Table Of ContentPerverse sheaves and graphs on surfaces
6
1
0 Mikhail Kapranov, Vadim Schechtman
2
n
January 11, 2016
a
J
8
] 1 Introduction
T
A
. The aim of this note is to propose a combinatorial description of the cate-
h
t gories Perv(S,N) of perverse sheaves on a Riemann surface S with possible
a
m singularities at a finite set of points N. Here we consider the sheaves with
[ values in the category of vector spaces over a fixed base field k. We al-
low (in fact, require) the surface S to have a boundary. Categories of the
1
v type Perv(S,N) can be used as inductive building blocks in the study of any
9
category of perverse sheaves, see [GMV]. Therefore explicit combinatorial
8
7 descriptions of them are desirable.
1
0 We proceed in a manner similar to [KS1]. To define a combinatorial
.
1 data we need to fix a Lagrangian skeleton of S. In our case it will be a
0
spanning graph K ⊂ S with the set of vertices Vert(K) = N (for the precise
6
1 meaning of the word ”spanning” see Section 3B). (In the case of a hyperplane
v: arrangement inCn discussed in [KS1] theLagrangianskeleton was Rn ⊂ Cn.)
i We denote by Ed(K) the set of edges of K. We suppose for simplicity in this
X
Introduction that K has no loops. As any graph embedded into an oriented
r
a
surface, K is naturally a ribbon graph, i.e., it is equipped with a cyclic order
on the set of edges incident to any vertex.
To any ribbon graph K we associate a category A whose objects are
K
collections {E ,E ∈ Vect(k), x ∈ Vert(K),e ∈ Ed(K)} together with linear
x e
maps
γ //
E oo E .
x e
δ
given for each couple (x,e) with x being a vertex of an edge e. Here Vect(k)
denotes the category of finite dimensional k-vector spaces. These maps must
1
satisfy the relations which use the ribbon structure on K and are listed in
Section 3C below,.
IfK ⊂ S isaspanning graphasabove, thenourmainresult (seeTheorem
3.6) establishes an equivalence of categories
∼
Q : Perv(S,N) −→ A
K K
For F ∈ Perv(S,N) the vector spaces Q (F) ,Q (F) are the stalks of
K x K v
the constructible complex R (F) = Ri! (F)[1] on K which, as we prove, is
K K
identified with a constructible sheaf in degree 0.
A crucial particular case is S = the unit disc D ⊂ C, N = {0}. Take
for the skeleton a corolla K with center at 0 and n branches. Thus, the
n
same category Perv(D,0) has infinitely many incarnations, being equivalent
to A := A , n ≥ 1.
n Kn
The corresponding equivalence Q is described in Section 2, see Theorem
n
2.1. The special cases of this equivalence are:
∼
(i) Q : Perv(D,0) −→ A : this is a classical theorem, in the form given
1 1
in [GGM].
∼
(ii) Q : Perv(D,0) −→ A is a particular case of the main result of [KS1],
2 2
see op. cit., §9A. In loc. cit. we have also described the resulting
equivalence
∼
A −→ A
1 2
explicitely. In a way, objects of A are ”square roots” of objects of
2
A , in the same manner as the Dirac operator is a square root of the
1
Schro¨dingeroperator. ThatiswhywecallA a”1/n-spin(parafermionic)
n
incarnation” of Perv(D,0).
Finally, in Section 3D we give (as an easy corollary of the previous discus-
sion) a combinatorial description of the category PolPerv(S,N) of polarized
perverse sheaves, cf. [S]. These objects arise ”in nature” as decategorified
perverse Schobers, cf. [KS2], the polarization being induced by the Euler
form (X,Y) 7→ χ(RHom(X,Y)).
The idea of localization on a Lagrangian skeleton was proposed by M.
Kontsevich in the context of Fukaya categories. The fact that it is also
applicable to the problem of classifying perverse sheaves (the constructions
of [GGM] and [KS1] can be seen, in retrospect, as manifestations of this
2
idea) is a remarkable phenomenon. It indicates a deep connection between
Fukaya categories and perversity. A similar approach will be used in [DKSS]
to construct the Fukaya category of a surface with coefficients in a perverse
Schober.
The work of M.K. was supported by World Premier International Re-
search Center Initiative (WPI Initiative), MEXT, Japan.
2 The “fractional spin” description of perverse
sheaves on the disk
A. Statement of the result. Let X be a complex manifold. By a perverse
sheaf on X we mean a C-constructible complex F of sheaves of k-vector
spaces on X, which satisfies the middle perversity condition, normalized so
that a local system in degree 0 is perverse. Thus, if k = C and M is a
holonomic D -module, then RHom (M,O ) is a perverse sheaf.
X DX X
Let D = {|z| < 1} ⊂ C be the unit disk. Let k be a field. We denote by
Perv(D,0) the abelian category of perverse shaves of k-vector spaces on D
which are smooth (i.e., reduce to a local system in degree 0) outside 0 ∈ D.
Let n ≥ 1 be an integer. Let A be the category of diagrams of finite-
n
dimensional k-vector spaces (quivers) Q , consisting of spaces E ,E ,··· ,E
0 1 n
and linear maps
γi //
E oo E , i = 1,··· ,n,
0 i
δi
satisfying the conditions (for n ≥ 2):
(C1) γ δ = Id .
i i Ei
(C2) TheoperatorT := γ δ : E → E (where i+1 isconsidered modulo
i i+1 i i i+1
n), is an isomorphism for each i = 1,··· ,n.
(C3) For i 6= j,j +1 mod n, we have γ δ = 0.
i j
For n = 1 we impose the standard relation:
(C) The operator T = Id −γ δ : E → E is an isomorphism.
E1 1 1 1 1
Theorem 2.1. For each n ≥ 1, the category Perv(D,0) is equivalent to A .
n
3
For n = 1 this is the standard (Φ,Ψ) description of perverse sheaves on
the disk [GGM], [Be]. For n = 2 this is a particular case of the description
in [KS1] (§9 there).
B. Method of the proof. For the
proof we consider a star shaped graph
R
3
K = K ⊂ D obtained by drawing n
n R
2
radii R ,...,R from 0 in the counter-
1 n
U
clockwise order, see Fig. 1. Then D−K 2
U
is the union of n open sectors U ,··· ,U 1
1 n
R
numbered so that Uν is bordered by Rν 0 1
U
and Rν+1. ... n
Proposition2.2. ForanyF ∈ Perv(D,0)
we have Hi (F) = 0 for i 6= 1. Therefore R
K n
the functor
Figure 1: The graph K = K .
n
R : Perv(D,0) −→ Sh , F 7→ R(F) = R (F) := H1 (F)
K K K
is an exact functor of abelian categories.
Proof: Near a point x ∈ K other than 0, the graph K is a real codimension
1 submanifold in D, and F is a local system in degree 0, so the statement
is obvious (“local Poincar´e duality”). So we really need only to prove that
the space Hj (F) = Hj (D,F) vanishes for j 6= 1. The case n = 0 is
Kn 0 Kn
known, H1 (D,F) being identified with Φ(F), see [GGM]. The general case
K1
is proved by induction on n. We consider an embedding of K into K so
n n+1
that the new radius R subdivides the sector U into two. This leads to a
n+1 n
morphism between the long exact sequences relating hypercohomology with
and without support in K and K :
n n+1
··· //Hj(D −K ,F) //Hj+1(D,F) //Hj+1(D,F) //···
n Kn
αj βj =
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
··· //Hj(D −K ,F) // Hj+1 (D,F) //Hj+1(D,F) //···
n+1 Kn+1
For j 6= 0 the map α is an isomorphism because its source and target are 0.
j
Indeed, D−K as well as D−K is the union of contractible sectors, and
n n+1
F is a local system in degree 0 outside 0, so the higher cohomology of each
sector with coefficients in F vanishes. This means that
Hj+1 (D,F) = Hj+1(D,F) = 0, j 6= 0
Kn+1 Kn
4
and so by induction all these spaces are equal to 0.
Remark 2.3.An alternative proof of the vanishing of H6=1(F) can be ob-
Kn 0
tainedby noticing that RΓ (F) [1] canbe identified with Φ (F), the space
Kn 0 zn
of vanishing cycles with respect to the function zn. It is known that form-
ing the sheaf of vanishing cycles with respect to any holomorphic function
preserves perversity.
The graph K = K is a regular cellular space with cells being {0} and
n
the open rays R ,··· ,R . For F ∈ Perv(D,0) the sheaf R(F) is a cellular
1 n
sheaf on K and as such is completely determined by the linear algebra data
of:
(1) Stalks at the (generic point of the) cells, which we denote:
E = E (F ) := R(F) = stalk at 0;
0 0 n 0
E = E (F) = R(F) = stalk at R , i = 1,··· ,n.
i i Ri i
(2) Generalizationmaps corresponding to inclusions of closures of the cells,
which we denote
γ : E −→ E , i = 1,··· ,n.
i 0 i
This gives “one half” of the quiver we want to associate to F.
C. Cousin complex. In order to get the second half of the maps (the δ ),
i
we introduce, by analogy with [KS1], a canonical “Cousin-type” resolution of
any F ∈ Perv(D,0).
Denote by
i : K ֒→ D, j : D −K ֒→ D
the embeddings of the closed subset K and of its complement D − K =
n U . For any complex of sheaves F on D (perverse or not) we have a
ν=1 ν
cFanonical distinguished triangle in DbShD:
(2.4) i i!F −→ F −→ j j∗F −→δ i i!F[1]
∗ ∗ ∗
(here and elsewhere j means the full derived direct image). Recalling that
∗
i! has the meaning of cohomology with support, and denoting j : U ֒→ D
ν ν
the embeddings of the connected components of D −K, we conclude, from
Proposition 2.2:
5
Corollary 2.5. Let F ∈ Perv(D,0). Then F is quasi-isomorphic (i.e., can
be thought of as represented by) the following 2-term complex of sheaves on
D:
n
E•(F) = j F| −→δ R(F) .
(cid:26) ν∗ Uν (cid:27)
Mν=1 (cid:0) (cid:1)
Here the grading of the complex is in degrees 0,1 and the map δ is induced
by the boundary morphism δ from (2.4).
We further identify
F| ≃ E = constant sheaf on U with stalk E
Uν νUν ν ν
as follows. As F is locally constant (hence constant) on U , it is enough to
ν
specify an isomorphism
H0(U ,F) −α→ν E = H1 (F) ,
ν ν K x
where x is any point on R . Taking a small disk V around x, we have
ν
E = H1 (V,F). This hypercohomology with support is identified, via the
ν V∩K
coboundary map of the standard long exact sequence relating cohomology
with and without support), with the cokernel of the map
H0(V,F) −→ H0(V ∩U ,F)⊕H0(V ∩U ,F) = H0(V −K,F)
ν ν−1
SinceF isconstant onV, theprojectionofH0(V ∩U ,F) = H0(U ,F)tothe
ν ν
cokernel, i.e., to E , is an isomorphism. We define α to be this projection.
ν ν
We now assume n ≥ 2. Then the closure of U is a proper closed sector
ν
U , and so we can rewrite the complex E•(F) as
ν
n
E•(F) = E −→δ R(F) .
(cid:26) νUν (cid:27)
Mν=1
Indeed,
j (F| = j (E ) = E .
ν∗ Uν ν∗ νUν νUν
D. Analysis of the morphism δ. We now analyze the maps of stalks over
various points induced by δ. Since the target of δ is supported on K, it is
enough to consider two cases:
6
Stalks over 0. We get maps of vector spaces
δ : E = E −→ R(F) = E , ν = 1,··· ,n.
ν νUν 0 ν 0 0
(cid:0) (cid:1)
These maps, together with the generalization maps γ , form the quiver
ν
γi //
Q = Q(F) = E oo E , i = 1,··· ,n
0 i
(cid:8) δi (cid:9)
which we associate to F.
Stalks over a generic point x ∈ R . As x lies in two closed subsets U and
ν ν
U , we have two maps
ν−1
δ : E = E −→ R(F) = E ,
Uν,Rν νUν x ν x ν
(cid:0) (cid:1)
δ : E = E −→ R(F) = E .
Uν−1,Rν ν−1Uν−1 x ν−1 x ν
(cid:0) (cid:1)
Proposition 2.6. We have the following relations:
δ = Id , δ = −T ,
Uν,Rν Eν Uν−1,Rν ν−1
where
T : E = h0(U ,F) −→ E = H0(U ,F)
ν−1 ν−1 ν−1 ν ν
is the counterclockwise monodromy map for the local system F| .
D−{0}
Proof: We start with the first relation. Recal that the identification α :
ν
H0(U ,F) → E was defined in terms of representation of E as a quotient,
ν ν ν
i.e., in terms of the coboundary map in the LES relating hypercohomology
with and without support. So the differential δ in E•(F), applied to a section
s ∈ H0(U ,F), gives precisely α (s), so after the identification by α , the
ν ν ν
map on stalks over x ∈ R , becomes the identity.
ν
We now prove the second relation. Representing e ∈ E by a section
ν−1
s ∈ H0(U ,F), we see that δ (e) is represented by the image of
ν−1 Uν−1,Rν
(0,s) ∈ H0(U ,F)⊕H0(U ,F)
ν ν−1
in the quotient
H0(U ,F)⊕H0(U ,F) H0(V,F).
ν ν−1
(cid:0) (cid:1)(cid:14)
ButtheidentificationofthisquotientwithE isviatheprojectiontothefirst,
ν
notsecond, summand, i.e., toH0(V ∩U ,F) = H0(U ,F). The element (t,0)
ν ν
projecting to the same element of the quotient as (0,s), has t = −T (s) =
ν−1
minus the analytic continuation of s to U .
ν
7
Proposition 2.7. The maps γ ,δ in the diagram Q(F) satisfy the condi-
ν ν
tions (C1)-(C3), i.e., Q(F) is an object of the category A .
n
Proof:| We spell out the conditions that the differential δ in the Cousin com-
plex E•(F) is a morphism of sheaves. More precisely, both terms of the
complex are cellular sheaves on D with respect to the regular cell decompo-
sition given by 0,R ,··· ,R ,U ,··· ,U . So the maps of the stalks induced
1 n 1 n
by δ, must commute with the generalization maps.
Consider the generalization maps from 0 to R . In thje following diagram
ν
the top row is the stalk of the complex E•(F) over 0, the bottom row is the
stalk over R , and the vertical arrows are the generalization maps:
ν
n E Pδµ //E
µ=1 µ 0
L
pν,ν−1 γν
(cid:15)(cid:15) (cid:15)(cid:15)
E ⊕E // E ,
ν ν−1 ν
Id−Tν−1
the lower horizontal arrow having been described in Proposition 2.6. We
now spell out the condition of commutativity on each summand E inside
µ
n E .
µ=1 µ
L
Commutativity on E : this means γ δ = Id.
ν ν ν
Commutativity on E : this means γ δ = −T , in particular, this
ν−1 ν ν−1 ν−1
composition is an isomorphism.
Commutativity on E for µ 6= ν,ν −1: This means that γ δ = 0, since
µ ν µ
the projection p annihilates E . The proposition is proved.
ν,ν−1 µ
E. Q(F) and duality. Recall that the category Perv(D,0) has a perfect
duality
(2.8) F 7→ F⋆ = D(F)[2],
i.e., the shifted Verdier duality normalized so that for F being a local system
(in our case, constant sheaf) in degree 0, we have that F⋆ is the dual local
system in degree 0. We will use the notation (2.8) also for more general
complexes of sheaves on D.
On the other hand, the category A also has a perfect duality
n
Q = E oo γν //E n 7→ Q∗ = E∗ oo δν∗ //E∗ n .
(cid:8) 0 δν ν (cid:9)ν=1 (cid:8) 0 γν∗ ν (cid:9)ν=1
8
Proposition 2.9. The functor Q : Perv(D,0) → A commutes with duality,
n
i.e., we have canonical identifications Q(F⋆) ≃ Q(F)∗.
Proof: We modify the argument of [KS1], Prop. 4.6. That is, we think of K
as consisting of n “equidistant” rays R , joining 0 with
ν
ζν−1, ζ = e2πi/n, ν = 1,2,··· ,n.
We consider another star-shaped graph K′ formed by the radii R′,··· ,R′
1 n
so that R′ is in the middle of the sector U . Thus, the rotation by eiπ/n
ν ν
identifies R with R′ and K′ with K.
ν ν
We can use K′ instead of K to define R(F) and R(F⋆).We will denote
the corresponding sheaves R (F) = H1 (F), and similarly for R (F⋆).
K′ K′ K′
The sheaves defined using K, will be denoted by R (F) and so on.
K
Since Verdier duality interchanges i! and i∗ (for i : K′ → D being the
embedding), we have
⋆ ⋆
R (F ) ≃ F|
K′ K′
(usual restriction). To calculate this restriction, we use the Cousin resolution
of F defined by using K and the U :
ν
n
F ≃ E• = E −→δ R (F) .
(cid:26) νUν K (cid:27)
Mν=1
So we restrict E• to K′. SInce K′∩K = {0} and R (F) is supported on K,
K
the restriction R (F)| = E is the skyscraper sheaf at 0 with stalk E .
K K′ 00 0
So
n
F| ∼ (E ) δ′|K−′=→Pδν (E ) .
K′ (cid:26) ν R′ν 0 0(cid:27)
Mν=1
On the other hand, the shifted Verdier dual to R (F⋆), as a sheaf on K′ is
K′
identified by, e.g., [KS1], Prop. 1.11 with the complex of sheaves
⋆
P(γF )∗
E (F⋆)∗ ⊗or(C) −C→′C E (F⋆)∗ ⊗or(C) .
C C
(cid:26) C C (cid:27)
CM⊂K′ CM⊂K′
codim(C)=0 codim(C)=1
Here C runs over all cells of the cell complex K′, and E (F⋆) is the stalk of
C
the cellular sheaf R (F⋆) at the cell C. Explicitly, C is either 0 or one of
K′
the R′, so
ν
n ⋆
P(γF )∗
R (F⋆)⋆ = E (F⋆)∗ −C→′C E (F⋆)∗ .
K′ ν 0
(cid:26)Mν=1 Rν 0(cid:27)
9
By the above, this complex is quasi-isomorphic to
n
PδF
F| = E (F) −→ν E (F) .
K′ ν 0
(cid:26)Mν=1 Rν 0(cid:27)
So we conclude that
E (F⋆) = E (F)∗, γF⋆ = (δF)∗.
ν ν ν ν
This proves the proposition.
Proof of Theorem 2.1. We already have the functor
Q : Perv(D,0) −→ A , F 7→ Q(F).
n
Let us define a functor E : A → DbSh . Suppose we are given
n D
γν // n
Q = E oo E ∈ A .
(cid:8) 0 δν ν (cid:9)ν=1 n
We associate to it the Cousin complex
n
E•(Q) = E −→δ R(Q) .
(cid:26) νUν (cid:27)
Mν=1
Here R(Q) is the cellular sheaf on K with stalk E at 0, stalk E at R and
0 ν ν
the generalization map from 0 to R given by γ . The map ulδ is defined on
ν ν
the stalks as follows:
Over 0: the map
(E ) = E −→ R(Q) = E
νUν 0 ν 0 0
is given by δ : E → E .
ν ν 0
Over R : The map
ν
n
E = E ⊕E −→ R(Q) = E
µ ν ν−1 Rnu ν
(cid:18)Mµ=1 Uµ(cid:19)Rν |
is given by
Id−T : E ⊕E −→ E .
ν−1 ν ν−1 ν
Reading the proof of Proposition 2.6 backwards, we see that the conditions
(C1)-(C3) mean that in this way we get a morphism δ of cellular sheaves on
D, so E•(Q) is an object of DbSh .
D
Further, similarly to Proposition 2.9, we see that E•(Q∗) ≃ E(Q)⋆.
10
