Perverse Schobers
Mikhail Kapranov, Vadim Schechtman
November 12, 2014
The notion of a perverse sheaf, introduced in [BBD], has come to play acentral role in algebraic geometry and representation theory. In particular,
appropriate categories of perverse sheaves provide “categorifications” of various representation spaces, these spaces being recovered as the Grothedieckgroups of the categories.
The goal of this paper is to suggest the possibility of categorifying thevery concept of a perverse sheaf. In other words, we propose to develop atheory of perverse sheaves not of vector spaces but of triangulated categories.
Given a complex manifold X, an analytic Whitney stratification S =(Xα )α∈A of X and a ground field k, one has the category Perv(X, S) of
perverse sheaves of k-vector spaces on X smooth with respect to S. Traditionally, there have been two ways of looking at Perv(X, S):(1) General definition: as an abelian subcategory in the triangulatedb(X, S) of constructible complexes of sheaves of k-vector
category Dconstr
spaces on X, smooth with respect to S.
(2) Quiver description (for some particular (X, S)): as a category of
diagrams of some given type formed by vector spaces (Vi )i∈I and maps
between them subject to certain relations. These diagrams have thefollowing features:/
Vj having the same ends but oppo(2a) Arrows come in pairs Vi osite directions. This reflects the (Verdier) self-duality of Perv(X, S).

(2b) In most cases, the relations contain 2 or 3 summands, with coefficients ±1.

1

So far, there is no obvious direct way to categorify the approach (1) since
it is not clear what are complexes of triangulated categories.
On the other hand, we observe that the features (2a) and (2b) are of
the kind that immediately suggest a categorical generalization.
We canreplace vector spaces Vi by triangulated categories Vi and arrows by exact
functors. The pairs of opposite arrows in (2a) can be interpreted as adjoint
pairs of functors, 2-term relations as isomorphisms of functors and 3-termones as exact triangles in appropriate functor categories (see Appendix forthe precise framework in which these make sense). By forming Grothendieckgroups Vi = K0 (Vi ) ⊗ k of such a diagram of categories, one would thenobtain a quiver in the original sense, i.e., a perverse sheaf. The idea thatthe cone can be seen as a categorical analog of the difference, lies, of course,
at the very foundations of algebraic K-theory, especially in the Waldhausen
approach.
This strongly suggests that there should be meaningful objects which canbe understood as “perverse sheaves of triangulated categories" and which giveusual perverse sheaves by passing to the Grothendieck groups. We propose tocall such hypothetical objects perverse Schobers (or, sometimes, for brevity,
simply Schobers), using the German analog1 of the English word “stack"which would be the correct (but overused) term for speaking of “sheaves ofcategories".
In this paper we work out several basic examples of quiver descriptions ofperverse sheaves and define, in an ad hoc way, what should be the perverse
Schobers in these situations. In the simplest case, we propose, in §1, to
identify perverse Schobers on a disk with one allowed singular point, withspherical functors of [A1][AL2].
For a disk with several allowed singular points we propose, in §2, a definition in terms of certain diagrams of spherical functors, and explain theinvariance properties of such a definition. Among other things, we reformulate the classical Picard-Lefschetz formula as a general statement about
perverse sheaves on a disk, and then lift it to a distinguished triangle associated to a perverse Schober and a certain configuration of paths. Such
“Picard-Lefschetz triangles" should therefore be considered as fundamental
features of perverse Schobers. It is natural to expect analogous features in
1

A literal Russian analog would be the word стог. We learned the term “Schober" from
W. Soergel.

2the case dim(X) > 1 and even consider them as “codimension 1 data" of aperverse Schober.
A series of examples of spherical functors is provided by representation
theory. More precisely, for a reductive group G we have spherical functors
acting in the derived category of sheaves on G/B and satisfying the relationsof the corresponding braid group Br(g). In §4 we review these examples andsuggest a conjectural interpretation in terms of perverse Schobers on h/W
the space of conjugacy classes of semisimple elements in the Lie algebra g.
Most of the known quiver descriptions of Perv(X, S) can be obtainedusing a choice of “cuts" which are certain totally real subvarieties K ⊂ X(of real dimension equal to dimC X). In §5 we summarize the features ofsuch cuts and note that Lagrangian varieties, used in constructing Fukaya
categories [FO3 ][Se], provide a reasonable class of candidates for cuts. One
can therefore expect Fukaya-categorical constructions to have a bearing on
the problem of classification of perverse sheaves.
Additionally, we discuss the idea of defining “Fukaya categories with coefficients". This idea was proposed by M. Kontsevich in order to study theusual Fukaya category of a manifold by fibering it over a manifold of smaller
dimension. We suggest that perverse Schobers should be considered as theright “coefficient data" for such a definition, just like sheaves are naturalcoefficient data for defining cohomology.
We would like to thank A. Bondal, V. Ezhov, M. Finkelberg, P. Schapira,W. Soergel, Y. Soibelman and B. Toën for useful discussions and correspondence. V.S. is grateful to the Kavli IPMU for hospitality and support during
the visit when this paper was finished. The work of M. K. was supported
by World Premier International Research Center Initiative (WPI Initiative),
MEXT, Japan.

3

1Perverse Schobers on a disk: spherical functors

A. Disk with one marked point. Let ∆ be the unit disk in C andPerv(∆, 0) be the category of perverse sheaves on ∆ with the only possible
singularity at 0. The most iconic example of a quiver description of perverse
sheaves is the following classical statement [Be] [GGM].
Theorem 1.1. Perv(∆, 0) is equivalent to the category P1 of quadruples(Φ, Ψ, u, v) where Φ, Ψ ∈ Vectk and
Φo

(1.2)

v
u

/

Ψ

are linear maps such that
(1.3)

TΨ := IdΨ − vu is an isomorphism.

Exercise 1.4. Show that TΨ is an isomorphism iff TΦ : IdΦ − uv is an isomorphism.
∆
K

0

b

Figure 1: Defining Φ and Ψ topologically.
One way of constructing an explicit equivalence is as follows [GGM]. Choosea base point b on the boundary of ∆ and connect it with 0 by a simple arc
K, see Fig. 1. Then to F ∈ Perv(∆, 0) we associate the spaces
Φ(F) := H1K (F)0 ' H1K (∆, F) (vanishing cycles),Ψ(F) := Fb ' H1K (F)b4(nearby cycles).

(We recall that F|∆−{0} is a local system in degree 0). The map v = vF is
the generalization map [GM][Cu] for the constructible sheaf H1K (F) on K,
and u is the composition
(1.5)

Fb

counterclockwise
continuation/ H 0 (∆ − K, F)δ/ H1 (∆, F).
K

Remark 1.6. Further, we have the following elementary statements which
we recall here as indicative of a certain pattern.
(1) HiK (F) = 0 for i 6= 1.
(2) The sheaf R(F) = H1K (F) on K is constant on K − {0} so it has only
two essentially different stalks R(F)0 = Φ(F) and R(F)b = Ψ(F).
(3) Each of the two stalks, considered as a functor Perv(∆, 0) → Vectk , isan exact functor which takes Verdier duality to vector space duality.
(4) The map u = uF is the dual uF = (vF ∗ )∗ .
B. Spherical functors. As a natural categorical analog of the data (1.2)(1.3) we would like to suggest the following remarkable concept introduced
by R. Anno [A1].
Definition 1.7. Let
S : D0 −→ D1
be an exact functor between triangulated categories (see Appendix for conventions). Assume that S admits a left adjoint L and a right adjoint R, sothat we have the unit and counit natural transformations
SR ⇒ IdD1 ,IdD0 ⇒ RS,

LS ⇒ IdD0 ,IdD1 ⇒ SL,

whose cones will be denoted by
T1 = Cone{SR ⇒ IdD1 }, T10 = Cone{IdD1 ⇒ SL}[−1] (the twist functors),
T0 = Cone{IdD0 ⇒ RS}[−1], T00 = Cone{LS ⇒ IdD0 } (the cotwist functors).
We call S a spherical functor, if:(SF1) T1 is an equivalence.
5

(SF2) The composition R → RSL → T0 L[1] is an isomorphism. In otherwords, composition with T0 identifies R and L.
In this case T10 is quasi-inverse to T1 and T00 is quasi-inverse to T0 .
More precisely, each of the pairs of adjoint functors
(1.8)

D0 o

S
R

/

D1 o

D1 ,

S
L

/ D0

can be regarded as an analog of (1.2). Further, (SF1) is an analog of (1.3),
the adjunction unit allowing us to take the “categorical difference", i.e., thecone. Further still, the categorical analog of Exercise 1.4 can be found in thefollowing result of Anno and Logvinenko [AL2].
Theorem 1.9. In addition to (SF1) and (SF2) consider the following two
conditions:(SF3) T0 is an equivalence.
(SF4) The composition LT1 [−1] → LSR → R is an isomoprhism.
Then, any two of the conditions (SF1)-(SF4) imply the other two.
So we can consider a diagram (1.8), i.e., the data of a spherical functor,
as the data defining a “perverse Schober” over (∆, 0). By passing to K0 (orto any Ki , or to any homological functor, Hochschild homology for example)
we get a perverse sheaf over (∆, 0).
B. Examples of spherical functors. We now give some examples, to be
used later.
Example 1.10. Let Sd be the d-dimensional sphere and q : Sd → pt be theprojection. We then have the functor
S=q −1D0 := Db (Vect) = Db (pt) −→ Db (Sd ) =: D1
with right adjoint R = Rq∗ and left adjoint L ' Rq∗ [−d]. The second adjunction is the Poincaré duality. Formaly, it comes from the adjoint pair
(Rq! , q ! ) by noticing that q ! ' q −1 [d] (since q is smooth orientable of relative dimension d) and that Rq! = Rq ∗ (since q is proper), see [KS2] forbackground. More intrinsically, L is canonically identified with the tensor
product of Rq∗ [−d] and H d (Sd , k), the 1-dimensional k-vector space spannedby global orientations of Sd .
6

Proposition 1.11. (a) S is a spherical functor.
(b) More generally, for any Sd -fibration q : Z → Y of CW-complexes, thefunctor
S=q −1D0 := Db (Y ) −→ Db (Z) =: D1
is a spherical functor.
Proof: We prove (a), since (b), being a relative version, is proved in the sameway.
The functor T1 = Cone{SR ⇒ IdD1 } is the endomorphism of Db (S d )defined as follows. Letj

U = (Sd × Sd ) − ∆ ,→ Sd × Sd
be the embedding of the complement of the diagonal, π1 , π2 : Sd × Sd → Sd
be the projections and p1 , p2 : U → Sd be their restrictions to U . Then
T1 (F) = Rp2! (p∗1 F),
a formula remindful of the Fourier-Sato transform relating sheaves on dual
spheres [SKK]. To see this, we write the functor in the RHS in terms of a“kernel", asRπ2∗ ((π1∗ F) ⊗k K),

K = j! kU [1] ∈ Db (Sd × Sd )and note the exact sequence
0 → K[−1] −→ kSd ×Sd −→ k∆ → 0,
in which kSd ×Sd is the kernel for SR while k∆ is the kernel for IdD1 .
The twist T10 = Cone{IdD1 ⇒ SL}[−1] can be found explicitly as
T10 (F) = Rp2∗ (p∗2 F))[−1] = Rπ2∗ ((π1∗ F) ⊗k K0 ),K0 = Rj∗ kU [−1].

Now, the condition (SF1), i.e., the fact that T1 and T10 are quasi-inverse toeach other, can be established directly by finding the “composition” of thekernels K and K0
−1
−1 0
K ),
K ∗ K0 = Rπ13∗ (π12K ⊗k π237

πij : Sd × Sd × Sd → Sd × Sd ,and showing that both K ∗ K0 and K0 ∗ K are isomorphic to k∆ in degree 0.
This amounts to the fact that for distinct points x, y ∈ Sd we have
H • Sd − {x}, j! kSd −{x,y} = 0,
while for x = y we have H • (Sd − {x}) = k and Hc• (Sd − {x}) = k[−d].
Further, the cotwist T0 is the shift by (−d) tensored with H d (Sd , k), the1-dimensional vector space of orientations of Sd . So it is an equivalence and(SF2) is also satisfied.
Example 1.12. Note the particular case d = 2, when S2 = CP1 . Proposition
1.11 implies that for any P1 -fibration q : Z → Y of complex algebraic varieties, q −1 : Db (Y ) → Db (Z) is a spherical functor. Another class of examples
is provided by quaternionic geometry, since HP1 = S4 .
Let us now mention some “coherent” examples.
Example 1.13. Recall that an n-dimensional smooth projective variety Z
over k is called Calabi-Yau (in the strict sense), if(
k, if i = 0, or i = n,
H i (Z, OZ ) =0, otherwise.
Let X be a smooth algebraic variety over k, and q : Z → X be a smoothproper family of Calabi-Yau manifolds. Then the pullback functor q ∗ :bb
Dcoh(X) → Dcoh(Z) is a spherical functor. The proof is similar to thatof Proposition 1.11.
Example 1.14. ([A1]) Consider a diagramiρY ←− D ,→ X
of smooth complex varieties X, D, Y , where ρ is a P1 -bundle and i an embedding of a divisor. We then have a diagram of adjoint functors
L=ρ! i∗←−b
D0 := Dcoh
(Y )S=i∗ ρ∗

−→

R=ρ∗

i!

←−

and R = T0 L where T0 = Cone{IdD0 ⇒ RS}.
8

b
Dcoh
(X)

Lemma 1.15. (R. Anno, [A1]). S is spherical iff the intersection index of
D with a generic fiber of ρ is (−2).
Example 1.16. A particular case of Example 1.14 and Lemma 1.15 is obtained foriρ
Y = pt ←− D = P1 ,→ X = T ∗ P1
The corresponding spherical functor can be seen as a “quasi-classical approximation” to that in Example 1.12.
Remark 1.17. More precisely, for a complex algebraic manifold Z, the cobherent derived category Dcoh
(T ∗ Z) can be thought of as a quasi-classicalapproximation to the derived category Db (Z) of arbitrary sheaves on Z.
Indeed, passing to solutions of D-modules gives a functor between derived
categories in the first line of the following table (a functor that restricts to
the Riemann-Hilbert equivalence between constructible and holonomic regular derived categories). It can be compared with the second line which is an
instance of Serre’s theorem for the affine morphism p : T ∗ Z → Z.
(Arbitrary) sheaves on Z
Coherent sheaves on T ∗ Z

Coherent DZ -modules on Z
Coherent p∗ OT ∗ Z = gr(DZ )modules on Z

Note that the functor S = q −1 on sheaf-theoretic derived categories in
Example 1.12 matches, after being interpreted in terms of D-modules andpassing to the associated graded, the functor S = i∗ ρ∗ on coherent derived
categories in Example 1.16.

9

2

Disk with several marked points

A. Quiver description of perverse sheaves. Let B = {b1 , · · · , bn } be a
finite set of marked points in the unit disk ∆ and Perv(∆, B) be the categoryof perverse sheaves on ∆ with possible singularities at B. We then have thefollowing [GMV, Prop. 1.2].
Proposition 2.1. Perv(∆, B) is equivalent to the category Pn of diagramsvi
formed by vector spaces Ψ, Φ1 , · · · , Φn and linear maps Φi o u / Ψ such thatieach TΨ,i := IdΨ − vi ui is an isomorphism.
The category Pn can be seen as an amalgamation of n copies of thecategory P1 from Theorem 1.1. To construct an equivalence, we choose abase point b ∈ ∂∆ and a “system of cuts" K, i.e., a set of simple arcs
{K1 , · · · , Kn } with Ki connecting b with bi and with different Ki meeting
only at a small common
interval near b, see Fig. 2. Notationally, we view K
S
as the union K = Ki ⊂ ∆.
∆
b1

K1

b2
Kn

K2

..
.

b

bn

Figure 2: Equivalence depending on a system of cuts.
Given K, we have an equivalence FK : Perv(∆, B) → Pn sending F to
(2.2)

1
ΦK
i (F) = HK (F)bi ,ΨK (F) = Fb ' H1Ki (F)b = H1K (F)b .

Remark 2.3. We have the following elementary statements, continuing thepattern of Remark 1.6.
10

(1) HiK (F) = 0 for i 6= 1.
(2) Each stalk of R(F), considered as a functor Perv(∆, B) → Vectk , is anexact functor which takes Verdier duality to vector space duality.
Remark 2.4. The space of cohomology with support H1K (∆, F) can be seen
as “uniting” all the spaces of vanishing cyclies ΦKi (F). As common in singularity theory, we can imagine that F obtained as a deformation of a perverse
sheaf G ∈ Perv(∆, 0) with only one (but more complicated) singular point at
0. Then H1K (∆, F) recovers Φ(F). Note that a categorification of the spaceof vanishing cycles of an isolated singular point of a function is provided
by the Fukaya-Seidel category [Se], and the method of construction adopted
in loc. cit. uses precisely a deformation into several Morse critical points.
Therefore, the space H1K (∆, F) for a perverse sheaf F ∈ Perv(∆, B), can beseen as a de-categorified analog of the Fukaya-Seidel category.
The equivalence FK depends only on the isotopy class of K. Unlike the one
point case, there are now many such classes, forming a set which we denote
C. It is acted upon simply transitively by the Artin braid groupBrn =s1 , · · · , sn−1 si si+1 si = si+1 si si+1 .

Indeed,Brn = π0 Diff + (∆; B, b)
is the group of isotopy classes of diffeomorphisms of ∆, preserving orientation,
preserving b as a point and B as a set. The equivalences FK for different
K ∈ C are connected by self-equivalences fσ of Pn :
(2.5)

fσ

Pn e

/
9 Pn ,FKσ ∈ Brn .

Fσ(K)Perv(∆, B)
The self-equivalence fsi corresponding to a generator si of Brn , is given by[GMV, Prop. 1.3]:
fsi Ψ, Φj , ui , vj = Ψ, Φ0j , u0j , vj0 ,(2.6)

Ψ0 = Ψ, Φ0j = Φj , u0j = uj , vj0 = vj ,j 6= i, i + 1,

Φ0i+1 = Φi , Φ0i = Φi+1 ,−10u0i = ui+1 , vi0 = vi+1 , u0i+1 = ui TΨ,i+1 , vi+1
= TΨ,i+1vi .
11Remarks 2.7. (a) Note that perverse sheaves being a topological concept,
the group Diff + (∆; B, b) naturally acts on the category Perv(∆, B) from thefirst principles, the action descending to that of Brn .
(b) We can turn Proposition 2.1 and formulas (2.6) around to produce anintrinsic (i.e., not tied to any particular K and manifestly Brn -equivariant)definition of Perv(∆, B) which does not appeal to any pre-existing conceptof a perverse sheaf. More precisely, we can define an object P ∈ Perv(∆, B)
to be a system of objects PK ∈ Pn , K ∈ C and compatible isomorphisms
fσ (PK ) → Pσ(K) , σ ∈ Brn so that each particular PK is just a particular“shadow” of a more intrinsic object P .
B. Schobers on a disk with several marked points. To give an “invariant" definition of a perverse Schober on (∆, B), we adopt the approachof Remark 2.7(b). That is, for each system of cuts K ∈ C we define a Kcoordinatized Schober to be a system of n spherical functors with a common
targetSK = Si : Di −→ D, i = 1, · · · , n .
Each SK gives rise to spherical reflection functors Ti = Cone{Si Ri ⇒ IdD },i = 1, · · · , n. According to our convention on working with triangulated
categories in terms of dg-enhancements, see Appendix, all K-coordinatized
Schobers form an ∞-category which we denote SchK (∆, B). We define equivalences fσ : SchK (∆, B) → Schσ(K) (∆, B), σ ∈ Brn on generators si ∈ Brn by
the direct analog of (2.6):

(2.8)



fsi Si : Di −→ D = Si0 : Di0 −→ D0 ,D0 = D, Dj0 = Dj , Sj0 = Sj , j 6= i, i + 1,
0
Di+1
= Di , Di0 = Di+1 ,−10
Si0 = Si+1 , Si+1
= Ti+1Si .

We then extend to arbitrary σ ∈ Brn by verifying the braid relations for thefi which is done in exactly the same way as for (2.6).
By definition, a perverse Schober on (∆, B) is a system S = (SK )K∈C
of coordinatized Schobers and compatible identifications fσ (SK ) → Sσ(K) .
The datum SK will be referred to as the K-shadow of S. We denote bySch(∆, B) the ∞-category of perverse Schobers on (∆, B).

12

C. The Picard-Lefschetz formula. Underlying classical Picard-Lefschetz
theory, there is a general statement about perverse sheaves on a disk which
we now formulate.
Let F ∈ Perv(∆, B). In the approach of (2.2), “the” space of vanishingcycles of F at some bi ∈ B can be defined in terms of a small segment of anarc terminating in bi (which does not, a priori, have to be a part of a systemof cuts). Let now γ be a simple arc joining two marked points bi and bk andnot passing through any other marked points, as in Figure 3.
∆
bi
K

β
bj
α

γ

γ

0

b

bk

Figure 3: The Picard-Lefschetz situation.
We can then define the spaces of vanishing and nearby cycles of F relativeto γ:
Φi,γ = H1γ (F)bi , Φk,γ = H1γ (F)bk ,Ψγ = H1γ (F)gen (generic stalk).
which are connected by the maps
vi,γΦi,γ o u

i,γ

/

vk,γ

Ψγ o uk,γ/ Φk,γ .

The definition of these maps is similar to §1A: the maps v are generalization
maps, and the maps u are obtained by counterclockwise continuation, as in(1.5). We define the transition map along γ as
Mik (γ) = uk,γ ◦ vi,γ : Φi,γ −→ Φk,γ .
13The data of the Φi (F), i = 1, · · · , n as local systems on the circles around bi
and of all the Mik (γ) describe the image of F in the localization of Perv(∆, B)
by the subcategory of constant sheaves [GMV, §2].
One can say that “abstract Picard-Lefschetz theory” is the study of how
Mik (γ) changes when we replace γ by a different (non-isotopic) arc γ 0 .
More precisely, assume that γ 0 is obtained from γ by an “elementarymove” past another marked point bj so that the bigon formed by γ 0 and γ
contains bj and two arcs α and β, but no other marked points as in Fig. 3.
In particular, we assume that the closed path obtained by following γ from bi
to bk and then γ 0 from bk to bi , has orientation compatible with the standard(counterclockwise) orientation of ∆. Note that homotopy inside the bigon
and the clockwise rotation around bj give identifications(2.9)

Φi,γ ' Φi,β ' Φi,γ 0 ,Φk,γ ' Φk,α ' Φk,γ 0 ,Φj,β ' Φj,α ,so we can consider them as single spaces denoted by Φi , Φk and Φj respectively.
Proposition 2.10 (Picard-Lefschetz formula for perverse sheaves). We havethe equality of linear operators Φi → Φk :Mik (γ 0 ) = Mik (γ) + Mjk (α)Mij (β).
This statement is a version of [GMV, Prop. 2.4], formulated in a moreinvariant way and without localizing by constant sheaves. It holds for perverse sheaves on any oriented surface. It is convenient to give two proofs of
Proposition 2.10.
Invariant proof: To eliminate the need for the first two identifications in(2.9), let us deform the paths α, β, γ, γ 0 so that:• γ, β, γ 0 have a common segment [bi , b0i ] near bi .
• γ, α, γ 0 have a common segment [b0k , bk ] near bk . See Fig. 4.
We denote by γ, γ 0 the parts of γ and γ 0 lying between b0i and b0k , by β thepart of β between b0i and bj , and by α the part of α between bk and b0j .

14

bi

α

γ

γ0
b0k

b0i
γ0
γ

β

bk

Figure 4: The Picard-Lefschetz situation, deformed.
The points b0i , b0k being smooth for F, both sides of our putative equality
factor through the mapsv[bi ,b0i ] : Φi −→ Fb0i ,u[b0k ,bk ] : Fb0i −→ Φk .

Our statement would therefore follow from the next lemma which is an invariant version of the statement that the map (1.3) is indeed the monodromy
around 0.
Lemma 2.11. We have an equality of operators Fb0i → Fb0k :Tγ 0 = Tγ + vα Ruβ ,where:(1) uβ : Fb0i → H1β (F)bj is the coboundary of the counterclockwise continuation map, cf. (1.5).
(2) R : H1β (F)bj → H1α (F)bj is the identification obtained by deforming, byclockwise rotation aroung bj , the path β into the path α, i.e., R is thethird identification in (2.9).
(3) vα : H1α (F)bj → H1α (F)gen ' Fb0k is the generalization map of the sheaf
H1α (F).
Proof of the lemma: Let U be a small disk around bj . We fix a ∈ Fb0 andcompare the two sides of the putative equality when applied to a. For this,let sa ∈ Γ(U − β, F) be the section obtained by continuing a on the left
side of β towards and around bj . Then uβ (a) is equal to the class of sa in
H1β (F)bj . Next, Ruβ (a) is similarly represented by the section ta obtained
15

from sa by continuously moving the branch cut clockwise from α to β. Thismeans that ta = sa on the left side of α ∪ β. Finally, vα Ruβ (a) is obtained
as the difference of the two boundary values if ta when continued along bothsides of α all the w ay to b0k . It remains to notice that these boundary values
are equal, in virtue of the above, to Tγ (a) and Tγ 0 (a).
.
Proof using shadows: Choose a system of cuts K (depicted by dotted lines
in Fig. 3) adopted to our situation. We assume that the arcs Ki , Kj and
Kk are positioned as in the figure, i.e., that γ together with Ki and Kk
form a triangle containing Kj , α and β and not containing any other marked
points. We orient each Kν to run from b to bν . Consider the quiver FK (F) =(Ψ, Φi , ui , vi ). Note that we have isotopies of oriented paths rel. B:
α ∼ Kk ∗ Kj−1 , β ∼ Kj ∗ Ki−1 , γ 0 ∼ Kk ∗ Ki−1 ,γ ∼ Kk ∗ ∂∆ ∗ Ki−1 .
Here ∗ means composition of the paths and ∂∆ is the boundary circle of ∆,oriented anticlockwise and run from b to b. We can use these isotopies tocalculate the transition maps, obtaining
Mjk (α) = uk,K vj,K , Mik (γ 0 ) = uk,K vi,K ,Mik (γ) = uk,K Tj,Ψ vi,k ,Mij (β) = uj,K vi,K ,and the claim follows from the identity Tj,Ψ = Id − vi uj .
D. The Picard-Lefschetz triangle. Let S be a perverse Schober on(∆, B). The transition maps constructed in n◦ C categorify to functors between triangulated categories. More precisely, if γ is an oriented arc joining
bi and bk as above, then we have a diagram of triangulated categories andspherical functors which depends “canonically" (i.e., up to a contractible setof choices) only on the isotopy class of γ rel. B:(2.12)

Φi,γ (S)Si,γ/ Ψγ (S) oSk,γΦk,γ (S).

To define it, we choose K 0 ∈ C so that Ki0 (oriented from b to bi ), together
with γ and (Kk0 )−1 form a positively oriented triangle not containing anyother marked points. (Note: theof K as in Fig.3 is not good.) We choice
0
0
0 n
0
K
K
consider the K -shadow SK 0 = Sν : Dν → DK ν=1 of S and define000Ψγ (S) = DK , Φi,γ (S) = DiK , Φk,γ (S) = DkK ,00Si,γ = SiK , Sk,γ = SkK .
16

It is straightforward to see (by looking at the subgroup in Brn permuting all
K 0 with our property) that this definition is indeed “canonical” in the sense
described. So we consider the data (2.12) as intrinsically associated to S andγ.
Denote Rk,γ the right adjoint to the spherical functor Sk,γ . Define now
the transition functor
Mik (γ) = Rk,γ ◦ Si,γ : Φi,γ (S) −→ Φk,γ (S).
Consider now a situation when we have arcs γ, γ 0 , α, β depicted in Fig. 3.
Using identifications of categories similar to (2.9), we can speak about triangulated categories Φi (S) and Φk (S).
Proposition 2.13 (Picard-Lefschetz triangle). We have a canonical triangleof exact functors Φi (S) → Φk (S):Mik (γ) −→ Mik (γ 0 ) −→ Mjk (α) ◦ Mij (β) −→ Mik (γ)[1].
Proof: This is obtained identically to the “shadow" proof of Proposition 2.10
with the identity Tj,Ψ = Id − vj uj replaced by the triangle coming from thefact that Tj = Cone{Id → Rj Sj }.
E. Schobers on a Riemann surface. Let Σ be an oriented topologicalsurface, possibly with boundary ∂Σ, and B ⊂ Σ is a finite set not meeting∂Σ. One can then define a perverse Schober on (Σ, B) by decomposing Σ as
∆ ∪C U , where ∆ ⊂ Σ is a closed disk with boundary circle C = ∂∆ whichcontains all points of B, and U is the closure of Σ − ∆. Then a perverse
Schober S is, by definition, a datum of:(1) A perverse Schober S∆ on (∆, B), defined as in n◦ B.
(2) A local system of triangulated categories SU on U identified with S∆
over C.
While one can work with objects thus defined (for instance, one can constructtransition functors and Picard-Lefschet triangles for arcs not necessarily contained in ∆), a more intrinsic definition is desirable.

17

3

Spherical functors and spherical pairs

A. Symmetric description of Perv(∆, 0). In [KS1, §9] we have given a
different quiver description of Perv(∆, 0) obtained as a particular case of ageneral result for real hyperplane arrangements:
Proposition 3.1. The category Perv(∆, 0) is equivalent to the category formedby diagrams of vector spaces
E−

(3.2)

o γ−
δ−

γ+

/ E0 o/E+δ+satisfying the two following conditions:(1) γ− δ− = IdE− , γ+ δ+ = IdE+ .
(2) The maps γ− δ+ : E+ → E− , γ+ δ− : E− → E+ are invertible.
This is obtained by choosing not one but two base points b+ , b− ∈ ∂∆and considering a cut K which joins b+ with b− and passes through 0, as indepicted in Fig. 5.
∆

b−

0

K

b+

Figure 5: A symmetric cut.
The spaces E± , E0 are obtained as the stalks of the sheaf H1K (F) at b± and
0 respectively, the maps γ± are the generalization maps for this sheaf, andthe δ± can be obtained by duality.
Remarks 3.3. (a) Note that in this description the maps P+ = δ− γ− andP+ = δ+ γ+ are projectors in E0 , that is P±2 = P± . We can consider E± as
subspaces in E0 which are the images of P± .
18

(b) The pattern of Remarks 1.6 and 2.3 continues here: HiK (F) = 0 for
i 6= 1 and each stalk of H1K considered as a functor into Vectk , takes Verdier
duality into vector space duality.
B. Reminder on semi-orthogonal decompositions. A categorificationof diagrams (3.2) is naturally formulated in the language of semi-orthogonal
decompositions of triangulated categories [BK1] [KuL] which we now recall.
Let B be a full triangulated subcategory of a triangulated category A,
with i : B → A being the embedding functor. We denote by⊥

B = {A ∈ A| Hom(A, B) = 0 ∀B ∈ B},

B ⊥ = {A ∈ A| Hom(B, A) = 0 ∀B ∈ B}
the left and right orthogonals to B We say that B is left admissible, resp.
right admissible, if i has a left adjoint ∗ i, resp. a right adjoint i∗ . If B is left
(resp. right) admissible, we have a semi-orthogonal decomposition
A = h⊥ B, Bi,

resp. A = hB, B ⊥ iwhich means that each object A ∈ A is included in functorial exact triangles
C −→ A −→ B → C[1],resp. B 0 −→ A −→ D0 → B 0 [1],C ∈⊥ B, B = ∗ i(A) ∈ B,
B 0 = i∗ (A) ∈ B, D0 ∈ B ⊥ .

In particular, ⊥ B = Ker(∗ i), resp. B ⊥ = Ker(i∗ ). We will call ∗ i the projection
on B along ⊥ B and i∗ the projection on B along B ⊥ .
If B is left admissible, then ⊥ B is right admissible, and (⊥ B)⊥ = B.
Similarly, for a right admissible B we have that B ⊥ is left admissible and⊥(B ⊥ ) = B.
C. Spherical pairs. Let E0 be a triangulated category and E+ , E− ⊂ E0
be a pair of left admissible subcategories, so that we have the diagrams ofembeddings
δ−
δ+
j−
j+
E− −→ E0 ←− E+ , ⊥ E− −→ E0 ←− ⊥ E+with δ± having a left adjoint γ± and j± having a right adjoint k± .
Definition 3.4. The pair of left admissible subcategories E± is called a spherical pair, if:19

(SP1) The compositionsγ+ δ− : E− −→ E+ ,γ− δ+ : E+ −→ E−

are equivalences.
(SP2) The compositions
k+ j− : ⊥ E− −→ ⊥ E+ ,k− j+ : ⊥ E+ −→ ⊥ E−

are equivalences.
Remark 3.5. Note that unlike a spherical functor, the definition of a spherical pair does not appeal to any enhancement of the triangulated category E0 ,as no functorial cones are taken. In the following (as elsewhere in the paper)
we will, however, assume that we are in an enhanced situation as describedin Appendix.
D. From a spherical pair to a spherical functor. Let E± be a sphericalpair. Consider the diagramS

D0 = ⊥ E− = Ker(γ+ ) −→ E+ = D1 ,where S = γ+ |⊥ E+ .
Proposition 3.6. S is a spherical functor.
This follows from Theorem 1.9 (which gives that (SF1) and (SF3) imply
sphericity) and from the next more precise statement.
Proposition 3.7.
δ+(a) The functor R : E+ → ⊥ E− defined as the composik−tion E+ −→ E0 −→ ⊥ E− is right adjoint to S.
(b) The functor
T1 = Cone{SR ⇒ IdD1 } : D1 = E+ −→ E+ = D1
is identified with the composition γ+ ◦ δ− ◦ γ− ◦ δ+ : E+ → E+ . In
particular, it is invertible.

20(c) The functor
T0 = Cone{IdD1 ⇒ RS}[−1] : D0 = ⊥ E− −→ ⊥ E− = D0
is identified with the composition k+ ◦ j− ◦ k− ◦ j+ : ⊥ E− → ⊥ E− . In
particular, it is invertible.
Proof: (a) S is the composition γ+ ◦j− . By definition, γ+ = ∗ δ+ and therefore
δ+ = γ+∗ . Therefore S ∗ = j−∗ ◦ γ+∗ = k− ◦ δ+ .
(b) Let us make k− = j−∗ : E0 → ⊥ E− into an endofunctor of E0 bycomposing it with the embedding j− :⊥ E− → E0 . Then the functorial triangle
for the orthogonal decomposition E0 = h⊥ E− , E− i gives that
(3.8)

j− ◦ k− = Cone{IdE0 ⇒ δ− γ− } [−1].

We now write SR = S ◦ k− ◦ δ+ and note that S = γ+ ◦ j− , so SR is thecomposition
δ+
k−
j−
γ+
E+ −→ E0 −→ ⊥ E− −→ E0 −→ E+ .
Identifying δ+ on the left and γ+ on the right with the corresponding arrows
in the proposed formula for T1 , we reduce part (b) of our proposition to thestatement thatδ− ◦ γ− = Cone{j− ◦ k− ⇒ IdE0 },which is equivalent to (3.8).
(c) is proved similarly to (b).
We do not know whether each spherical functor can be obtained from aspherical pair. but the following construction provides some geometric hint.
Example 3.9 (Polar coordinates). Let Y be a CW-complex and p : V →Y be a real vector bundle. Denote by i : Y ,→ V the embedding of the zerosection, and by j : V ◦ ,→ V the embedding of the complement of the zerosection. Let q : S = V ◦ /R∗>0 → Y be the spherical bundle associated to V ,and τ : V ◦ → S the natural projection. Let also π = qτ : V ◦ → Y be thecomposite projection.
b
Let E0 = E0 (V ) := Dconic(V ) ⊂ Db (V ) be the full subcategory of R∗>0 conic complexes, i.e., of complexes F such that each H i (F) is constant on21

each orbit of R∗>0 in V , see [KS2]. Consider the subcategories E± = E± (V ) ⊂E0 defined as follows. First, let
E+ = E+ (V ) := j! π −1 Db (S) ' Db (S), so that ⊥ E+ = i∗ Db (Y ) ' Db (Y ).
Second, let E− = E− (V ) consist of complexes F with Rp∗ F = 0, so that⊥E− = p−1 Db (Y ) ' Db (Y ). Then (E+ , E− ) is a spherical pair, and thefunctor
S=γ+ j−Db (Y ) ' ⊥ E− −→ E+ ' Db (S)is identified with the spherical functor q −1 from Example 1.10.
Remark 3.10. In the situation of Example 3.9, let V ∗ be the vector bundle
dual to V , so that the Fourier-Sato transform [KS2] gives an identificationFbb
E0 (V ) = Dconic(V ) −→ Dconic(V ∗ ) = E0 (V ∗ ).

This identification takes the category E± (V ) to E∓ (V ∗ ).

22

4

Derived categories on the G/P and Schobers
on symmetric products h/W .

A. The braid group action on Db (G/B) and Dcoh (T ∗ (G/B)). Let g be a
split reductive Lie algebra over C, with Cartan subalgebra h and Weyl group
W . We denote by hR ⊂ gR the real parts of h and g. Let ∆sim ⊂ ∆+ ⊂ ∆ be
the systems of simple and positive roots of g inside the set of all roots. The
complex vector space h has an arrangement of hyperplanes {α⊥ }α∈∆+ and so
has a natural stratification by flats of this arrangements, see [KS1].
We consider the complex manifold h/W . It has the induced stratification,
denote it S. The open stratum (h/W )0 , is the classifying space of the braid
group of g, which we denote by Br = Br(g). By definition, Br is generated
by elements sα , α ∈ ∆sim subject to the relations defining the Weyl group W
with the exception of the relations s2α = 1.
Example 4.1. In the case g = gln , the manifold h/W is space of monicpolynomials f (z) of degree n in one variable. The stratification S is thestratification by types of coincidence of roots of f (z) and is labelled by (unordered) partitions of n. The group Br(gln ) is the usual Artin braid groupBrn .
Let B ⊂ G be the standard Borel subgroup corresponding to the choiceof ∆+ . For any I ⊂ ∆sim let PI ⊃ B be the standard parabolic subgroup
generated by B and the exponents of the Chevalley generators associatedto (−α), α ∈/ I. Thus P∅ = G and P∆sim = B. We write Pα = P∆sim −{α} ,α ∈ ∆sim , for the “next to minimal" parabolic subgroup associated to {α}.
We have the P1 -fibrationqα
G/B −→ G/Pα
and therefore, by Proposition 1.11, a spherical functor−1
Sα =qαDb (G/Pα ) −→ Db (G/B)
and the corresponding twist functor Tα : Db (G/B) → Db (G/B) written
directly as(4.2)

pα,i :Tα (F) = R(pα,2 )! p∗1,α F[−1],(G/B) ×G/Pα (G/B) − rel. diagonal −→ G/B, i = 1, 2.

The following fact was known before the concept of a spherical functor
was discovered, cf. [BBM].
23

Proposition 4.3. The functors Tα defined by (4.2), are equivalences which
satisfy the relations of Br(g), i.e., define an action of Br(g) on the derived
category Db (G/B).
Remarks 4.4. (a) Proposition 4.3 for g = gl(n) has a quaternionic analog.
We consider the space HFn of complete flags of (left) quaternionic subspaces
V1 ⊂ · · · ⊂ Vn−1 ⊂ Hn , dimH (Vi ) = i. It fits into the HP1 -fibrations
qi : HFn −→ HFn(i) = (V1 ⊂ · · · ⊂ Vi−1 ⊂ Vi+1 ⊂ · · · ⊂ Hn
over the spaces of next-to-complete flags. We note that HP1 = S4 is the 4sphere. By Proposition 1.11 this gives, for each i = 1, · · · , n − 1, a spherical(i)functor SiH = qi−1 : Db (HFn ) → Db (HFn ) and the induced twist automorphism TiH of Db (HFn ). One can see directly by analyzing the Schubertcorrespondences in HFn , that the TiH satisfy the relations of the Artin braid
group Brn .
(b) One has also a real analog for any g, using the real loci of the flag
varieties which fit into RP1 = S1 -fibrations.
A quasi-classical analog of the action in Proposition 4.3 has been constructed in [KhT, BR]. It uses the diagramραiαT ∗ (G/B) := X
Yα := T ∗ (G/Pα ) ←− Dα := T ∗ (G/Pα ) ×G/Pα G/B −→which produces a spherical functorbbSαcoh = iα∗ ρ∗α : Dcoh
T ∗ (G/Pα ) → Dcoh
T ∗ (G/B)

It was proved in [KhT] for g = gln and in [BR] for arbitrary g, that thebcorresponding twist functors Tαcoh define an action of Br(g) on Dcoh
T ∗ (G/B)(which, in fact, extends to an action of the affine braid group).
These constructions can be seen as giving local systems of triangulated
categories on (h/W )0 = K(Br(g), 1) with general stalk being Db (G/B), resp.
b
Dcoh(T ∗ (G/B)). We would like to suggest that these local systems extend
to natural perverse Schobers on the entire h/W . For this, we review somefeatures of usual perverse sheaves in this situation.
B. Perverse sheaves on h/W and double cubical diagrams. Denote
Perv(h/W ) the category of perverse sheaves on h/W smooth with respect tothe stratification S from n◦ A. A complete quiver description on Perv(h/W ) is24not yet available. However, the results of [KS1] provide the following partial
picture which aligns with the examples we considered earlier.
Let hR ⊂ h be the real form of h. Inside h/W we consider the “realskeleton" (or “cut”)
K = hR /W ⊂ h/W.
It can be thought of as a “curvilinear cone", the image of the dominantsim
Weyl chamber h+| be the rank of g. Let SR be theR ⊂ hR . Let r = |∆rstratification of K by the 2 strata which are the images of the faces of h+R.
We denote these strata by SI , I ⊂ ∆sim , so that dimR (SI ) = |I|.
Note that each SI is contractible. Therefore (cf. [GM]) a sheaf G on K
constructible with respect to SR can be recovered from the cubical diagram ofvector spaces GI = Γ(SI , G) and generalization maps γII 0 : GI → GI 0 , I ⊂ I 0
sim
which are required to form a representation of the poset 2∆ (so that the
diagram is commutative).
The following is deduced by pulling F back to a perverse sheaf on hsmooth with respect to an arrangement of hyperplanes {α⊥ }α∈∆+ and applying the results of [KS1].
Proposition 4.5. (a) For F ∈ Perv(h/W ) we have HiK (F) = 0 for i 6=
dim(K)
dimR (K), and the sheaf RF = HK(F) on K is constructible with respect
to the stratification SR .
(b) Let I ⊂ ∆sim . Denoting EI (F) = Γ(SI , RF ), we have that EI :Perv(h/W ) → Vectk is an exact functor which takes the Verdier duality to
the vector space duality.
Therefore we can associate to any F ∈ Perv(h/W ) a double cubical diagram E(F) formed by the vector spaces EI (F) and the maps
(4.6)

EI (F) o

γIJ
δJI

/

EJ (F) ,I ⊂ J.

Here γIJ is the generalization map for EF , and δJI is the dual to the generalization map for EF ∗ , where F ∗ is the perverse sheaf Verdier dual to F.
Each of the collections (γIJ ), (δJI ) forms a commutative cube. It seems very
plausible that the functor F 7→ E(F) from Perv(h/W ) to the category of
double cubical diagrams is fully faithful, i.e., F can be recovered from E(F).
Example 4.7. For g = sl2 we have h/W = C and K = R≥0 so the above
reduces, very precisely, to the construction of §1A, except with the disk
replaced by C.
25

C. A double cubic diagram related to flag varieties. We now note thatgeometry of flag varieties provides a natural double cubic diagram of categories of the same shape as (4.6). More precisely, we have an ordinary cubical diagram of algebraic varieties G/PI and projections qIJ : G/PJ → G/PI ,I ⊂ J. We have then the double cubical diagram formed by triangulated
categories Db (G/PI ) and the adjoint pairs of functors(4.8)

Db (G/PI ) o

−1
qIJ

/

Db (G/PJ ) ,I ⊂ J.

RqIJ∗−1
As in Example 1.10, we can also consider the left adjoint to qIJ
which differsfrom the right adjoint by a shift.
At the “quasi-classical level” one has a similar diagram formed by the catb
egories Dcoh
(T ∗ (G/PI )) and the functors between them obtained by trans−1
lating qIJ
and RqIJ∗ into the language of D-modules and passing to theassociated graded modules.

We would like to suggest that the diagram (4.8) comes from a more fundamental object: a perverse Schober on h/W extending the local system
discussed in n◦ A.
bSimilarly for the quasi-classical analog with the Dcoh
(T ∗ (G/PI )). In thissituation we have in fact more: an actions the affine braid group. Thesecan possibly come from Schobers not on h/W but on T /W where T is the
maximal torus in the algebraic group corresponding to g.

26

5“Fukaya-style” approach to perverse sheaves:cuts, real skeletons and Langangian varieties

A. Maximally real cuts. Since our preliminary definitions of a perverse
Schobers were based on quiver descriptions of perverse sheaves, let us lookat some general features of such descriptions.
Let (X, S) be a stratified complex manifold of dimension n. Obtaining
a quiver description of Perv(X, S) requires, in particular, construction ofmany exact functors Perv(X, S) → Vectk . Indeeed, any component of theputative quiver must be such a functor. Arrows of the quiver are then naturaltransformations between these exact functors.
The common tool for that, used in examples in this paper, is a choice ofa closed subset K ⊂ X with the following property:(Cut) For any F ∈ Perv(X, S) we have HiK (F) = 0, i 6= n.
This property implies that the functor of abelian categories
R : F 7−→ R(F) := HnK (F),

Perv(X, S) −→ ShK

is exact. So stalk of R(F) at any point x ∈ K can be used as a component ofa quiver, while generalization maps between the stalks provide some of thearrows.
We will call each K satisfying (Cut) an (admissible) cut for (X, S) and denote by C(X, S) the set of all such cuts. It is natural to look for a descriptionof Perv(X, S) in terms of some data associated to all the cuts.
Recall that by the Riemann-Hilbert correspondence we can realize eachF ∈ Perv(X, S) as R HomDX (M, OX ) for a left DX -module M whose characteristic variety satisfies the inclusion[
TX∗ α X.
Ch(M) ⊂ ΛS :=α

We denote by D Mod(X, S) the category of such DX -modules. In these terms,
a more detailed scenario (sufficient condition) for (Cut) to hold would be for
K to satisfy the following two properties:(Cut1) We have HiK (OX ) = 0 for i 6= n.
27

(Cut2) Assuming (Cut1), the sheaf BK = HnK (OX ) (which, considered as asheaf on X, is automatically a sheaf of left DX -modules), satisfies
ExtjDX (M, BK ) = 0,

∀M ∈ D Mod(X, S), j > 0.

The property (Cut1) is not related to a choice of S and holds for any
totally real subset of X, see [Har]. More precisely, we recall, see [BER] forbackground:
Definition 5.1. (a) Let V be a C-vector space of finite dimension n. A
real subspace L ⊂ V is called totally real, if L ∩ iL = 0. We say that L is
maximally real, if it is totally real of dimension n.
(b) Let X be a complex manifold of dimension n. A C ∞ -submanifold
K ⊂ X is called totally real, resp. maximally real if for each x ∈ K thesubspace Tx K ⊂ Tx X is totally real resp. maximally real.
The results of Harvey [Har] imply:
Proposition 5.2. Any closed subset of a totally real submanifold of X satisfies (Cut1).
More precisely, the result of loc. cit. is for arbitrary totally real subsetsof X, a class of sets which includes totally real submanifolds [Har, §3.6 Ex.
2] and is closed under passing to closed subsets [Har, Cor. 3.2].
For example, Rn , as well as Rn≥0 satisfies (Cut1). The sheaf BRn is thesheaf of hyperfunctions of Sato [SKK].
We now consider the condition (Cut2). Sufficient criteria for it to holdwere given by Lebeau [Le] and Honda-Schapira [HS]. The criterion of [HS]is based on the concept of positive position of two real Lagrangian submanifolds in a complex symplectic manifold such as T ∗ X. We do not recall thisconcept here, referring to [Sch] and references therein for more background.
Informally, the essense of the criterion can be formulated like this.

(5.3)

For (Cut2) to hold, K must be “maximally real with respect tothe stratification S”, in particular, the intersection of K
with each stratum Xα should be maximally real in Xα .

Here is a precise but more restrictive statement which is a reformulation of
Example 1 of [HS].
28Proposition 5.4. Suppose S is such that ΛS is contained in the union ofTZ∗β X for a collection of smooth closed complex submanifolds Zβ ⊂ X. Suppose K ⊂ X is a maximally real analytic submanifold such that each K ∩ Zβ
is maximally real in Zβ . Then K satisfies (Cut2).
B. Maximally real vs. Lagrangian cuts. Let V be a complex vector
space of dimension n and GR (n, V ) the Grassmannian of real n-dimensional
subspaces in V . We denote by Gmax (V ) the open subset in GR (n, V ) formedby maximally real subspaces.
Suppose V is equipped with a positive definite hermitian form h. Separating the real and imaginary parts h = g +iω, we have that ω is a symplectic
form on V . Let LGω (V ) be the closed subset in GR (n, V ) formed by subspaces
Lagrangian with respect to ω. The following is well known.
Proposition 5.5. LGω (V ) is contained in Gmax (V ) and the embedding isa homotopy equivalence. In other words, LGω (V ) can be seen as a compact
form of Gmax (V ).
Proof: For V = Cn with the standard hermitian form we haveLGω (V ) = O(2n)/U (n) ⊂ GL(2n, R)/GL(n, C) = Gmax (V ).
Let now X be a complex manifold equipped with a Kähler metric h =g + iω. Then ω makes X into a symplectic manifold, and we have
Corollary 5.6. Any Lagrangian submanifold of X is maximally real.
Note that all the cuts used in this paper as well as in [GGM] [KS1], are
Lagrangian. This suggests a possibility of describing more general Perv(X, S)
in terms of data coming from Lagrangian cuts.
On the other hand, Lagrangian submanifolds of a Kähler manifold X areorganized into a far-reaching structure: the Fukaya category F(X) (as well as
its modifications such as the Fukaya-Seidel and wrapped Fukaya categories),see [FO3 ] [Se] for background. We briefly recall that F(X) is an (a priori
partially defined) C-linear A∞ category whose objects are, in the simplest
setting, compact Lagrangian submanifolds K ⊂ X. The space Hom(K1 , K2 )is defined when K1 and K2 meet transversely and in this case is formally29

spanned by the set K1 ∩ K2 (with appropriate grading, see loc. cit.). The
A∞ -compositionµK1 ,··· ,Km :m−1
O

Hom(Ki , Ki+1 ) −→ Hom(K1 , Km ),m ≥ 1,

i=1

is given by counting holomorphic2 disks D ⊂R X with boundary on K1 ∪ · · · ∪Km . Each D gives a contribution exp − D ω to the appropriate matrix
element of µK1 ,··· ,Km .
One can expect that the Fukaya structure on the collection of admissible Lagrangian cuts for (X, S) has some significance for explicit descriptionof Perv(X, S). In particular, a holomorphic disk D with boundary on theunion of admissible cuts K1 , · · · , Km may provide a link between the sheaves
HnKν (F), F ∈ Perv(X, S) for different ν, via the structure of perverse sheaves
on a disk (§2).
Remarks 5.7. (a) Any collection K1 , · · · , Km of maximally real (not necessarily Lagrangian) submanifolds in X provides a natural boundary condition
for holomorphic disks. It has been noticed [Se] that many ingredients of the
Fukaya category construction have an “intrinsic" meaning and can be definedwithout explciit reference to the symplectic structure. In particular:• The role of the grading on Hom(K1 , K2 ) is to make the count of disks
possible by ensuring that in evaluating matrix elements of µ between
basis vectors of the same degree, the index of the linearlized elliptic
problem is 0 (so the corresponding linear ∂-operator is, generically,
invertible).
R • The quantity exp − D ω can be intepreted as the determinant of theinvertible ∂-operator above.
They could therefore make sense for more general maximally real submanifolds. The usual technical reason for restricting to Lagrangian submanifolds
in defining F(X) is that the Gromov compactness theorem (which, via theproperties of 1-dimensional moduli spaces, is used to prove the A∞ -axioms)
2
Here we leave aside the additional complication that it may be necessary to deformthe complex structure on X to ensure generic position.

30

has originally been established only in that setting. However, see [FZ] for a
recent generalization to the maximally real case.
(b) the above mentioned role of cuts (in particular, Lagrangian cuts)for description of perverse sheaves on X, is different from the classical “microlocal” point of view [KS2] which empasizes complex, conic Lagrangiansubvarieties in T ∗ X rather than real ones in X itself. For the relation of the
microlocal approach to the Fukaya category (of T ∗ X), see [NZ][Na]. A moregeneral idea that the Fukaya category of any symplectic manifold should havean interpretation in terms of its geometric quantization, was proposed earlierin [BS].
C. Perverse Schobers as coefficients data for Fukaya categories.
The Fukaya category F(X) of a symplectic manifold X can be seen as acategorification of its middle (co)homology (or, rather, of the part represented
by Lagrangian cycles). Here “cohomology” is understood as H • (X, C), thecohomology with constant coefficients, a particular case of a more general
concept of H • (X, F), the cohomology with coefficients in a sheaf F or evenmore generally, in a complex of sheaves.
This point of view leads to the idea of introducing coefficients into thedefinition of the Fukaya category as well. It was proposed by M. Kontsevich
with the goal of understanding the (usual) Fukaya category of a manifold byprojecting it onto a manifold of smaller dimension.
In this direction we would like to suggest that perverse Schobers are theright coefficient data for defining Fukaya categories. That is, to a perverse
Schober S on a Kähler manifold X there should be naturally associated a
triangulated category F(X, S), which for the constant Schober Db (VectC )reduces to F(X). If we think about perverse sheaves in terms of some vector
space data associated to Lagrangian cuts and then categorify these data todefine Schobers, then it is natural to try to define F(X, S) in terms of thesecategorified data.
For example, when X is a Riemann surface with marked points, theFukaya category of X with coefficients in a constant (Z/2-graded) triangulated category A was defined in [DK]. One can easily modify this definition
when A is replaced by a local system of triangulated categories on X, i.e.,by the next simplest instance of a perverse Schober. We leave the case of anarbitrary perverse Schober on a Riemann surface (§2E) for future work.
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A

Conventions.

For the considerations of this paper to make sense, we should work in someframework of “refined”, or “enhanced” triangulated categores having functorialcones. Let us describe one such framework, to be followed in the main body.
(Another one would be that of stable ∞-categories [Lu].)All our categories will be k-linear, where k is a fixed base field. We recall
that Tabuada [Ta1, Ta2] has introduced a Morita model structure on the
category of dg-categories. Dg-categories fibrant with respect to this model
structure are called perfect, see also [TV]. Perfect dg-categories form a subclass of pre-triangulated categories in the sense of [BK2]. That is, a perfect
dg-category A gives rise to a triangulated category H 0 (A) which is, in addition, idempotent complete.
In the main body of this paper the word “triangulated category" will always mean “a triangulated category D together with an identification D 'H 0 (A) where A is a perfect dg-category", and “exact functor" will mean “anexact functor of triangulated categories obtained, by passing to H 0 , from a
dg-functor between perfect dg-categories" and similarly for “natural transformation". With this understaning we will speak about “the” exact functor
Cone{T : F ⇒ G} where T is a natural transformation of exact functors.
For a finite CW-complex Z we denote Db (Z) the bounded derived category of all sheaves of k-vector spaces on Z. For a smooth complex algebraicb(X) the bounded derived category of coherent
variety X we denote by Dcohsheaves on X.
The condition of perversity for constructible complexes on a complex
manifold X is normalized in such a way that a constant sheaf in degree 0 isperverse.

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References[A1] R. Anno Spherical functors, arXiv:0711.4409.
[A2] R. Anno. Affine tangles and irreducible exotic sheaves, arXiv:0802.1070.
[AL1] R. Anno, T. Logvinenko. Orthogonally spherical objects and spherical fibrations, arXiv:1011.0707.
[AL2] R. Anno, T. Logvinenko. Spherical DG-functors, arXiv:1309.5035.
[BER] M. Salah Bauendi, P. Ebenfeld, L. P. Rotschild. Real Submanifolds in
Complex Space and Their Embeddings. Princeton Univ. Press, 1999.
[Be]

A. Beilinson. How to glue perverse sheaves. In: K-theory, arithmetic andgeometry (Moscow, 1984), Lecture Notes in Math. 1289, Springer-Verlag,
1987, 42 - 51.

[BBD] A. Beilinson, I. Bernstein, P. Deligne. Faisceaux Pervers, Astérisque 100,
1982.
[BBM] A. Beilinson, R. Bezrukavnikov, I. Mirkovic. Tilting exercises. Mosc. Math.
J. 4 (2004), 547–557.
[BR] R. Bezrukavnikov, S. Riche. Affine braid group actions on derived categories
of Springer resolutions. Ann. Sci. Éc. Norm. Supér. 45 (2012), 535-599.
[BK1] A. I. Bondal, M. M. Kapranov. Representable functors, Serre functors andmutations. Math. USSR Izv. 35 (1990), 519-541.
[BK2] A. I. Bondal, M. M. Kapranov. Enhanced triangulated categories. Math.
USSR Sbornik, 70 (1991) 93-107.
[BS] P. Bressler, Y. Soibelman. Mirror symmetry and deformation quantization.
arXiv hep-th/0202128.
[BT] C.Brav, H.Thomas, Braid groups and Kleinian singularities, Math. Ann. 351(2011), 1005 - 1017.
[CG] Neil Chriss, Victor Ginzburg, Representation Theory and Complex Geometry. Birkhäuser, 1997.
[Cu] J. Curry. Sheaves, Cosheaves and Applications. arXiv:1303.3255.

33

[DK] T. Dyckerhoff, M. Kapranov. Triangulated surfaces in triangulated categories. ArXiv: 1306.2545.
[FZ] U. Frauenfelder, K. Zehmisch. Gromov compactness for holomorphic discs
with totally real boundary conditions. arXiv:1403.6139.
[FO3 ] K. Fukaya, Y.-G. Oh, H. Ohta, K. Ono. Lagrangian Intersection Floer
Theory. Amer. Math. Soc. Publ. 2009.
[GGM] A. Galligo, M. Granger, P. Maisonobe. D-modules et faisceaux pervers dont
le support singulier est un croisement normal. Ann. Inst. Fourier Grenoble,
35 (1985), 1-48.
[GM] S. Gelfand, R. D. MacPherson. Verma modules and Schubert cells: a dictionary. in: “Paul Dubreil and Marie-Paule Malliavin Algebra Seminar",
34th Year (Paris, 1981), pp. 1-50. Lecture Notes in Math. 924, Springer,
Berlin-New York, 1982.
[GMV] S. Gelfand, R. D. MacPherson, K. Vilonen. Perverse sheaves and quivers.
Duke Math. J. 3 (1996), 621-643.
[Har] R. Harvey. The theory of hyperfunctions on totally real subsets of a complex
manifold with application to extension problems. Amer. J. Math. 91 (1969),
853-873.
[HS] N. Honda, P. Schapira. A vanishing theorem for holonomic modules withpositive characteristic varieties. Publ. RIMS Kyoto Univ. 26 (1990), 529-534.
[KS1] M.Kapranov, V.Schechtman, Perverse sheaves over real hyperplane arrangements, arXiv:1403.5800.
[KS2] M. Kashiwara, P. Schapira. Sheaves on manifolds. Springer, 1990.
[KhT] M. Khovanov, R. Thomas. Braid cobordisms, triangulated categories, andflag varieties, arXiv:math/0609335.
[KuL] A. Kuznetsov, V. Lunts. Categorical resolution of irrational singularities,
arXiv:1212.6170.
[Le]

G. Lebeau. Annulation de la cohomologie hyperfonction de certains modules
holonomes. C.R. Acad. Sci. Paris, Sér. A, t.290 (1980), 313-316.

[Lu]

J. Lurie. Stable ∞-categories, arXiv math/0608228.

34

[Na] D. Nadler. Fukaya categories as categorical Morse homology. SIGMA Symmetry Integrability Geom. Methods Appl. 10 (2014), Paper 018.
[NZ] D. Nadler, E. Zaslow. Constructible sheaves and the Fukaya category. J.
Amer. Math. Soc. 22 (2009) 233-286.
[SKK] M. Sato, T. Kawai, M. Kashiwara. Hyperfunctions and pseudo-differential
equations. Lecture Notes in Math. 287 (1973), 265 - 529.
[Sch] P. Schapira. Conditions de positivité dans une variété symplectique complexe. Application a l’ étude des microfonctions. Ann. Sci. ENS 14 (1991),
121-139.
[Se]

P. Seidel, Fukaya categories and Picard-Lefschetz theory, European Mathematical Society, Zurich, 2008.

[ST] P. Seidel, R. Thomas, Braid group actions on derived categories of coherent
sheaves, arXiv:math/0001043.
[Ta1] G. Tabuada. Une structure de catégorie de modèles de Quillen sur la catégoriedes dg-catégories. C. R. Math. Acad. Sci. Paris 340 (2005) 15-19.
[Ta2] G. Tabuada. Théorie homotopique des DG-catégories, arXiv:0710.4303.
[TV] B. Toën, M. Vaquié. Moduli of objects in dg-categories. Ann. Sci. École
Norm. Sup. (4) 40 (2007) 387-444.

M.K.: Kavli Institute for Physics and Mathematics of the Universe (WPI), 5-1-5
Kashiwanoha, Kashiwa-shi, Chiba, 277-8583, Japan. Email: mikhail.kapranov@ipmu.jp
V.S.: Institut de Mathématiques de Toulouse, Université Paul Sabatier, 118 route deNarbonne, 31062 Toulouse, France, schechtman@math.ups-toulouse.fr

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