SPHERICAL ADJUNCTION AND SERRE FUNCTOR FROM MICROLOCALIZATION AN APPROACH BY CONTACT ISOTOPIES

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SPHERICAL ADJUNCTION AND SERRE FUNCTOR FROM

MICROLOCALIZATION

– AN APPROACH BY CONTACT ISOTOPIES –

CHRISTOPHER KUO AND WENYUAN LI

Abstract. For a subanalytic Legendrian Λ ⊆S∗M, we prove that when Λ is either swappable

or a full Legendrian stop, the microlocalization at inﬁnity mΛ: ShΛ(M)→µshΛ(Λ) is a spherical

functor, and the spherical cotwist is the Serre functor on the subcategory Shb

Λ(M)0of compactly

supported sheaves with perfect stalks. This is a sheaf theory counterpart (with weaker assumptions)

of the results on the cap functor and cup functor between Fukaya categories. When proving spher-

ical adjunction, we deduce the Sato-Sabloﬀ ﬁber sequence and construct the Guillermou doubling

functor for microsheaves on any open subsets of the Legendrian and with respect to any Reeb ﬂow.

Contents

1. Introduction 2

1.1. Context and background 2

1.2. Main results and corollaries 3

1.3. Sheaf theoretic wrapping, small and large 7

Acknowledgement 7

2. Preliminary 7

2.1. Microlocal sheaf theory 8

2.2. Microsheaves 11

2.3. Various sheaf categories 13

3. Isotopy of sheaves 15

3.1. Continuation maps 16

3.2. Isotopies of sheaves 17

4. Doubling and ﬁber sequence 19

4.1. Sato-Sabloﬀ ﬁber sequence 20

4.2. Sabloﬀ-Serre duality 23

4.3. Doubling from Sato-Sabloﬀ sequence 24

4.4. Functorial local adjoints of microlocalization 30

4.5. Functorial global adjoints of microlocalizations 35

4.6. Relative doubling and ﬁber sequence 40

5. Spherical adjunction from microlocalization 41

5.1. Spherical adjunction and spherical functors 42

5.2. Natural transform between adjoints 43

5.3. Criterion for spherical adjunction 45

5.4. Spherical adjunction on subcategories 49

5.5. Serre functor on proper subcategory 50

6. Spherical pairs and perverse sch¨obers 52

6.1. Semi-orthogonal decomposition 52

6.2. Spherical pairs from variation of skeleta 56

7. Example that wrap-once is not equivalence 58

arXiv:2210.06643v4 [math.SG] 24 May 2024

2 CHRISTOPHER KUO AND WENYUAN LI

References 60

1. Introduction

1.1. Context and background. Our goal in the series of paper [43, 44] is to investigate non-

commutative geometry structure of the category of sheaves arising from the symplectic geometry

structure on the Lagrangian skeleton of the Weinstein pair (T∗M, Λ), where Λ ⊆S∗Mis a sub-

analytic Legendrian subset in the ideal contact boundary S∗Mof the exact symplectic manifold

T∗M.

Following Kashiwara-Schapira [37], given a real analytic manifold M, one can deﬁne a stable

∞-category ShΛ(M) of constructible sheaves on Mwith subanalytic Legendrian singular support

Λ⊆S∗M, which is invariant under Hamiltonian isotopies [30]. On the other hand, one can also

deﬁne another stable ∞-category µshΛ(Λ) of microlocal sheaves on the Legendrian Λ ⊆S∗M

[28, 50], and there is a microlocalization functor

mΛ: ShΛ(M)→µshΛ(Λ).

From the perspective of non-commutative geometry, the categories of (microlocal) sheaves, as

sheaves on the Lagrangian skeleton of the Weinstein pair (T∗M, Λ), should be understood as a non-

commutative manifold with boundary, abiding Poincar´e-Lefschetz duality and ﬁber sequence with

the presence of a relative Calabi-Yau structure [11], whose existence is proved by Shende-Takeda

using arborealization [64]. In this paper, we will study some non-commutative geometry structures,

which are related but diﬀerent from the duality ﬁber sequence in Calabi-Yau structures.

The category of (micro)sheaves is closely related to a number of central topics in symplectic

geometry and mathematical physics [49, 53]. Recently Ganatra-Pardon-Shende [24] showed that

the partially wrapped Fukaya category is equivalent to the category of compact objects in the

unbounded dg category of sheaves. In particular, for contangent bundles with Legendrian stops,

Perf W(T∗M, Λ)op ≃Shc

Λ(M).

The homological mirror symmetry conjecture [7,40] predicts an equivalence of the Fukaya categories

and categories of coherent sheaves on the mirror complex variety.

We will now explain the non-commutative geometry structures that will be investigated, namely

Serre duality and spherical adjunctions, which arise from various predictions from symplectic ge-

ometry, mirror symmetry and mathematical physics.

1.1.1. Context of spherical adjunction. Spherical adjunctions are introduced by Anno-Logvinenko

[5] in the dg setting and then generalized [16] in the stable ∞setting, as a generalization of the

notion of spherical objects [63]. Like spherical objects, spherical adjunctions provide interesting

ﬁber sequences and autoequivalences of the categories called spherical twists and cotwists.

In algebraic geometry, when we have a smooth variety Xwith a divisor i:D →X, the push

forward functor and pull back functor

i∗: Coh(D)⇋Coh(X) : i∗

form a spherical adjunction between the dg categories of coherent sheaves, where the spherical twist

is −⊗OX(D).

In symplectic geometry, as is suggested by Kontsevich-Katzarkov-Pantev [38] and Seidel [59],

we have another interesting class of spherical adjunctions inspired by long exact sequences in

Floer theory [17, 56, 57, 60]. For a symplectic Lefschetz ﬁbration π:X→Cwith regular ﬁber

F=π−1(∞), let FS(X, π) be the Fukaya-Seidel category associated to Lagrangian thimbles in X

and F(F) the Fukaya category of closed exact Lagrangians in F[58]. The cap functor

∩F:FS(X, π)→ F(F)

SPHERICAL ADJUNCTION AND SERRE FUNCTOR FROM MICROLOCALIZATION 3

deﬁned by intersection of the Lagrangians with Fadmits a left adjoint ∪called the (left) cup

functor [3] (see also [4, Appendix A]). In an unpublished work, Abouzaid-Ganatra proved that ∩

and ∪form a spherical adjunction for general symplectic Landau-Ginzburg models [3]. On the

other hand, using the formalism of partially wrapped Fukaya categories [23, 65], Sylvan considered

the Orlov cup functor

∪F:W(F)→ W(X, F )

associated to any Weinstein pair (X, F ) and showed that the ∪is a spherical functor1as long

as the Weinstein stop F⊂∂∞Xis a so called swappable stop [66]. In this case, the spherical

twists/cotwists are the monodromy functors deﬁned by wrapping around the contact boundary.

In microlocal sheaf theory, Nadler has also shown that functors between the pair of microsheaf

categories over the symplectic Landau-Ginzburg model (Cn, π =z1. . . zn), aftering (heuristically

speaking) adding additional ﬁberwise stops, form a spherical adjunction. Then, by removing the

ﬁberwise stops, the spherical adjunction for the original pair is also obtained [51], but it is unclear

how general this argument is in sheaf theory.

The structure of spherical adjunctions has appeared in a number of previous works and leads

to interesting applications in homological mirror symmetry [1, 21, 34, 51]. However, there are still

important problems that remain. Namely, the cap functor ∩is only deﬁned in the setting of a

symplectic Landau-Ginzburg model instead of a general Weinstein pair, since it is completely not

clear whether there are enough Lagrangian submanifolds asymptotic to a general Legendrian stop

[24]. This means that even though the cup functor ∪is proved to be spherical as long as the stop

is swappable [66], it is diﬃcult to characterize the adjoint functors geometrically.

1.1.2. Context of Serre functor. On the other hand, in algebraic geometry, the dualizing sheaf

induces Serre duality. When Xis smooth, the Serre functor is given by the canonical bundle

−⊗OX(KX).

From the perspective of Fukaya categories, following a proposal of Kontsevich, Seidel has con-

jectured [59] that for a symplectic Lefschetz ﬁbration, the spherical cotwist is the inverse Serre

functor

σ:FS(X, π)→ FS(X, π),

and proved partial results [60–62], while from the perspective of Legendrian contact homology,

Ekholm-Etnyre-Sabloﬀ have proved Sabloﬀ duality [17, 56] between linearized homology and co-

homology. These results predict an inverse Serre functor, which should be the Poincar´e-Lefschetz

duality on the category of constructible sheaves with perfect stalks

S+

Λ: Shb

Λ(M)→Shb

Λ(M).

Little is known for either the sheaf categories or Fukaya categories of general Weinstein pairs. .

1.2. Main results and corollaries. We state our main result which provides a general criterion

for the microlocalization functor mΛ: ShΛ(M)→µshΛ(Λ) to be spherical. Under the equivalence

of Ganatra-Pardon-Shende [24], the left adjoint of microlocalization ml

Λis equivalent to the Orlov

cup functor on wrapped Fukaya categories, while we expect that the microlocalization mΛis the

cap functor on Fukaya-Seidel categories (see Remark 1.6 and 1.7).

1.2.1. Spherical adjunction and Serre functor. Let Mbe a real analytic manifold. Consider a ﬁxed

Reeb ﬂow Tt:S∗M→S∗M. Recall that a (time-dependent) contact isotopy ϕt:S∗M×R→S∗M

is called a positive isotopy if α(∂tϕt)≥0. In the deﬁnition, we use the word stop for any compact

subanalytic Legendrians (following [23, 65]), meaning that Hamiltonian ﬂows are stopped by the

Legendrian.

1The data of a spherical functor is equivalent to the data of a spherical adjunction, as will be explained in Section

5.1. Here we use spherical functors because the adjoint functor is not explicitly constructed Sylvan’s work.

4 CHRISTOPHER KUO AND WENYUAN LI

The geometric notion of a swappable subanalytic Legendrian originates from positive Legendrian

loops that avoid the Legendrian at the base point [14], and is explicitly introduced by Sylvan [66].

Here our deﬁnition is slightly diﬀerent from [66].

Deﬁnition 1.1. A compact subanalytic Legendrian Λ⊆S∗Mis called a swappable stop if there

exists a compactly supported positive Hamiltonian on S∗M\Λsuch that the ﬂow sends Tϵ(Λ) to an

arbitrary small neighbourhood of T−ϵ(Λ), and the backward ﬂow sends T−ϵ(Λ) to an arbitrary small

neighbourhood of Tϵ(Λ).

We also introduce the notion of geometric and algebraic full stops, both called full stops for

simplicity. We will see in Proposition 5.15 that a geometric full stop is always an algebraic full

stop.

Deﬁnition 1.2. Let Mbe compact. A compact subanalytic Legendrian Λ⊆S∗Mis called an

algebraic full stop if ShΛ(M)is proper. Λ⊆S∗Mis called a geometric full stop if for a collection

of generalized linking spheres at inﬁnity Σ⊆S∗Mof Λ, there exists a compactly supported positive

Hamiltonian on S∗M\Λsuch that the ﬂow sends Σto an arbitrary small neighbourhood of T−ϵ(Λ).

Example 1.3. There is a large class of examples of swappable stops and full stops in Section 5.3.

Here are two cimple classes of examples. (1) For a subanalytic triangulation S={Xα}α∈I, the

union of unit conormal bundles Sα∈IN∗

∞Xαis a algebraic full stop (we suspect that it is also a

geometric full stop and a swappable stop, but we cannot prove that). (2) For an exact symplectic

Landau-Ginzburg model π:T∗M→C, the Lagrangian skeleton cFof a regular ﬁber at inﬁnity

F=π−1(∞)is a swappable stop and when πis a Lefschetz ﬁbration it is a geometric full stop.

We are able to state our main result, which provides a general criterion for the microlocalization

functor mΛto be spherical.

Theorem 1.4 (Theorem 5.1).Let Λ⊆S∗Mbe a compact subanalytic Legendrian. Suppose Λis a

full stop or a swappable stop. Then the microlocalization functor along Λand its left adjoint

mΛ: ShΛ(M)⇋µshΛ(Λ) : ml

form a spherical adjunction.

Restricting attention to the pair of sheaf categories of compact objects, and the corresponding

pair of sheaf categories of proper objects when the manifold is compact, which are the sheaf theoretic

models of suitable versions of Fukaya categories, we can show the following corollary.

Corollary 1.5. Let Λ⊆S∗Mbe a closed subanalytic Legendrian. Suppose Λis either a swappable

stop or a geometric full stop. Then the microlocalization functor along Λon the sheaf category of

objects with perfect stalks

mΛ: Shb

Λ(M)→µshb

Λ(Λ)

is a spherical functor. Respectively, the left adjoint of the microlocalization functor on the sheaf

category of compact objects

Λ:µshc

Λ(Λ) →Shc

Λ(M)

is also a spherical functor.

Remark 1.6. According to [24, Proposition 7.24] there is a commutative diagram between mi-

crolocal sheaf categories and wrapped Fukaya categories

W(F)∼//

∪F



µshc

cF(cF)

m∗



W(T∗M, F )∼//Shc

cF(M).

SPHERICAL ADJUNCTION AND SERRE FUNCTOR FROM MICROLOCALIZATION 5

Therefore the second part of our theorem recovers the result by Sylvan [66] that

∪F:W(F)→ W(X, F )

is spherical in the case X=T∗M. However, diﬀerent from [66], we are able to explicitly construct

the left and right adjoint functors in the proof.

Remark 1.7. For a Lefschetz ﬁbration π:T∗M→Cwith regular ﬁber at inﬁnity F=π−1(∞),

W(T∗M, F ) is generated by Lagrangian thimbles [25, Corollary 1.14] and is a proper category

[24, Proposition 6.7] (when M=Tnand Fis the Weinstein thickening of the FLTZ skeleton

[19, 55], this can also be proved using mirror symmetry [45]), and hence

Shb

cF(M)≃Shc

cF(M)≃ W(T∗M, F ).

Since it is expected that W(T∗M, F )≃ FS(T∗M, π)2, there should be an equivalence Shb

cF(M)≃

FS(T∗M, π) (when M=Tn, Zhou has sketched a proof in his thesis [69]). Therefore, our theorem

should be viewed as a sheaf theory version of the result [3] that

∩F:FS(T∗M, π)→ F(F)

is spherical. However, since we do not know a commutative diagram, that result [3] does not

directly follow from ours.

We can write down the spherical twists and cotwists as follows. Previous work of the ﬁrst author

[42] has deﬁned the positive wrapping functor W+

Λ(resp. negative wrapping functor W−

Λ) sending

an arbitrary sheaf in Sh(M) to ShΛ(M) by a colimit (resp. a limit) of positive (resp. negative)

wrappings into Λ. The spherical cotwist (resp. the dual cotwist) ShΛ(M)→ShΛ(M) for mΛis the

functor deﬁned by wrapping positively (resp. negatively) around S∗Monce along the Reeb ﬂow.

Proposition 1.8. Let Λ⊆S∗Mand Tt:S∗M→S∗Mbe a Reeb ﬂow. Then the spherical cotwist

and dual cotwist are the negative and positive wrap-once functor (where  > 0is suﬃciently small)

S−

Λ=W−

Λ◦T−ϵ, S+

Λ=W+

Λ◦Tϵ.

Remark 1.9. By [24, Proposition 7.24], we know that the cup functor ∩Fon partially wrapped

Fukaya categories is isomorphic to the left adjoint of microlocalization ml

Λon sheaf categories.

Therefore, we have a commutative diagram

W(T∗M, F )∼//

S±



Shc

cF(M)

S±



W(T∗M, F )∼//Shc

cF(M),

such that our wrap-once functor S±

cFis isomorphic to the wrap-once functor of Sylvan [66].

We will also write down a formula for spherical twists and dual twists in Section 6.1 Corollary

6.12, which can be interpreted as the monodromy functors.

Finally, the following statement shows that the negative wrap-once functor S−

Λis the Serre

functor on the subcategory of proper objects up to a twist by the dualizing sheaf ωM.

Proposition 1.10 (Proposition 5.29).Let Λ⊆S∗Mbe a full or swappable subanalytic compact

Legendrian stop. Then S−

Λ⊗ωMis the Serre functor on Shb

Λ(M)0of sheaves microsupported on Λ

with perfect stalks and compact supports. In particular, when Mis orientable, S−

Λ[−n]is the Serre

functor on Shb

Λ(M)0.

2As pointed out in [24, Footnote 2], if one takes W(T∗M, F ) as the deﬁnition of the Fukaya-Seidel category then

this is tautological. However a comparison result between W(T∗M, F ) and the Fukaya-Seidel category deﬁned in

[58, Part 3] is not yet in the literature.

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SPHERICALADJUNCTIONANDSERREFUNCTORFROMMICROLOCALIZATION–ANAPPROACHBYCONTACTISOTOPIES–CHRISTOPHERKUOANDWENYUANLIAbstract.ForasubanalyticLegendrianΛ⊆S∗M,weprovethatwhenΛiseitherswappableorafullLegendrianstop,themicrolocalizationatinfinitymΛ:ShΛ(M)→µshΛ(Λ)isasphericalfunctor,andthesphericalcotwististhe...

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