Non-resonance D-modules over arrangements of hyperplanes

Sergei Khoroshkin and Vadim Schechtman

1 Introduction

The aim of this note is a combinatorial description of a category of D-modules over an affine space, smooth along the stratification defined by an arrangement of hyperplanes. These D-modules are assumed to satisfy certain non-resonance condition. The main result, see Theorem 4.1, generalizes [3].

2 Quivers of an arrangement

2.1. Let A = {H_{i}}, i ∈ I be a collection of affine hyperplanes in a complex affine space X = C^{N}. Hyperplanes H_{i} define on X the structure of a stratified space which we denote by the same letter A or by X_{A}. The closure of a stratum X_{α} ⊂ C^{N} is an intersection of some hyperplanes

\overline{X}_{α} = ⋂_{i∈J⊂I} H_{i} ,  (2.1)

and the stratum X_{α} is an interiour of \overline{X}_{α}:

X_{α} = \overline{X}_{α} \ ( ⋃_{\overline{X}_{β}⊂\overline{X}_{α},\overline{X}_{β}≠\overline{X}_{α}} \overline{X}_{β}) .

The combinatorial structure of the adjacements of the strata can be described by a connected oriented graph Γ_{A}. The vertices α of the graph correspond to the strata X_{α}, the set of the vertices we denote by ver(A); an arrow α → β ∈ arr(A) corresponds to the adjacement in codimension 1:

α → β \overline{X}_{α} ⊃ \overline{X}_{β} , codim _{\overline{X}_{α}} \overline{X}_{β} = 1 .

Inclusions of closed strata define the partial order in the set ver(A): α > β ⇔ \overline{X}_{α} ⊇ \overline{X}_{β} ⇔ there exists an oriented path from α to β in graph Γ_{A}. This partial order is in agreement with integervalued function codim : Γ_{A} ⇒ N, codim α = codim _{C^{N}} \overline{X}_{α} which means that α > β ⇒ codim α < codim β. The partial order allows to attach to any vertice α of the graph Γ_{A} the full subgraph Γ_{A}^{α} ⊂ Γ_{A} whose vertices are the vertices β ∈ ver(A) which satisfy the condition β < α. One can easily see that the graph Γ_{A}^{α} describes the combinatorial structure of affine subspace \overline{X}_{α} with induced stratification. By X_{∅} we denote the the complement to all hyperplanes H_{i} which is an open in C^{N} stratum.

Definition 2.1 A quiver V_{A} = {V_{α}, A_{α,β}} of the arrangement A is a collection of linear spaces V_{α} over C, α ∈ ver(A) together with a collection of linear maps A_{α,β} : V_{β} → V_{α}, α, β ∈ ver(A) such that A_{α,β} = 0 unless α and β are connected by an arrow ( A_{α,β} ≠ 0 ⇒ α → β or β → α) , and which satisfy the following quadratic relations:

∑_{β} A_{α,β}A_{β,γ} = 0  for any α, γ ∈ ver(A), α ≠ γ .  (2.2)

Remark. One can see that the relation (2.2) is nontrivial only in two cases: (i) | codim α − codim γ |= 2 and α > γ or α < γ;

(ii) codim α = codim γ, codim _{X_{α}}X_{α} ∩ X_{γ} = 1.

In the last case there are one or two summands in the relation (2.2).

The quivers V_{A} compose the category Qui(A) with natural morphisms. In other words, Qui(A) is the category of finite-dimensional modules over graded algebra B_{A}. The algebra B_{A} is generated by orthogonal idempotents e_{α}, α ∈ ver(A) of degree 0, ∑ e_{α} = 1 and by degree one generators a_{α,β}, α, β ∈ ver(A) such that a_{α,β} = 0 if α and β are connected by an arrow in Γ_{A} and, moreover,

a_{α,β}e_{γ} = δ_{β,γ}a_{α,β}, e_{γ}a_{α,β} = δ_{γ,α}a_{α,β} ,

∑_{β} a_{α,β}a_{β,γ} = 0, α ≠ γ.

Let V_{A} = {V_{α}, A_{α,β}} be a quiver of the arragement A. We denote by V_{A}^{t} = {V_{α}^{t}, A_{α,β}^{t}} the following dual quiver:

V_{α}^{t} = V_{α}^{∗},  A_{α,β}^{t} = (−1)^{(codim α+codim β−1)/2}A_{β,α}^{∗} .  (2.3)

The correspondence V_{A} → V_{A}^{t} defines the contravariant functor of duality

t : Qui(A) → Qui(A) .

3 Quiver D-modules

3.1. For any stratum X_{α} we denote by Ω_{α} one-dimensional space of codim X_{α}-forms ω_{α} ∈ det T_{X_{α}}^{∗}, which are constant along X_{α}, and by Ω̂_{α} denote one-dimensional space of top degree forms of the conormal bundle of X_{α}, ω̂_{α} ∈ T_{(X_{α}|X)}^{∗} constant along X_{α}.

Let V_{A} be a quiver of an arrangement A. We relate to it a D_{X}-module EV_{A} by the following rules:

(i) EV_{A} is generated by the linear space ⨁_{α}(V_{α} ⊗ Ω_{a}) as D_{X}-module,

(ii) for any α and for any vectorfield L_{α} along X_{α} with constant coefficients

L_{α}(v_{α} ⊗ ω_{α}) = ∑_{β:α→β} \frac{L_{α}(ω̂_{β}) ∧ ω_{α}}{ω̂_{β} ∧ ω_{β}} A_{β,α}v_{α} ⊗ ω_{β},  v_{α} ∈ V_{α}, ω_{γ} ∈ Ω_{γ}, ω̂_{γ} ∈ Ω̂_{γ} , (3.1)

(iii) for any α and for any affine function f_{α}, f_{α}|_{X_{α}} = 0
f_{α}(v_{α} ⊗ w_{α}) = ∑_{β:β→α} \frac{df_{α} ∧ ω̂_{β} ∧ ω_{α}}{ω̂_{β} ∧ ω_{β}} A_{β,α}v_{α} ⊗ ω_{β},  v_{α} ∈ V_{α}, ω_{γ} ∈ Ω_{γ}, ω̂_{γ} ∈ Ω̂_{γ} , (3.2)

In other words, EV_{A} is a factor of free D_{X}-module D_{X} ⊗ (⊕_{α}V_{α} ⊗ Ω_{α}) modulo the relations (3.1)–(3.2).

The closure \overline{X}_{α} of any stratum is affine subspace of X = C^{N}. We can choose local affine coordinates \{\overline{z}^{α}, \overline{w}^{α}\} = \{z_{1}^{α}, …, z_{dimX_{α}}^{α}, w_{1}^{α}, …, w_{codimX_{α}}^{α}\} such that \overline{X}_{α} is defined by the equations:

w_{1}^{α} = w_{2}^{α} = … = w_{codimX_{α}}^{α} = 0 .  (3.3)

Proposition 3.1 (i) EV_{A} is holonomic D_{X}-module with regular singularities flat along the stratification A (that is, ss.EV_{A} ⊂ ⋃_{α} T_{(X_{a}|X)}^{∗});

(ii) The global sections of EV_{A} are:

Γ(EV_{A}) ≡ ⊕_{α}C[\{\overline{z}^{α}\}, \{\frac{∂}{\overline{w}^{α}} \}]V_{α} ⊗ Ω_{α} .  (3.4)

Proof . (ii) Due to the relaions (3.1) and (3.2) the space of global sections is isomorphic to a factorspae of linear space

U = ⊕_{α}U^{α} = ⊕_{α}C[\{\overline{z}^{α}\}, \{\frac{∂}{\overline{w}^{α}} \}]V_{α} ⊗ Ω_{α} .

So it is sufficient to define the structure of D_{X} module in linear space U. We can equip U with a structure of graded space by the relations deg (v_{α}⊗w_{a}) = 0, deg (z_{i}^{α}) = deg (∂_{w_{j}^{α}}) = 1 (∂_{w_{j}^{α}} = \frac{∂}{\overline{w}^{α}}) and define an action of the generators x_{i}, ∂_{x_{i}} of the ring D_{X} by induction over the grading. The generators ∂_{z_{i}^{α}} and w_{j}^{α} act on the elements of degree 0 v_{α} ⊗ ω_{α} according to the relations (3.1) and (3.2), whereas the generators ∂_{w_{j}^{α}} and z_{i}^{α} act on them by free multiplication. If we already define the action of the generators x_{i}, ∂_{x_{i}} on the elements of degree ≤ (n−1), the action of x_{i}, ∂_{x_{i}} on the elements of degree n from U_{α} is uniquely defined by the following conditions:

(a) the elements ∂_{w_{j}^{α}} and z_{i}^{α} act on subspace U^{α} by free multiplication, in particular,

[z_{i}^{α}, z_{k}^{α}] = [z_{i}^{α}, ∂_{w_{j}^{α}}] = [∂_{w_{j}^{α}}, ∂_{w_{l}^{α}}] = 0 ,  (3.5)

(b) the operators ∂_{z_{i}^{α}} and w_{j}^{α} commute with ∂_{w_{j}^{α}} and z_{i}^{α} according to the relations in D_{X}, that is,

[∂_{z_{i}^{α}}, ∂_{w_{j}^{α}}] = [w_{j}^{α}, z_{i}^{α}] = 0 , [∂_{z_{i}^{α}}, z_{k}^{α}] = δ_{i,k}, [w_{j}^{α}, ∂_{w_{l}^{α}}] = −δ_{j,l} .  (3.6)

We have to prove that this action satisfied all the relation in D_{X}, so we are to check the relations

[∂_{z_{i}^{α}}, ∂_{z_{k}^{α}}] = [w_{j}^{α}, w_{l}^{α}] = [∂_{z_{i}^{α}}, w_{j}^{α}] = 0 .  (3.7)

The relations (3.5) and (3.6) allow to reduce the check of (3.7) to the space of degree zero, where it is valid due to (3.1), (3.2) and (2.1) (this is a simple but crucial calculation).

(i) The description of the global sections of the module EV_{A} enables one to describe the concrete good filtration of EV_{A}. Remind that a filtration M_{k} of D_{X}-module X is good, if it is an argeement with a filtration D_{k} of D_{X} over the degree of differential operator: D_{k}M_{l} ⊂ M_{k+l}; D_{k}M_{l} = M_{k+l} for all l ≥ l_{0}, k ≥ 0; and all M_{k} are finite O_{X}-modules.

Let, as beforeU^{α} = ⊕_{α}C[\{\overline{z}^{α}\}, \{\frac{∂}{\overline{w}^{α}} \}]V_{α} ⊗ Ω_{α} .. We showed that, globally, Γ(EV_{A}) = U = ⊕_{α}U_{α}. We equip U_{α} with a filtration putting deg(V_{α} ⊗ Ω_{α}) = codim α , deg z_{i}^{α} = 0 , deg ∂_{ω_{i}^{α}} = 1, U_{k}^{α} = {m ∈ U_{α}, deg m ≤ k} and let U_{k} = ⊕_{α}U_{k}^{α}. Then, due to the relations (3.1) and (3.2), the filtration 0 ⊂ U_{0} ⊂ … ⊂ U is a good filtration. Moreover, the same relations show that the product

∏_{α} (∏_{i} p_{i}^{α} · ∏_{j} w_{j}^{α})

annihilate gr EV_{A} (here p_{i}^{α} = gr _{\frac{∂}{∂_{z^{α}_{i}}}}). It means that EV_{A} is holonomic D_{X} module with

singular support contained in ∪_{α}T_{(X_{α}|C^{N})}^{∗}. It has regular singularities due to (3.1), (3.2)

Thus we have a functor

E : Qui(A) ⟶ M(A) ,  (3.8)

where M(A) is a category of holonomic RS D_{X}-modules flat anong the stratification A. Note that the presentation of the module EV_{A} allows one in addition easily describe the sections over the open sets

X_{H} = X \ H

where H is a hyperplane. Let, for instance, H_{β} be a generic hyperplane which contains the stratum X_{β}. Then the global sections over X_{H_{β}} constitue the linear space

U_{β} = ⊕_{α:β↛α}U^{α} = ⊕_{α}C[\{\overline{z}^{α}\}, \{\frac{∂}{\overline{w}^{α}} \}]V_{α} ⊗ Ω_{α} ,

where the polynomial over z_{i}^{α} should be replaced by the regular functions of \overline{X}_{β} ∩ U.

3.2. One can also localize the whole construction to the complement of certain hyperlanes, that is, take as an initial space X the complement to some hyperplanes instead of affine space. For instance, the configuration {A, H} of hyperplanes A = {H_{i}}, i ∈ I in C^{N} \ H defines the quivers which consist of linear spaces V_{α}, α ∈ Γ_{A}, together with linear maps A_{α,β} : V_{β} → V_{α}, which are zero unless α and β are connected by an arrow, and linear maps N_{α} : V_{α} → V_{α} which are not zero only for the strata X_{α} which intersect with H over codimension one. These operators are subjected to the relations:

∑_{β} A_{α,β}A_{β,γ} = 0 for any α, γ ∈ ver(A), α ≠ γ .  (3.9)

N_{β}A_{β,α} = A_{β,α} (N_{α} + ∑_{{γ≠β: \atop \overline{X}γ ∩H=\overline{X}_{β} ∩H}} A_{α,γ}A_{γ,α}) ,  α → β ,  (3.10)

A_{α,β}N_{β} = (N_{α} + ∑_{{γ≠β: \atop \overline{X}γ ∩H=\overline{X}_{β} ∩H}} A_{α,γ}A_{γ,α}) A_{α,β} ,  α → β ,  (3.11)

The corresponding D_{X}-module is described by analogous relations:

L_{α}(v_{α} ⊗ ω_{α}) = ∑_{β:α→β} \frac{L_{α}(ω̂_{β}) ∧ ω_{α}}{ω̂_{β} ∧ ω_{β}} A_{β,α}v_{α} ⊗ ω_{β}+

+ϕ_{L_{α}}(v_{α} ⊗ ω_{α}),  v_{α} ∈ V_{α}, ω_{γ} ∈ Ω_{γ}, ω̂_{γ} ∈ Ω̂_{γ} ,  (3.12)

for any α and for any vectorfield L_{α} along X_{α} with constant coefficients

f_{α}(v_{α} ⊗ w_{α}) = ∑_{β:β→α} \frac{df_{α} ∧ ω̂_{β} ∧ ω_{α}}{ω̂_{β} ∧ ω_{β}} A_{β,α}v_{α} ⊗ ω_{β},  v_{α} ∈ V_{α}, ω_{γ} ∈ Ω_{γ}, ω̂_{γ} ∈ Ω̂_{γ} , (3.13)

for any α and for any affine function f_{α}, f_{α}|_{X_{α}} = 0.

The summand ϕ_{L_{α}}(v_{α} ⊗ ω_{α}) is nonzero only for those \overline{X}_{α} which intersect with H over in the codimension 1 and looks like

ϕ_{L_{α}}(v_{α} ⊗ ω_{α}) = \frac{L_{α}(f)}{f} (N_{α}v_{α} ⊗ ω_{α}−

∑_{γ:\overline{X}_{γ}∩H=\overline{X}_{α}∩H} \frac{df ∧ ω̂_{γ} ∧ ω_{Hα}}{ω̂_{γ} ∧ ω_{γ}} · \frac{ω̂_{α} ∧ ω_{α}}{df ∧ ω̂_{α} ∧ ω_{Hα}} A_{γ,δ}A_{δ,α}v_{α} ⊗ ω_{γ})

Here ω_{Hα} is an arbitrary top form on H ∩ X_{α} with constant coefficients, the symbol δ stands for a stratum which coincides with X_{α} + X_{γ}; if X_{α} + X_{γ} is not a stratum then the corresponding summand in r.h.s. of the above formula is zero (X_{α} + X_{γ} means a sum of linear subspaces in linear space with an origin in X_{α} ∩ X_{γ}). The statement of Proposition remains true for such D-modules.

3.3. A free resolution of quiver D-modules.

Let T_{α} = T′_{α} ⊕ T′′_{α}, dim T_{α} = N be a direct sum of the space of the sections of tangent bundle T_{X_{α}} to a stratum X_{α} constant along X_{α} and of the space of sections of conormal bundle T_{(X_{α}|C^{N})}^{∗} constant along X_{α}. In local coordinates \overline{z}^{α}, \overline{w}^{α} (3.3) the spaces T′_{α} and T′′_{α} are: T′_{α} = C < ∂_{z_{i}^{α}} >, T′′_{α} = C < dw_{j}^{α} >. In other way the space T_{α} can be identify with the space of constant sections of tangent bundle T_{T_{(Xα|C^{N})}^{∗}} to the space of conormal bundle T_{(X_{α}|C^{N})}^{∗} ⊂ T_{C^{N}}^{∗}.

There is a natural embedding I_{α} of the space T_{α} into the space of first order differential operators:

I_{α} : T_{α} ↪ D_{≤1} .

In local coordinates (3.3) I_{α}(∂_{z_{i}^{α}} = \frac{∂}{∂z_{i}^{α}}, I_{α}(dw_{j}^{α} = w_{j}^{α}. The map I_{α} does not depend on ths choise of affine coordinates \overline{z}^{α}, \overline{w}^{α} satisfying (3.3).

For any quiver V_{A} the relations (3.1) and (3.2) define also a natural map

Q_{α} : T_{α} ⊗ V_{α} ⊗ Ω_{α} → ⊕_{β}V_{β} ⊗ Ω_{β}

which looks as follows:

Q_{α}(ξ ⊗ v_{α} ⊗ ω_{α}) = ∑_{β:α→β} \frac{ξ(ω̂_{β}) ∧ ω_{α}}{ω̂_{β} ∧ ω_{β}} A_{β,α}v_{α} ⊗ ω_{β} ,  ξ ∈ T′_{α}, ω̂_{β} ∈ Ω̂_{β} ,  (3.14)

Q_{α}(df_{α} ⊗ v_{α} ⊗ ω_{α}) = ∑_{β:β→α} \frac{df_{α} ∧ ω̂_{β} ∧ ω_{α}}{ω̂_{β} ∧ ω_{β}} A_{β,α}v_{α} ⊗ ω_{β} ,  df_{α} ∈ T′′_{α}, ω̂_{β} ∈ Ω̂_{β} . (3.15)

Let E^{n}V_{A}, n = 1, 2, …, N be a free left D-module

E^{n}V_{A} = ⊕_{α}D ⊗ (⋀^{n}(T_{α}) ⊗ V_{α} ⊗ Ω_{α})

and d^{(n)} : E^{n}V_{A} → E^{n−1}V_{A} be the following D-linear map of free D-modules:

d^{(n)}(1 ⊗ t_{1} ∧ … ∧ t_{n} ⊗ v_{α} ⊗ ω_{α}) = ∑_{k=1}^{n}(−1)^{k+1}I_{α}(t_{k}) ⊗ t_{1} ∧ … ∧ t̂_{k} ∧ … ∧ t_{n} ⊗ v_{α} ⊗ ω_{α}−

− ∑_{k=1}^{n}(−1)^{k+1}1 ⊗ t_{1} ∧ … ∧ t̂_{k} ∧ … ∧ t_{n} ⊗ Q_{α}(t_{k} ⊗ v_{α} ⊗ ω_{α}) .  (3.16)

We state that d^{(n−1)}d^{(n)} = 0 and

Proposition 3.2 (E^{•}V_{A}, d^{(•)}) is a free resolution of D-module EV_{A}.

Proof. The relation d^{(n−1)}d^{(n)} = 0 is equivalent to basic relations (2.2) on the operators A_{α,β}. The proof of the exactness of the complex is the same as for standard Koszul resolution of the ring O_{X}.

Proposition 3.3 The dual complex

⋀^{N}(T_{C^{N}}^{∗})[−N] ⊗_{O} Hom_{D} (E^{•}V_{A}, d^{(•)})

is a free resolution of the module EV_{A}^{t}.

Proof. One can see that

Hom_{D}(D ⊗ ⋀^{n}(T_{α}) ⊗ V_{α} ⊗ Ω_{α}, D) ≈ D ⊗ ⋀^{N−n}(T_{α}) ⊗ V_{α}^{∗} ⊗ Ω_{α} ⊗ ⋀^{N}(T_{C^{N}})  (3.17)

and the dual maps d^{(n)∗} of right D-modules D ⊗ ⋀^{N−n}(T_{α}) ⊗ V_{α}^{∗} ⊗ Ω_{α} ⊗ ⋀^{N}(T_{C^{N}}) are of the form analogous to (3.16):

d^{(n)∗}(1⊗t_{1} ∧…∧t_{n} ⊗v_{α}^{∗} ⊗ω_{α} ⊗π_{∅}) = ∑_{k=1}^{n}(−1)^{k+1}I_{α}(t_{k})⊗t_{1} ∧…∧t̂_{k} ∧…∧t_{n} ⊗v_{α}^{∗} ⊗ω_{α} ⊗π_{∅}−

− ∑_{k=1}^{n}(−1)^{k+1}1 ⊗ t_{1} ∧ … ∧ t̂_{k} ∧ … ∧ t_{n} ⊗ Q_{α}^{∗}(t_{k} ⊗ v_{α}^{∗} ⊗ ω_{α}) ⊗ π_{∅} ,

where Q_{α}^{∗} is given by the relations (3.14), (3.15) with v_{α}^{∗}, v_{β}^{∗} and A_{β,α}^{∗} substituted instead of v_{α}, v_{β} and A_{α,β}, π_{∅} is an N-vector on C^{N}. The passage from the right D-modules to the left D-modules gives the sign (−1)^{(codim α−codim β−1)/2} before A_{β,α}^{∗}.

Corollary 3.1 The D-module EV_{A}^{t} is the dual to EV_{A}.

Note that the appearence of N-vector in (3.17) instead of the top form shows that it is more natural to treate the quiver’s D-modules as right D-modules but the presense of affine structure in the stratified space allows one to pass from the left to right D-modules just by simple change of signes in their defining relations.

4 Weighted quivers and weighted D-modules

4.1. The Category Qui_{λ}(A) of weighted quivers.

Let us assign to any hyperplane H_{i} of the arrangement A a complex number (weight) λ_{i}, and set

λ_{α} = ∑_{i:H_{i}⊃\overline{X}_{α}} λ_{i}  (4.1)

for any stratum X_{α} of the arrangement, and

λ_{α,β} = ∑_{i:H_{i}⊃\overline{X}_{β},H_{i}⊅X_{α}} λ_{i}  (4.2)

for any arrow α → β of the graph Γ_{A}. Let us fix a stratum \overline{X}_{α}. Then we can attach to this stratum a subalgebra B_{α}^{0} ⊂ B_{A} of the path algebra B_{A} (see section 1) generated by the elements A_{α}^{β} = A_{α,β}A_{β,α}, where β runs all the vertices of Γ_{A} satisfying the condition α → β. The algebra B_{α}^{0} can be described as an algebra with generators A_{α}^{β}, α → β subject to quadratic relations

[ ∑_{β:α→β→γ} A_{α}^{β}, A_{α}^{δ}] = 0  (4.3)

which are enumerated by the flags α → δ → γ. One can easily see that for any quiver V_{A} the linear space V_{α} is equiped with a structure of B_{α}^{0} module. We define a subcategory Qui_{λ}(A) ⊂ Qui(A) as follows:

Definition 4.1 A quiver V_{A} ∈ Qui(A) belongs to the subcategory Qui_{λ}(A) if for any stratum X_{α}, the linear space V_{α} considered as a B_{α}^{0}-module admits a finite filtration by B_{α}^{0}-submodules with the factors isomorphic to C_{λ,α}, where C_{λ,α} is one-dimensional B_{α}^{0}-module in which the generators A_{α}^{β} act by multiplication over λ_{α,β}:

ρ_{λ,α}(A_{α}^{β}) = λ_{α,β} · Id.  (4.4)

In other words, the action of the operators A_{α,β} in the space V_{α} can be given in some basis by triangular matrices with diagonal λ_{α,β} · Id. The proposition below describes the properties of the category Qui_{λ}(A).

Proposition 4.1 (i) The category Qui_{λ}(A) is a full abelian subcategory of Qui(A) closed under extensions; (ii) The category Qui_{λ}(A) is stable with respect to the involution ^{t}; (iii) The category Qui_{λ}(A) has finite number of irreducibles; the irreducibles L_{λ,α} are labeled by the vertices α of the graph Γ_{A} and are supported on the subgraph Γ_{A}^{α} of the closure of the stratum X_{α}, that is (L_{λ,α})_{β} ≠ 0 ⇒ α > β and (L_{λ,α})_{α} ≠ 0.

The conjecture that the category Qui_{λ}(A) containes the projective covers and has homological dimension at most N also makes sense but we did not try to prove it.

Proof. The statements (i) and (ii) follow directly from the definitions and from the description of the involution. The most nontrivial statement (iii) is based on the explicit construction of an analog of Verma modules M_{λ,α} ⊂ Qui_{λ}(A) which we give below. The quiver M_{λ} = M_{λ,∅} has the unique irreducible factor L_{λ} = L_{λ,∅} and is characterized by the property

Hom_{Qui(A)} (M_{λ}, V_{A}) ≈ Hom_{B_{∅}^{0}} (C_{λ,∅}, (V_{A})_{∅})  (4.5)

for any quiver V_{A} ∈ Qui(A). The B_{∅}^{0} character of C_{λ,∅} is defined by the relation (4.4):

ρ_{λ,∅}(A_{∅,i}A_{i,∅}) = λ_{i} · Id .  (4.6)

The construction of the quivers M_{λ,α} and of their irreducible factors uses the inclusions j_{α} : \overline{X}_{α} ↪ C^{N}. The inclusion j_{α} defines the induced stratification A^{α} in affine space \overline{X}_{α} which is described by the graph Γ_{A}^{α}. It also defines the full exact functor (which we denote by the same letter)

j_{α} : Qui(A^{α}) → Qui(A) .

A collection of weights λ = {λ_{i}} for the stratification A naturally defines a collection of weights λ^{α} = {λ_{α,β}} where β runs all the vertices of Γ_{A}^{α} satisfying the condition α → β. Simple linear algebra calculations show that the functor j_{α} maps the category Qui_{λ^{α}}(A^{α}) into Qui_{λ}(A):

j_{α} (Qui_{λ^{α}}(A^{α})) ⊂ Qui_{λ}(A) .  (4.7)

Thus the irreducible quiver L_{λ,α} with a support Γ_{A}^{α} is

L_{λ,α} = j_{α}(L_{λ^{α}})

and the proof of the statement (iii) of the proposition follows by induction over the codimension of the support of the irreducibles using (4.5). Let us describe now the quiver M_{λ}.

Construction of the quiver M_{λ}.

The quiver M_{λ} which represents the functor

F(V_{A}) = Hom_{B_{∅}^{0}} (C_{λ,∅}, (V_{A})_{∅})

can be defined in terms of the flag complex [4].

Definition 4.2 A (complete) flag L_{α_{0},...,α_{n}} in the stratified space X_{A} is a chain of the closures of strata

\overline{X}_{α_{0}} ⊃ \overline{X}_{α_{1}} ⊃ … ⊃ \overline{X}_{α_{n}} ,  codim\overline{X}_{α_{j}} = j .  (4.8)

In terms of the graph Γ_{A}, the complete flag L_{α_{0},...,α_{n}} is a chain α_{0} → … → α_{n}, α_{0} = ∅. The linear space (M_{λ})_{γ} of the quiver M_{λ} we identify with the factorspace of linear space with a basis given by all flags L_{α_{0},...,α_{n}}, α_{n} = γ modulo the relations

∑_{β:α_{k−1}→β→α_{k+1}} L_{α_{0},...,α_{k−1},β,α_{k+1},...α_{n}} = 0  (4.9)

valid for all partial flags α_{0} → … → α_{k−1} > α_{k+1} → … → α_{n}, α_{0} = ∅, α_{n} = γ, codim α_{k+1} = codim α_{k−1} + 2. For any arrow α → β the linear operators

A_{β,α} : (M_{λ})_{α} → (M_{λ})_{β}, A_{α,β} : (M_{λ})_{β} → (M_{λ})_{α}

have the following form:

A_{β,α}(L_{α_{0},...,α_{n−1},α}) = L_{α_{0},...,α_{n−1},α,β} ,  (4.10)
A_{α,β}L_{α_{0},...,α_{n},β} =
∑_{k=0}^{n}(−1)^{k}λ_{α_{n−k},α_{n−k+1}} ∑_{α′_{n−k+1},...,α′_{n}:α′_{j}→α_{j+1},α′_{j}≠α_{j},α′_{n}=β} L_{α_{0},...,α_{n−k},α′_{n−k+1},...,α′_{n}} .  (4.11)

One can verify by linear algebra calculations that the operators (4.10) and (4.11) are correctly determined maps of corresponding factorspaces and define a quiver from Qui_{λ}(A) satisfying the condition (4.5).

4.2. The category M_{λ}(A) of weighted D-modules.

For any collection λ = {λ_{i}, i ∈ I} of weights we define also a category M_{λ}(A) ⊂ M(A) of weighted D-modules.

Let f_{γ}, γ ∈ Γ_{A} stands for a generic linear (affine) function over C^{N}, such that f_{γ}|_{\overline{X}_{γ}} = 0. Let Φ_{f_{α}}(M)|_{X_{α}} be the restriction to the open part X_{α} of affine subspace \overline{X}_{α} of D-module of vanishing cycles Φ_{f_{α}}(M). If M ∈ M(A) then Φ_{f_{α}}(M)|_{X_{α}} is a flat D_{X_{α}}-module which one can describe by a flat connection in trivial bundle over X_{α} with regular singularities on \overline{X}_{α}. Let us fix for a moment a collection of generic functions f_{α}, α ∈ Γ_{A}.

Definition 4.3 A D-module M from M(A) belongs to the category M_{λ}(A), if for any stratum X_{α} D_{X_{α}}-module Φ_{f_{α}}(M)|_{X_{α}} admits D_{X_{α}}-filtration with factors isomorphic to the modules given by a flat connection in one-dimensional bundle with a form

ν_{α} = ∑_{β:α→β} λ_{α,β}d log f_{β}

In other words, for some affine coordinates D-module Φ_{f_{α}}(M)|_{X_{α}} is given by a connection in trivial bundle with a form

ν_{α} = ∑_{β:α→β} L_{α,β}d log f_{β}

where L_{α,β} is a triangular matrix with constant coefficients and diagonal entries being equal to λ_{α,β}. One can see that the definition of the category M_{λ}(A) does not depend on the choise of generic linear functions f_{α}, α ∈ Γ_{A} and depends only on the projection π̃(λ) of the collections of weights to C/Z, π̃(λ) = exp(2πiλ).

Definition 4.4 A collection {λ} of weights is called non-resonance if for any α, β, γ, δ ∈ Γ_{A}, the following conditions hold:

λ_{α,β} ∉ Z \ {0}  and  λ_{α,β} − λ_{γ,δ} ∉ Z .  (4.12)

The non-resonance property also essentially depends only on the exponents of weights in a sense that one can or cannot get the condition (4.12) by shifting some of λ_{i} by an integer. One of the main properties of weighted D-modules is given in the following theorem.

Theorem 4.1 For any non-resonance collection {λ} of weights, the functor E establishes an equivalence of categories Qui_{λ}(A) and M_{λ}(A).

The statement of the theorem is a direct generalization of the results of [3]. Let us describe the scheme of the proof. It consists of the inductive (over the codimension of strata) construction of an inverse functor Ẽ : M_{λ}(A) → Qui_{λ}(A) with simultenious proof of basic statement that the functor E coincides with Beilinson’s functor [1] of glueing the D-modules. The functor Ẽ is as follows. The linear space (ẼM)_{α} for M ∈ M_{λ}(A) is isomorphic to the space of flat sections of Φ_{f_{α}}(M)|_{X_{α}} and as B_{α}^{0}-module the space (ẼM)_{α} is equivalent to Φ_{f_{α}}(M)|_{X_{α}} as π_{1}(\overline{X}_{α} \ X_{α})-module, the operators A_{α,β} and A_{β,α}, α → β are proportional to canonical maps can and var between D_{X_{α}}-modules Φ_{f_{α}}(M)|_{X_{α}} and Ψ_{f_{α}}(M)|_{X_{α}}. Thus, the crucial calculations are the calculation of the vanishing cycles functor Ψ_{f_{α}}(M)|_{X_{α}} with the action of monodromy operator on it. These calculations were made in [3], under unipotent monodromy assumption, that is, for zero weights λ_{i}.

As a result of a more general calculation of the specialization functors, we show in the next section that the vanishing cycles calculations remain valid also under a weaker nonresonance assumption. This allows to establish an equivalence of categories for non-resonance weighted D-modules as well.

5 Specialization of quivers and D-modules

5.1. Specialization of quivers.

Let X_{α} be a stratum of stratification X_{A}. Its closure \overline{X}_{α} is an affine subspace of X = C^{N}, and the total space of normal bundle T_{\overline{X}_{α}}X inherits the affine structure and can be naturally identified with a product of the affine space \overline{X}_{α} and the linear space X/\overline{X}_{α}: T_{\overline{X}_{α}}X ≈ \overline{X}_{α} × X/\overline{X}_{α}. Both \overline{X}_{α} and X/\overline{X}_{α} are stratified spaces: the strata of \overline{X}_{α} are X_{β}, X_{β} ⊂ \overline{X}_{α}, β ∈ Γ_{A}, and the strata of X/\overline{X}_{α} are the projections p_{α}(X_{β}), β ∈ Γ_{A}, X_{β} ∩ \overline{X}_{α} ≠ 0. Here p_{α} : C^{N} = X → X/\overline{X}_{α} is a natural projection. Let us denote strata of \overline{X}_{α} as X_{β′} and strata of X/\overline{X}_{α} as X_{β′′}. Note that the stratification of X/\overline{X}_{α} is linear: the closure of any stratum contains zero vector.

The stratification T_{\overline{X}_{α}}A of the space T_{\overline{X}_{α}}X identified as affine space with \overline{X}_{α} × X/\overline{X}_{α} is described by certain direct products of strata of \overline{X}_{α} and X/\overline{X}_{α}: (X_{β′} × X_{β′′}) ∈ T_{\overline{X}_{α}}A if there exists such a stratum X_{β} of stratification A for C^{N} that

X_{β′} = \overline{X}_{α} ∩ X_{β}, X_{β′′} = p_{α}(X_{β}) .

One can see that the stratification T_{\overline{X}_{α}}A is also described by intersections of hyperplanes H̃_{i}, i ∈ I : H̃_{i} = (H_{i} ∩\overline{X}_{α})×p_{α}(H_{i}). The passage from stratification A to T_{\overline{X}_{α}}A is illustrated by a two-dimensional example below.

[figure: \overline{X}_{α}; H_{1}; H̃_{1}; H_{2}; H_{3}; H_{4}; H̃_{2} = H̃_{3}; \overline{X}_{α}; A; T_{\overline{X}_{α}}A]

Correspondingly, the graph Γ_{T_{\overline{X}_{α}}A} is a subgraph of direct product of the graph Γ′_{A} of the stratification of the space \overline{X}_{α} and of the graph Γ′′A of the stratification of the space X/\overline{X}_{α}: Γ_{T_{\overline{X}_{α}}A} ∈ Γ′_{A} × Γ′′_{A}. It means that the vertices of Γ_{T_{\overline{X}_{α}}A} are numerated by some pairs (β′, β′′) of the vertices of Γ′A and of Γ′′_{A}, and the arrows could be of two possible types:

(β′, β′′) → (γ′, β′′) , if β′ → γ′ ∈ arr(Γ′_{A}) ,
(β′, β′′) → (β′, γ′′) , if β′′ → γ′′ ∈ arr(Γ′′_{A}) .

Since the stratification T_{\overline{X}_{α}}A is given by intersections of hyperplanes H̃_{i}, we have by definition 1.1 the quiver category Qui(T_{\overline{X}_{α}}A).

Let now V_{A} = {V_{β}, A_{β,γ}} be a quiver from the category Qui(A). We define a quiver Sp_{α}V_{A} = {V_{(β′,β′′)}, A_{(γ′,γ′′),(β′,β′′)}} as follows. The space V_{(β′,β′′)} is the following direct sum of linear spaces V_{β}:

V_{(β′,β′′)} = ⊕_{β:X_{β}∩\overline{X}_{α}=X_{β′},p_{α}(X_{β})=X_{β′′}}V_{β} .  (5.1)

The operators A_{(β′,β′′),(γ′,γ′′)} coincide with operators from the quiver V_{A}:

A_{(γ′,β′′),(β′,β′′)}(v_{β}) = ⊕_{γ:X_{γ}∩\overline{X}_{α}=X_{γ′},p_{α}(X_{γ})=X_{β′′}}A_{γ,β}(v_{β}) ,  (5.2)
A_{(β′,γ′′),(β′,β′′)}(v_{β}) = ⊕_{γ:X_{γ}∩\overline{X}_{α}=X_{β′},p_{α}(X_{γ})=X_{γ′′}}A_{γ,β}(v_{β}) ,  (5.3)

where the vector v_{β} belongs to linear space V_{β} such that X_{β} ∩ \overline{X}_{α} = X_{β′}, p_{α}(X_{β}) = X_{β′′}. A map Sp_{α} is also defined on the weights. For a collection of weights {λ} = λ_{i} = λ(H_{i}) of stratification A the collection {Sp_{α}(λ)} is defined by the rule Sp_{α}(λ)_{i} = ∑_{j:H̃_{j}=H̃_{i}} λ(H_{i}).

Proposition 5.1 (i) The relations (5.1)–(5.3) define an exact functor

Sp_{α} : Qui(A) → Qui(T_{\overline{X}_{α}}A)

which commutes with the duality ^{t}; (ii) For any collection λ of weights for the configuration A,

Sp_{α}(M_{λ}) = M_{Sp_{α}(λ)} .  (5.4)

Proof. In order to check (i) it is suficient to check that the operators defined by the relations (5.2) and (5.3) satisfy the relations (2.2) for the configuration T_{\overline{X}_{α}}A. These are linear algebra calculations. Analogously, for the proof of (ii) one has to check the formulas (4.10), (4.11) or to use the functorial definition of the modules M_{λ} and M_{Sp_{α}(λ)}.

Let now \overline{X}_{α_{1}} ⊂ \overline{X}_{α_{2}} ⊂ X = C^{N} be a flag of inclusions of closed strata to X. Then the spaces T_{\overline{X}_{α_{1}}}(T_{\overline{X}_{α_{2}}}) and T_{T_{\overline{X}_{α_{1}}} \overline{X}_{α_{2}}}(T_{\overline{X}_{α_{1}}}X) are canonically isomorphic to a direct product

\overline{X}_{α_{1}} × \overline{X}_{α_{2}}/\overline{X}_{α_{1}} × X/\overline{X}_{α_{2}}. Let us call it the space of flags of normal covectors and denote by T_{\overline{X}_{α_{2}}⊂\overline{X}_{α_{2}}}. It is a stratified space with a stratification analogous to described above. We denote it by T_{\overline{X}_{α_{2}}⊂\overline{X}_{α_{2}}}A. One can easily see that the compositions

Sp_{α_{1}}Sp_{α_{2}} : Qui(A) → Qui(T_{\overline{X}_{α_{2}}⊂\overline{X}_{α_{2}}}A)

and

Sp_{α_{2}}Sp_{α_{1}} : Qui(A) → Qui(T_{\overline{X}_{α_{2}}⊂\overline{X}_{α_{2}}}A) ,

are canonically isomorphic. Here the functor Sp_{α_{2}} in the second composition is the functor of specialization to the stratum \overline{X}_{α_{1}} × \overline{X}_{α_{2}}/\overline{X}_{α_{1}}.

Analogously, for a three-flag \overline{X}_{α_{1}} ⊂ \overline{X}_{α_{2}} ⊂ \overline{X}_{α_{3}} ⊂ X = C^{N} the morphism of associativity is identical on the linear spaces of a quiver, so for any flag

\overline{X}_{α_{1}} ⊂ \overline{X}_{α_{2}} ⊂ … ⊂ \overline{X}_{α_{n}} ⊂ X = C^{N}

of closed strata we have a specialization functor

Sp_{α_{1},...α_{n}} : Qui(A) → Qui(T_{\overline{X}_{α_{1}}⊂...⊂\overline{X}_{αn}})

which is described by itereted relations (5.1)–(5.3) and satisfy the properties mentioned in Proposition 4.1.

5.2. Specialization of quiver D-modules.

Let us first consider the case of linear stratification in linear space C^{N}, that is we suppose that all hyperplanes H_{i} contain an origin. Let I_{α} be an ideal of \overline{X}_{α}: f ∈ I_{α} ⇔ f|_{\overline{X}_{α}} = 0 and let F_{α}^{•}(D_{X}) be a filtration of the ring D_{X} corresponding to ideal I_{α} [2]:

F_{α}^{k}(D_{X}) = {P ∈ D_{X} : P (I_{α}^{j} ⊂ I_{α}^{j+k}for any j} .

Here I_{α}^{k} = C[X] for k ≤ 0. Then F_{α}^{k}(D_{X})/F_{α}^{k+1}(D_{X}) is isomorphic to a space of differential operators over T_{\overline{X}_{α}}X homogenuous of order k and the ring Gr F_{α}(D_{X}) is isomorphic to the ring D_{T_{\overline{X}_{α}}X}.

Let now V_{A} be a quiver from Qui(A) and EV_{A} be corresponding D_{X}-module. We define a filtration F_{α}^{·}EV_{A} of EV_{A} in the following way:

F_{α}^{n}EV_{A} = ∑_{k=0}^{N} ∑_{β:codim_{X_{β}} X_{β}∩\overline{X}_{α}=k} F_{α}^{n−k}(D_{X}) · (V_{β} ⊗ Ω_{β}) .  (5.5)

One can check that the filtration F_{α}^{•}EV_{A} is good, that is satisfy the following conditions: (i) F_{α}^{k}(D_{X}) · F_{α}^{j}EV_{A} ⊂ F_{α}^{k+j}EV_{A};

(ii) F_{α}^{k}(D_{X}) · F_{α}^{j}EV_{A} = F_{α}^{k+j}EV_{A} for j, k ≥ 0 and for j ≤ −N, k ≤ 0;

(iii) ∪_{j}F_{α}^{j}EV_{A} = EV_{A};
(iv) F_{α}^{j}EV_{A} is coherent F_{α}^{0}(D_{X})-module.

The following proposition describes Gr F_{α}(EV_{A}) as D_{T_{\overline{X}_{α}}X}- module. It follows directly from the definition of quiver’s D-module EV_{A} and from the definition of the filtration.

Proposition 5.2 There is a canonical isomorphism of D_{T_{\overline{X}_{α}}X}-modules:

Gr F_{α}(EV_{A}) ≈ ESp_{α} V_{A} .  (5.6)

As it follows from the relations (5.1)–(5.3), the isomorphism (5.6) means that D_{T_{\overline{X}_{α}}X}- module Gr F_{α}(EV_{A}) is generated by vectors v_{α} ⊗ ω_{β′} ∧ ω_{β′′} where β ∈ ver(Γ_{A}), v_{β} ∈ V_{β}, ω_{β′} is a top form for X_{β} ∩ \overline{X}_{α}, ω_{β′′} is a top form for X_{β}/X_{β} ∩ \overline{X}_{α} = p_{α}(X_{β}) with the following relations:

L_{β′}(v_{β} ⊗ ω_{β′} ∧ ω_{β′′}) = ∑_{{γ:β→γ \atop Xγ ∩\overline{X}α≠X_{β} ∩\overline{X}α}} \frac{L_{β′}(ω̂_{γ′}) ∧ ω_{β′}}{ω̂_{γ′} ∧ ω_{γ′}} A_{γ,β}(v_{b}) ⊗ ω_{γ′} ∧ ω_{β′′}  (5.7)

where L_{β′} is linear vectorfied along stratum X_{β} ∩ \overline{X}_{α}, ω̂_{γ′} is a top form for \overline{X}_{α}/\overline{X}_{γ} ∩ \overline{X}_{α}, ω_{γ′} is a top form for \overline{X}_{γ} ∩ \overline{X}_{α};

L_{β′′}(v_{β} ⊗ ω_{β′} ∧ ω_{β′′}) = ∑_{{γ:β→γ \atop Xγ ∩\overline{X}α=X_{β} ∩\overline{X}α}} \frac{L_{β′′}(ω̂_{γ′′}) ∧ ω_{β′′}}{ω̂_{γ′′} ∧ ω_{γ′′}} A_{γ,β}(v_{b}) ⊗ ω_{β′} ∧ ω_{γ′′}  (5.8)

where L_{β′′} is linear vectorfied in X/\overline{X}_{α} along X_{β}/X_{β}∩\overline{X}_{α}, ω̂_{γ′′} is a top form for (X/\overline{X}_{α})/(X_{γ}/\overline{X}_{γ}∩ \overline{X}_{α}), ω_{γ′′} is a top form for X_{γ}/\overline{X}_{γ} ∩ \overline{X}_{α};

f_{β′}(v_{β} ⊗ ω_{β′} ∧ ω_{β′′)} = ∑_{{γ:γ→β \atop Xγ ∩\overline{X}α≠X_{β} ∩\overline{X}α}} \frac{df_{β′} ∧ ω̂_{γ′} ∧ ω_{β′}}{ω̂_{γ′} ∧ ω_{γ′}} A_{γ,β}(v_{b}) ⊗ ω_{γ′} ∧ ω_{β′′}  (5.9)

where f_{β′} is linear function on \overline{X}_{α} equal to zero on X_{β} ∩ \overline{X}_{α}, with the same ω̂_{γ′}, ω_{γ′} as for (5.7);

f_{β′′}(v_{β} ⊗ ω_{β′} ∧ ω_{β′′}) = ∑_{{γ:γ→β \atop Xγ ∩\overline{X}α=X_{β} ∩\overline{X}α}} \frac{df_{β′′} ∧ ω̂_{γ′′} ∧ ω_{β′′}}{ω̂_{γ′′} ∧ ω_{γ′′}} A_{γ,β}(v_{b}) ⊗ ω_{β′} ∧ ω_{γ′′}  (5.10)

where f_{β′′} is linear function on X/\overline{X}_{α} equal to zero on X_{β}/X_{β} ∩ \overline{X}_{α} with the same ω̂_{γ′′} and ω_{γ′′} as in (5.8).

We can drop the assumption of linearity for the stratification if we pass first to analytic D-modules, and thus kill the strata which do not intersext with \overline{X}_{α}, and then use the well known equivalence between analytic and algebraic D-modules. Alternatively, we could first localize the initial D-module to a Zariski open set which does not contain strata not intersecting with \overline{X}_{α}, and then use the construction above.

In order to compare D_{T_{\overline{X}_{α}}X}- module Gr F_{α}(EV_{A}) with the Malgrange-Kashiwara specialization of the D_{X}-module EV_{A}, we should investigate the eigenvalue of the target vectorfiled θ^{α} which satisfy the relations θ^{α}|_{I_{α}/I_{α}^{2}} = id [2]. In local coordinates \{\overline{z}^{α}, \overline{w}^{α}\} = \{z_{1}^{α}, …, z_{dim X_{α}}^{α}, w_{1}^{α}, …, w_{codim X_{α}}^{α}\} satisfying (3.3) it looks like θ^{α} = ∑_{j} w_{j}^{α} \frac{∂}{∂w_{j}^{α}}. Let also λ = {λ_{1}, …, λ_{n}} be a collection of weights and M_{λ} be a quiver constructed in section 3.1 (see (4.5)).

Proposition 5.3

θ^{α}|Gr^{k}F_{α}EM_{λ} = λ_{α} + k .  (5.11)

The proof follows from the relations (5.7), (5.8).

Correspondingly, for D_{X}-module E(M_{λ,β}) supported on the stratum \overline{X}_{β} (\overline{X}_{β} ∩ \overline{X}_{α} ≠ 0), we have

θ^{α}|Gr^{k}F_{α}EM_{λ,α} = λ_{β′,β} + k ,

where X_{β′} = X_{β} ∩ \overline{X}_{α}.

Corollary 5.1 (i) Let the number λ_{α} = ∑_{i:H_{i}⊃X_{α}} ∉ Z \ 0. Then

Sp_{\overline{X}_{α}}E(M_{λ}) ≈ E Sp_{α}(M_{λ}) ;  (5.12)

(ii) Let the nonresonance condition takes place: λ_{β,γ} − λ_{ν,ε} ∉ Z \ 0. Then

Sp_{\overline{X}_{α}}E(M) ≈ E Sp_{α}(M)  (5.13)

for any M ∈ Qui(A).

Here Sp_{\overline{X}_{α}} denotes the Malgrange-Kashiwara specialization functor, [2].

Proof of Theorem 4.1. Following the scheme of [3], we choose a collection of generic linear functions f_{α}, f_{α}|_{\overline{X}_{α}} = 0 and perform the glueing procedure starting from a local system on the open stratum X_{∅} and adding step by step strata of bigger codimension. In order to glue the stratum X_{α} we add hyperplane f_{α} = 0 to the stratification with zero weight and with a condition that the support of D-module intersects with hyperplane f_{α} = 0 only by stratum \overline{X}_{α}. Part (ii) of the Corollary 4.1 allows to calculate the linear algebra data

Ψ_{f_{α}}(M)\overleftarrow{→}Φ_{f_{α}}(M)

in terms of Malgrange-Kashiwara speciaization functor assuming we start from quiver’s Dmodules with nonresonance condition.

Corollary 4.1 guarantees that they coincide with the linear algebra data calculated by means of specialization functors in category Qui_{λ}(A) (and thus coincide with those calculated by Beilinson’s technique for the unipotent case in [3]). Then we can refer to [3] since we drop unipotent restriction. Alternatively, one can get an independent proof after the formulation of an analog of the Beilinson-Kashiwara equivalence theorem for the category Qui(A) and functors Sp_{α} from section 4.1, which is purely combinatorial.

References

[1] A. Beilinson, How to glue perverse sheaves, in: K-theory, arithmetic and geometry, Yu.I. Manin (Ed.), Lect. Notes in Math., 1289, Springer-Verlag, Berlin et al., 1987; 42-51.

[2] M. Kashiwara, Vanishing cycles and holonomic systems of differential equations, in: Algebraic Geometry (Tokyo/Kyoto, 1982), M. Raynaud, T. Shioda (Eds.), Lect. Notes in Math., 1016, Springer-Verlag, Berlin et al., 1983; 134-142.

[3] S. Khoroshkin, D-modules over arrangements of hyperplanes, Comm. in Algebra 23(9) (1995), 3481-3504.

[4] V. Schechtman, A. Varchenko, Arrangements of hyperplanes and Lie algebra homology, Inv. Math., 106 (1991), 134-194.

S.K.: Institute for Theorerical and Experimental Physics, Cheremushkinskaya 25, 117259 Moscow, Russia; khor@heron.itep.ru

V.S.: Max-Planck-Institut für Mathematik, Gottfried-Claren-Straße 26, 53225 Bonn, Germany; vadik@mpim-bonn.mpg.de