows:

(λ, µ) · (~x, u) :=(

λ~x, λ−(n+2)µu)

, (λ, µ) ∈ G, (~x, u) ∈ Y.

Let χ± : G→ C∗ be characters defined as χ(λ, µ) := λ±. By taking the GITquotients with respect to the characters χ±, we obtain

Y + = [V (OPn+1(−n− 2))/C∗], Y − = [Cn+2/C∗],

where C∗ acts on the fiber of V (OPn+1(−n− 2)) by weight one, and C∗ actson Cn+2 by weight one. We have natural embeddings Y ± → [Y/G].

We further take a general homogeneous polynomial W ∈ C[x1, · · · , xn+2]of degree n+ 2, and define a function QW : Y → C by QW (~x, u) := uW (~x).Then QW is a η-semi-invariant function, where η : G → C∗ is a characterdefined as η(λ, µ) = µ. Indeed, we have

QW ((λ, µ) · (~x, u))) = (λ−n−2µu)λn+2W (~x) = µuW (~x) = µQW (~x, u).

Hence the data (Y, η,QW )G defines a gauged Landau-Ginzburg model.

Definition 8.1. (1) For an interval I, we define the window subcate-gory GI ⊂ Dabs[factG(Y, η,QW )] to be a triangulated subcategorygenerated by factorizations (4.1) with terms

Ej ≃⊕

i∈I∩Z

O(i)lij , j = 1, 0, lij ∈ Z≥0

as C∗-equivariant sheaves on Y , where C∗ acts on Y via the inclusionC∗ ⊂ G into the first factor.

(2) For an integer w ∈ Z, we put

Gw := G[w,w+n+2), Gw := G[w,w+n+2].

The following is a special case of the main theorem of [BFK19, HL15]:


Theorem 8.2 ([BFK19, HL15]). For each integer w ∈ Z, the compositions

r+w : Gw → Dabs[factG(Y, η,QW )]res+−−−→ Dabs[factC∗(V (OPn+1(−n− 2)), χ1, QW )],

r−w : Gw → Dabs[factG(Y, η,QW )]res−−−−→ Dabs[factC∗(Cn+2, χn+2,W )],

are equivalences, where res± denote the restriction functors to the semi-stable loci Y ±. In particular, we have an equivalence

ψw := r−w ◦ (r+w )−1 : Dabs[factC∗(V (OPn+1(−n− 2)), χ1, QW )]

→ Dabs[factC∗(Cn+2, χn+2,W )].

Definition 8.3. For each integer w ∈ Z, we put

Φw := ψ−1w−1 ◦ ψw ∈ Auteq

(

Dabs[factC∗(V (OPn+1(−n− 2)), χ1, QW )])

and call it as the window shift autoequivalence.

In the following, we will interpret the window shift autoequivalences asautoequivalences of Db(X) under the Knorror periodicity, where X := (W =0) ⊂ Pn+1 is a smooth Calabi-Yau hypersurface of dimension n. We havethe following diagram:

(8.1) X � � iX//

� _

γX��

Pn+1� _

γ

��

V (OPn+1(−n− 2))|X� � i //

p

��

V (OPn+1(−n− 2))QW

//

q

��

C

X � �

iX// Pn+1,

where p, q denote the projections, γ, γX are the inclusions as zero sections,and i, iX are the natural inclusions.

We use the following result:

Proposition 8.4 ([HLS16, Proposition 3.4]). The window shift autoequiv-alence Φw fits into the following exact triangle(8.2)RHom(γ∗OPn+1(−w − n− 1),−)⊗ γ∗OPn+1(−w − n− 1) → (−) → Φw(−).

In other words, Φw is the spherical twist around γ∗OPn+1(−w − n− 1).

Recall from Theorem 4.16 that we have an equivalence

(8.3) i∗p∗ : Db(X)

∼−→ Dabs[factC∗(V (OPn+1(−n− 2)), χ1, QW )].

Proposition 8.5. Let w ∈ Z be an integer. Under the Korror periodicityequivalence (8.3), the window shift autoequivalence Φw corresponds to thespherical twist STOX(w+2n+3) on D

b(X).

Proof. To simplify the notation, we put V := V (OPn+1(−n − 2)). Since wehave an isomorphism

Φk−n−1 ≃ ((−)⊗OV (k)) ◦ Φ−n−1 ◦ ((−)⊗OV (−k))

of functors (see e.g., [HLS16, Lemma 3.2]) for each k ∈ Z, we may assumew = −n− 1.


We prove a functorial isomorphism Φ−n−1◦i∗p∗(E) ≃ i∗p

∗◦STOX(n+2)(E)

for each object E ∈ Db(X). By applying the functor i∗p∗ to the defining

triangle

RHom(OX(n+ 2), E) ⊗OX(n+ 2) → E → STOX(n+2)(E),

we obtain(8.4)RHom(OX (n+ 2), E) ⊗ i∗OV |X (n+ 2) → i∗p

∗E → i∗p∗ ◦ STOX(n+2)(E).

On the other hand, by the triangle (8.2), we have the exact triangle

(8.5) RHom(γ∗OPn+1 , i∗p∗E)⊗ γ∗OPn+1 → i∗p

∗E → Φ−n−1(i∗p∗E).

We compute the complex RHom(γ∗OPn+1 , i∗p∗E) as follows:

(8.6)

RHom(γ∗OPn+1 , i∗p∗E) ≃ RHom(i∗γ∗OPn+1 , p∗E)

≃ RHom(γX∗i∗XOPn+1 , p∗E)

≃ RHom(i∗XOPn+1 , γ∗Xp∗E(−n− 2)[1]),

≃ RHom(OX (n+ 2), E)[1],

where the first isomorphism follows from the adjunction; the second iso-morphism follows from Lemma 8.6 (2) below; the third isomorphism followsfrom the adjunction and the fact that ωX/V |X ≃ OX(−n − 2); the last iso-morphism follows from p ◦ γX = idX .

Moreover, we have an isomorphism

(8.7) i∗OV |X (n+ 2)[−1] ≃ γ∗OPn+1

by Lemma 8.6 (1) below.By the isomorphisms (8.6) and (8.7), it follows that the triangles (8.4)

and (8.5) are isomorphic, as required. �

We have used the following lemma in the above proof:

Lemma 8.6. Put L := OPn+1(−n−2), V := V (L). The following statementshold:

(1) We have an isomorphism

γ∗OPn+1 ≃ i∗OV |X (n + 2)[−1].

(2) We have an isomorphism

i∗γ∗OPn+1 ≃ γX∗i∗XOPn+1 .

Proof. Put s := q∗W : q∗L → OV , and let t : OV → q∗L(χ1) be the tauto-logical section. By Lemma 4.15, we have

γ∗OPn+1 ≃ K(t∨, s∨) ≃ K(s, t)∨ ≃ i∗OV |X (n2)[−1],

which proves the first assertion.Similarly, we have

i∗γ∗OPn+1 ≃ i∗K(t∨, s∨) = K(t∨|V |X , 0) ≃ γX∗i∗XOPn+1 ,

where the second equality follows from the vanishing s|V |X = 0, hence thesecond assertion holds. �

We can now give the proof of Orlov’s theorem:


Proof of Theorem 5.1. As before, we put V := V (OPn+1(−n − 2)). Theexistence of the equivalence

Db(X) ≃ HMFgr(W )

follows from Theorem 8.2 together with the equivalences (8.3) and (4.6).For the second statement, consider the following commutative diagram:

(8.8)

Dabs[factC∗(V, χ1, QW )]

ψ−2n−4

**

⊗O(1)��

G−n−3

r+−2n−4oo

r−−2n−4

//

⊗O(1)

��

Dabs[factC∗(Cn+2, χn+2,W )]

⊗O(1)��

Dabs[factC∗(V, χ1, QW )]

ψ−2n−3

55G−2n−3

r+−2n−3

oo

r−−2n−3

// Dabs[factC∗(Cn+2, χn+2,W )],

where O(1) ∈ Dabs[factG(Y, η,QW )] denotes the C∗-equivariant line bundleof weight one, with respect to the C∗-action via the inclusion C∗ ⊂ G to thefirst factor.

By (4.6), the autoequivalence

(−)⊗O(1) ∈ Auteq(Dabs[factC∗(Cn+2, χn+2,W )])

corresponds to the degree shift equivalence τ under the natural equivalence

Dabs[factC∗(Cn+2, χn+2,W )]) ≃ HMFgr(W ).

On the other hand, the autoequivalence ⊗O(1) on Dabs[factC∗(V, χ1, QW )]corresponds to the autoequivalence ⊗OX(1) on D

b(X) via the Knorror pe-riodicity (8.3).

By the above observations, the commutativity of the diagram (8.8) impliesthat

(ψ−2n−4)−1 ◦ τ ◦ ψ−2n−4 = (ψ−2n−4)

−1 ◦ ψ−2n−3 ◦ (−⊗OX(1))

= Φ−2n−3 ◦ (−⊗OX(1))

= STOX ◦(− ⊗OX(1)),

where the last equality follows from Proposition 8.5. �

8.2. Spherical pairs from VGIT. We end this section by constructingan example of spherical pairs using the theory of [BFK19, HL15], following[Don19a]. Let

S+ := {0} × C, S− := Cn+2 × {0} ⊂ Y = Cn+2 ×C

be the unstable loci with respect to the characters χ+, χ−, respectively. Letj± : S± → Y denote the inclusions.

Proposition 8.7 ([Don19a, HLS16]). We have a spherical pair

(8.9) G−n−2 = 〈Db(X),Db(pt)〉 = 〈HMFgr(W ),Db(pt)〉

such that the induced autoequivalence on Db(X) is isomorphic to STOX .


Proof. By [HLS16, Equation (3)], applied to the two different KN stratifi-cations

Y = (Y \ S+) ⊔ S+ = (Y \ S−) ⊔ S−,

we have a pair of semi-orthogonal decompositions:

(8.10) G−2n−4 = 〈G−2n−4, j+∗ OS+(−2n− 4)〉 = 〈G−2n−3, j

−∗ OS−(−n− 2)〉.

The proofs of [Don19a, Theorem 4.4, Proposition 4.5] show that the pair(8.10) defines a spherical pair. Moreover, by Theorem 8.2 and the equiva-lences (8.3) and (4.6), we have

G−2n−4 ≃ Db(X), G−2n−3 ≃ HMFgr(W ).

Hence we obtain the spherical pair as in (8.9).It remains to compute the induced autoequivalence on Db(X). By con-

struction, the induced autoequivalence on Dabs[factC∗(V, χ1, QW )] ≃ G−2n−4

is the spherical twist around the object j−∗ OS−(−n−2)|V ≃ γ∗OPn+1(−n−2).By Propositions 8.4 and 8.5, it corresponds to the spherical twist STOX un-der the Knorror periodicity equivalence (8.3). �

Corollary 8.8. The perverse schober in Theorem 5.7 upgrades to the spher-ical pair (8.9) around the point 0 ∈ P1.

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School of Mathematics, University of Edinburgh, JCMB, Peter Guthrie

Tait Road, Edinburgh EH9 3FD, UK.

Email address: [email protected]

Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku,

Nagoya, Japan, 464-8602

Email address: [email protected]

arXiv:2201.05902v1 [math.AG] 15 Jan 2022

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