Regularity and b-functions for D-modules

0.18 MB·English

by  Yves Laurent

REGULARITY AND B-FUNCTIONS FOR D-MODULES
YVES LAURENT
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2 Abstract

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AholonomicD-moduleonacomplexanalyticmanifoldadmitsalwaysab-functionalong
u
any submanifold. If the module is regular, it admits also a regular b-function, that is a
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8 b-function with a condition on the order of the lower terms of the equation. There is a
weaker condition of regularity: regularity along a submanifold. We prove that a module

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which is regular along a submanifold admits a regular b-function along this submanifold.
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2000 Mathematics Subject Classification. 35A27.
Key words and phrases. D-module, regular, b-function.
2 YVESLAURENT
Introduction
Let f be a holomorphic function on an open set X of Cn. A b-function for f is a
polynomial b ∈ C[s] such that there exists a differential operator P on U with parameter
s satisfying an equation:
(1) b(s)f(x)s = P(s,x,D )f(x)s+1
x
Thegenerator oftheidealoftheb-functionsassociated tof isusuallycalled theBernstein-
Sato polynomial of f (see[1] for details).
This definition has been extended by Kashiwara [3] to holonomic D-modules. Let X
be a complex manifold and Y a smooth hypersurface. Let D be the sheaf of differential
X
operators on X. Let M be a holonomic D -module and u a section of M. A b-function
X
for u along Y is an equation
(2) b(tD )u= tP(t,x,tD ,D )u
t t x
satisfied by u. Here (t,x) are local coordinates of X such that t is an equation for Y. A
similar definition exits for submanifolds of X of any codimension.
The b-function is called regular if P may be chosen so that its order as a differential
operator is not greater than the degree of b.
It has been proved by Kashiwara [3] that a holonomic D-module admits a b-function
along any smooth hypersurface and by Kashiwara-Kawa¨ı [5] that a regular holonomic
D-module admits a regular b-function along any smooth hypersurface.
Thereis another notion of regularity, theregularity of aD-modulealong asubmanifold.
The definition will be given in definitions 1.3 and 3.1. In the case of a hypersurface,
this is equivalent to the fact that solutions in the formal completion transversally to the
hypersurface are convergent. It has been proved that a regular holonomic module is
regular along any submanifold in [6]. It is also a direct consequence of the definition that
if a D-module admits a regular b-function along a submanifold, it is regular along it. We
will prove here that a module which is regular along a submanifold Y admits a regular
b-function along Y.
The problem is better understood after microlocalization, that is for modules over the
sheaf E of microdifferential operators. Regularity and b-function may be defined for a
X
coherent E -module M along a conic lagrangian submanifold Λ of the cotangent bundle
X
[8]. All definitions are invariant under quantized canonical transformation. Concerning
D-modules, this will prove that the result is true not only for hypersurfaces but may be
extended to any smooth subvariety of X.
AregularholonomicE -moduleisregularalonganysmoothconiclagrangian subvariety
X
of T∗X. Conversely, in the definition of Kashiwara-Kawa¨ı [6], a holonomic E -module is
X
regular if it is regular along the smooth part of its characteristic variety.
In the first section of this paper, we briefly recall the definitions of b-functions and
state the result for D -modules. In the second section, we study the microlocal case. We
X
define the filtrations and bifiltrations on holonomic E -modules In the third section, we
X
study the equivalent definitions of regularity along a lagrangian manifold and we prove
the existence of a regular b-function as a product of θd by the b-function b(θ).
1. The case of differential equations
In this first section, we briefly recall the definitions of b-functions and regularity in the
framework of D -modules.
X
REGULARITY AND B-FUNCTIONS FOR D-MODULES 3
1.1. Filtrations and b-functions. Let X be a complex analytic manifold, O the sheaf
X
of holomorphic functions on X and D the sheaf of differential operators on X with
X
coefficients in O . Let T∗X be the cotangent bundle to X with canonical projection
X
π : T∗X → X. Let Y be a submanifold of X.
The sheaf D is provided with two filtrations. The first one is the filtration (D )
X X,m m≥0
bytheusualorder,thatisthedegreeinthederivations. Here, wewilldenotethisfiltration
by FD , i.e. F D = D . The corresponding graded ring grFD is identified to
X m X X,m X
π∗O[T∗X] the sheaf of holomorphic functions on T∗X with polynomial coefficients in the
fibers of π.
The second one is the V-filtration which has has been defined by Kashiwara in [4] as:
(1.1) V D = {P ∈ D | | ∀ℓ ∈Z,PIℓ ⊂ Iℓ−k}
k X X Y Y Y
where I is the ideal of definition of Y and Iℓ = O if ℓ ≤ 0.
Y Y X
Let τ : T X → Y bethe normal bundleto Y in X and O the sheaf of holomorphic
Y [TYX]
functions on T X which are polynomial in the fibers of τ. Let O [k] be the subsheaf
Y [TYX]
of O of homogeneous functions of degree k in the fibers of τ. There are canonical iso-
[TYX]
morphisms between Ik/Ik−1 and τ O [k], between Ik/Ik−1 and τ O . Hence
Y Y ∗ [TYX] L Y Y ∗ [TYX]
the graded ring grVD associated to the V-filtration on D acts naturally on O . An
X X [TYX]
easy calculation [14] shows that as a subring of End(τ O ) it is identified to τ D
∗ [TYX] ∗ [TYX]
the sheaf of differential operators on T X with coefficients in O .
Y [TYX]
The Euler vector field θ of T X is the vector field which acts on O [k] by multipli-
Y [TYX]
cation by k. Let ϑ be any differential operator in V D whose image in grVD is θ. Let
0 X 0 X
M be a coherent D -module and u a section of M.
X
Definition 1.1. A polynomial b is a b-function for u along Y if there exists a differential
operator Q in V D such that (b(ϑ)+Q)u = 0.
−1 X
Definition 1.2. A polynomial b of degree m is a regular b-function for u along Y if there
exists a differential operator Q in V D ∩F D such that (b(ϑ)+Q)u = 0.
−1 X m X
The set of b-functions is an ideal of C[T], if it is not zero we call a generator of this
ideal ”the” b-function of u along Y.
Remark that the set of regular b-function for u is not always an ideal of C[T].
1.2. Regularity. A holonomic module is regular its formal solutions converge at each
point. More precisely, if x is a point of X, O the ring of germs of O at x and m
X,x X
the maximal ideal of O , let us denote by O the formal completion of O for the
X,x X,x X,x
m-topology. The holonomic D -module M isbregular if and only if:
X
j j
∀j ≥ 0, ∀x ∈ X, Ext (M,O )= Ext (M,O )
DX bX,x DX X,x
This is the direct generalization of the definition in dimension 1 (Ramis [12]). An
algebraic and microlocal definition is given by Kashiwara-Kawa¨ı in [6] and they prove
that, for a D -module, it is equivalent to this one.
X
A weaker notion is the regularity along a submanifold. This has been studied in several
papers [7],[8],[11]. The definition will be given in section 3 using microcharacteristic va-
rieties. Here we give a more elementary definition for D -modules which works only for
X
hypersurfaces.
4 YVESLAURENT
Let Y be a smooth hypersurface of X, ϑ is the vector field of section 1.1 and M be a
holonomic D -module M defined in a neighborhood of Y.
X
Definition 1.3. Theholonomic D -moduleMis regularalong Y ifany section uof Mis
X
annihilatedbyadifferentialoperatoroftheformϑN+P+QwhereP isinF D ∩V D
N−1 X 0 X
and Q is in F D ∩V D
N X −1 X
Let us denote by O the formal completion of O along Y, that is
X|Y X
b
O = limO /Ik
X|Y ←− X Y
b
k
We proved in [9] and [11] that M is regular along Y if and only if
j j
(1.2) ∀j ≥ 0, Ext (M,O ) = Ext (M,O )
DX bX|Y DX X
We can still state definition 1.3 and equation (1.2) when Y is not a hypersurface but
they are not equivalent. Correct statements will be given in section 2 in the microlocal
framework.
It has been proved by Kashiwara in [3], that if M is a holonomic D -module, there
X
exists a b-function for any section u of M along any submanifold Y of X. Kashiwara and
Kawa¨ı proved in [5] that if M is a regular holonomic D -module, there exists a regular
X
b-function for any section u of M along any submanifold Y of X.
We will prove in section 3.2 that if M is a holonomic D -module which is regular along
X
a submanifold Y then it admits a regular b-function along Y.
1.3. Explicit formulas in local coordinates. Consider local coordinates (x ,...,x ,t)
1 p
ofX suchthatY isthehypersurfaceY = {(x,t) ∈ X | t = 0}. ThenT X hascoordinates
Y
(x ,...,x ,t˜).
1 p
The Euler vector field θ of T X is θ = t˜D and we can choose ϑ = tD . Then a
Y t˜ t
b-function is an equation
b(tD )+tQ(x,t,D ,tD )
t x t
and it is a regular b-function if the order of Q is less or equal to the degree of b.
The module M is regular along Y if any section of M is annihilated by an equation
(tD )N +P(x,D ,tD )+tQ(x,t,D ,tD )
t x t x t
with P of order less or equal to N −1 and Q of order less or equal to N.
2. Microdifferential equations
In this section, we review basic definitions of filtration, V-filtration and bifiltrations on
E -modules. Details may be founded in [1] or [14], and [8].
X
2.1. V-filtration on microdifferential operators. We denote by E the sheaf of mi-
X
crodifferential operators of [13]. It is filtered by the order, we will denote that filtration
by E = F E and call it the usual filtration.
X S k X
Let O(T∗X) be the sheaf of holomorphic functions on T∗X which are finite sums of
homogeneousfunctionsinthefibersofπ :T∗X → X. ThegradedringgrFE isisomorphic
X
to π∗O(T∗X) [14].
In [8], we extended the definitions of V-filtrations and b-functions to microdifferential
equations and lagrangian subvarieties of the cotangent bundle.
REGULARITY AND B-FUNCTIONS FOR D-MODULES 5
Let Λ be a lagrangian conic submanifold of the cotangent bundle T∗X and M be a
Λ
holonomic E -module supported by Λ. A section u of MΛ is non degenerate if the ideal
X
of OT∗X generated by the principalsymbols of the microdifferential operators annihilating
u is the ideal of definition of Λ. The module MΛ is a simple holonomic E -module if it is
X
generated by a non degenerate section u . Such a module always exists locally [13].
Λ
Let M = E u . The V-filtration on E along Λ is defined by:
Λ,k X,k Λ X
(2.1) V E = {P ∈E | | ∀ℓ ∈Z,PM ⊂ M }
k X X Λ Λ,ℓ Λ,ℓ+k
This filtration is independent of the choices of M and u , so it is globally defined [8,
Λ Λ
Prop. 2.1.1.].
Let O [k] be the sheaf of holomorphic functions on Λ homogeneous of degree k in the
Λ
fibersofΛ→ X andO = O [k]. ThereisanisomorphismbetweenM /M
(Λ) Lk∈Z Λ Λ,k Λ,k−1
and O . By this isomorphism the graded ringgrVE acts on O and may beidentified
Λ,k X (Λ)
to the sheaf D of differential operators on Λ with coefficients in O .
(Λ) (Λ)
All these definitions are invariant under quantized canonical transformations [8].
The restriction to the zero section T∗X of T∗X of the sheaf E is the sheaf D of
X X X
differential operators. If Λ is the conormal bundle T∗X to a submanifold Y of X then
Y
the V-filtration induced on D by the V-filtration of E is the same than the V-filtration
X X
defined in section 1.1. The associated graded ring is the restriction to Y of D which is
(Λ)
the sheaf D of differential operators with coefficients polynomial in the fibersof Λ → X.
[Λ]
The correspondence between the isomorphism grVD ≃ D of section 1.1 and the
X [TYX]
isomorphism grVDX ≃ D[T∗X] is given by the partial Fourier transform associated to the
Y
duality between the normal bundle T X and the conormal bundle T∗X.
Y Y
Remark 2.1. Consider local coordinates (x ,...,x ,t ,...,t ) of X and Y = {(x,t) ∈X |
1 p 1 d
t = 0}. Let (x,t˜) be the corresponding coordinates of T X and (x,τ) the corresponding
Y
coordinates of T∗X. Let θ = d t˜D be the Euler vector field of the fiber bundleT X
Y Pi=1 i t˜i Y
and θ be the Euler vector field of Λ= T∗X.
Λ Y
The Fourier transform is given by t˜ 7→ −D and D 7→ τ . So θ is mapped to −d−θ
i τi t˜i i Λ
where d is the codimension of Y.
2.2. Filtrations on E-modules. Let M be a (left) coherent E -module. A F-filtration
X
(resp. a V-filtration) of M is a filtration compatible with the F-filtration (resp. V-
filtration) of E .
X
N
A good filtration of M is a filtration which is locally of the form M = E u
k Pi=1 X,k−ki i
for (u ,...,u ) local sections of M and (k ,...,k ) integers. In the same way, a good
1 N 1 N
N
V-filtration of M is locally of the form V M= V E u .
k Pi=1 k−ki X i
If M is provided with a good F-filtration, the associated graded module grFM is a
coherent module over grFEX = π∗O(T∗X). Then OT∗X ⊗π−1grFEX π−1grFM is a coherent
OT∗X-module which defines a positive analytic cycle on T∗X independent of the good
filtration. This cycle is the characteristic cycle of M and denoted by Ch(M). Its support
is thecharacteristic variety ofMdenotedCh(M), itis equaltothesufpportofthemodule
M itself.
The characteristic variety of a coherent E -module is a homogeneous involutive subva-
X
riety of T∗X. When its dimension is minimal, that is when it is a lagrangian, the module
is holonomic.
6 YVESLAURENT
If M is provided with a good V-filtration, the graded module grVM is a coherent
module over grVE = D hence grVM is a coherent D -module. The characteristic
X (Λ) (Λ)
cycle of this module is a positive analytic cycle on T∗Λ and its support is involutive.
According to [8] and [11], this characteristic cycle is called the microcharacteristic cycle
of M of type ∞ and is denoted by ChΛ(∞)(M). The corresponding microcharacteristic
f
variety is ChΛ(∞)(M). We have the following fundamental result:
Theorem 2.2 (Theorem 4.1.1 of [8]). The dimension of the characteristic variety of
grVM is less or equal to the dimension of the characteristic variety of M. In particular,
grVM is holonomic if M is holonomic.
As grVE is identified to the subsheaf of D of differential operators P satisfying
k X (Λ)
[θ,P] = kP, the Euler vector field θ acts on grVM, so we may define a morphism Θ by
Θ = θ−k on grVM. As [θ,P]= +kP on grVE , Θ commutes with the action of grVE
k k X X
that is defines a section of End (grVM).
grVEX
Definition 2.3. The set of polynomials b satisfying
b(Θ)grV(M) = 0
is an ideal of C[T]. When this ideal is not zero, a generator is called the b-function of M
for the V-filtration along Λ.
The b-function depends on the good V-filtration, its roots are shifted by integers by
change of V-filtration.
Theb-functionisinvariantunderquantizedcanonicaltransform. LetY beasubmanifold
of X of codimension d. If we want to identify the b-function of definition 2.3 with that of
definition 1.1, we have to replace ϑ in definition 1.1 by −d−ϑ because of remark 2.1.
Theorem 2.4. If M is a holonomic E -module, there exists a b-function for M along
X
any lagrangian submanifold Λ of T∗X.
This theorem has been proved first by Kashiwara for D-modules in [3]. It has been
proved for E -module in [8] as a corollary of theorem 2.2. In fact, as grVM is holonomic,
X
End (grVM) is finite dimensional and the b-function is the minimal polynomial of Θ
grVEX
on grVM.
2.3. Bifiltration. From the two filtrations on E , we get a bifiltration:
X
(2.2) ∀(k,j) ∈ Z×Z W E = F E ∩V E
k,j X j X k X
To this bifiltration is associated the bigraded ring:
(2.3) grWEX = MgrkW,jEX with grkW,jEX = Wk,jEX/(Wk−1,jEX +Wk,j−1EX)
(k,j)
This bigraded ring is equal to the graded ring of grVE = D (this latter with the
X (Λ)
standard filtration). So it is isomorphic to πΛ∗O[T∗Λ], the sheaf of holomorphic functions
on T∗Λ which are polynomial in the fibers of π : T∗Λ → Λ and sum of homogeneous
Λ
functions for the second action of C∗ on T∗Λ. ”Second action” means action induced on
T∗Λ by the action of C∗ on Λ.
Let M be a (left) coherent E -module. A good bifiltration of M is a bifiltration com-
X
patible with the bifiltration of E which is locally generated by local sections (u ,...,u )
X 1 N
REGULARITY AND B-FUNCTIONS FOR D-MODULES 7
ofMThismeansthatthereareintegers (µ ,...,µ )and(ν ,...,ν )suchthatW M =
1 N 1 N k,j
N
W E u .
Pν=1 k−kν,j−jν X ν
The bigraded module associated to the bifiltration is defined by
(2.4) grWM = MgrkW,jM with grkW,jM= Wk,jM/(Wk−1,jM+Wk,j−1M)
(k,j)
Proposition 2.5.
(1) The bigraded module associated to a good bifiltration of a coherent E -module is a
X
coherent grWEX = πΛ∗O[T∗Λ] module.
(2) The bifiltration induced on a coherent submodule by a good bifiltration is still a
good bifiltration.
This result may be found in [2] or [8], let us briefly recall its proof.
We denote by E the sheaf grVE provided with the filtration induced by that of E
V 0 X X
(cf [6]).
Lemma 2.6. Outside of the zero section of Λ, there isan equivalence between the following
data:
• a good bifiltration of M
• a coherent sub-E -module of M which generates M and is provided with a good
V
filtration.
Proof. Outsideofitszerosection, wemaytransformΛintotheconormaltoahypersurface
by a canonical transformation, that is in local coordinates:
Λ= {(x,t,ξ,τ) ∈ T∗X | t = 0,ξ = 0,τ 6= 0}
Then the microdifferential operator D is invertible. It is an element of W E .
t 1,1 X
ConsideragoodbifiltrationWMofMwhichislocallygeneratedbysectionsu ,...,u
1 N
ofM. MultiplyingthembysuitablepowersofD wemay assumethat(u ,...,u )belong
t 1 N
to
N = [ W0,jM
j∈Z
Then N is the sub-E -module of M generated by (u ,...,u ) and, by [6, Prop. 1.1.10.],
V 1 N
N is an E -coherent submodule of M. The sections (u ,...,u ) define a good filtration
V 1 N
of N and M= E ⊗N.
X
Conversely, let N be coherent sub-E -module of M which generates M and is provided
V
with a good filtration. The good filtration is given locally by sections (u ,...,u ) which
1 N
define a good bifiltration of M. (cid:3)
Proof of proposition 2.5. From [7, §2.6.] (see also [2]), we know that he filtration of E is
V
”a good noetherian filtration” that is a zariskian filtration in the notation of [14]. Then
the results of [14] show that good E -filtrations induce good E -filtrations on submodules
V V
and that the graded ring associated to a good E -filtration is a coherent grE -module.
V V
The result may be then deduce from the previous lemma out of the zero section of Λ.
We get the result on the zero section of Λ by adding a dummy variable as usual (cf [6,
§A1]). (cid:3)
8 YVESLAURENT
Let OT∗Λ be the sheaf of holomorphic functions on T∗Λ. As grWM is a πΛ∗O[T∗Λ]
coherent module, OT∗Λ ⊗O[T∗Λ] πΛ−1grM is a coherent OT∗Λ-module. Its support is in-
dependent of the good bifiltration. It is called the microcharacteristic variety of type
(∞,1) and is denoted by ChΛ(∞,1)(M). This microcharacteristic variety is an involutive
bihomogeneous subvariety of T∗Λ [7],[8].
Let us remark that the same microcharacteristic variety has been defined by Teresa
Monteiro Fernandes in [2], where it is denoted by C (M), and in [7] as the support of the
Λ
tensor product of the module by a sheaf of 2-microdifferential operators.
2.4. Local coordinates. Assume that the conic lagrangian manifold Λ is the conormal
to a submanifold Y of X. (By a canonical transformation we may always transform Λ
into a conormal).
Let (x ,...,x ,t ,...t ) be local coordinates of X such that Y = {(x,t) ∈ X | t = 0}.
1 p 1 d
Then Λ = {(x,t,ξ,τ) ∈ T∗X | t = 0,ξ = 0} and T∗Λ has coordinates (x,τ,x∗,τ∗).
N
The Euler vector field of Λ is θ = τ D and its principal symbol is the function
Pi=1 i τi
ϕ = N τ τ∗ =< τ,τ∗ >. The hypersurface S of T∗Λ defined by the equation ϕ is a
Pi=1 i i Λ
canonical hypersurface which will be considered in next section.
For the W bifiltration, the operator x , i = 1,...,p, is of bi-order (0,0), t , j = 1,...,d
i j
is of bi-order (−1,0), D is of bi-order (0,1), D is of bi-order (1,1).
xi tj
The operator ϑ may be chosen here as t D , it is of bi-order (0,1).
P j tj
3. Regularity along a lagrangian conic submanifold
3.1. Equivalentdefinitionsofregularity. TheEulervectorfieldθ ofΛisadifferential
Λ
operator on Λ, its characteristic variety is a canonical subvariety of T∗Λ which will be
denoted by S . As in section 2.1, M is a simple holonomic E -module supported by Λ.
Λ Λ X
Definition 3.1. Let Λ be a lagrangian conic submanifold of T∗X and M be a holonomic
E -module. The module M is regular along Λ if and only if it satisfies the following
X
equivalent properties:
i) The microcharacteristic variety ChΛ(∞,1)(M) is contained in SΛ.
ii) The microcharacteristic variety ChΛ(∞,1)(M) is lagrangian.
iii) The microcharacteristic variety ChΛ(∞,1)(M) is equal to ChΛ(∞)(M)
iv) ∀j ≥ 0, Extj (M,M )= Extj (M,E∞ ⊗ M )
EX Λ EX X EX Λ
We recall that E∞ is the sheaf of microdifferential operators of infinite order.
X
If Λ is the conormal bundle to a submanifold Y of X, we may take M = C the
Λ Y|X
sheaf of holomorphic microfunctions of [13]. Then (iv) is reformulated as:
(iv)’ ∀j ≥ 0, Extj (M,C ) = Extj (M,C∞ )
EX Y|X EX Y|X
Let us now briefly remain how the equivalence between the items of definition 3.1 is
proved.
By theorem 2.2, if M is holonomic, grVM is holonomic hence its characteristic variety
ChΛ(∞)(M)is lagrangian. Soif ChΛ(∞,1)(M)is equal to ChΛ(∞)(M) itis lagrangian that
is (iii)⇒(ii).
The microcharacteristic variety ChΛ(∞,1)(M) is defined by the coherent grBM-module
πΛ∗O[T∗Λ] hence is bihomogeneous, that is homogeneous in the fibers of T∗Λ and homo-
geneous for the action of C∗ induced by the action of C∗ on Λ. By section 4.3. of [8],
REGULARITY AND B-FUNCTIONS FOR D-MODULES 9
any lagrangian bihomogeneous submanifold of T∗Λ is contained in S . In fact, if Σ is
Λ
lagrangian the canonical 2-form Ω of T∗Λ vanishes on Σ and if it is bihomogeneous the
vectorfieldsv andv associatedtothetwoactionsofC∗ aretangenttoΣ. HenceΩ(v ,v )
1 2 1 2
vanishes on Σ and an easy calculation of [8] shows that Ω(v ,v ) is an equation of S . So
1 2 Λ
(ii)⇒ (i).
Finally, the main parts of the result are (i) ⇒(iv) which is given by corollary 4.4.2. of
[10] and (iv) ⇒(iii) given by theorem 2.4.2 of [11].
3.2. Regular b-function. Let Λ be a conic lagrangian submanifold of the cotangent
bundle T∗X. As before, θ is the Euler vector field of Λ and ϑ is a microdifferential
Λ Λ
operator in V E whose image in grVE is θ .
0 X 0 X Λ
Definition 3.2. Let M be a coherent E -module with a good bifiltration WM relative
X
to Λ. A regular b-function for WM is a polynomial b such that:
∀(k,j) ∈ Z×Z, b(ϑ −k)W M⊂ W M
Λ k,j k−1,j+n
where n is the degree of b.
if u is a section of M, a regular b-function for u is a regular b-function for the canonical
bifiltration of the module N = E u that is W N = (Wk,jE )u.
X k,j X
Remark that if WM is a good bifiltration, we get a good V-filtration by setting:
VkM = [ Wk,jM
j∈Z
So, consideringnonregular b-function,wemay defineab-functionforW tobeab-function
for the corresponding V-filtration.
Proposition 3.3. Let M be a holonomic E -module which a good bifiltration admitting
X
a regular b-function, then M is regular along Λ.
Indeed, the existence of a regular b-function implies immediately the point (i) of defini-
tion 3.1.
By the following theorem and its corollary, the converse is true which gives another
condition equivalent to the regularity along Λ.
Theorem 3.4. Let M be holonomic E -module regular along a the conic lagrangian man-
X
ifold Λ. Any section u of M admits locally a regular b-function.
If b(θ) is the b-function of u, there exits an integer d such that θdb(θ) is a regular
b-function for u.
Proof. Let θ be the Euler vector field of Λ and ϑ an operator of E whose image in grVE
X X
isθ. Bydefinition, theprincipalsymbolϕofθ isanequation ofthecanonicalhypersurface
S of T∗Λ.
Λ
If M is regular along Λ, the submodule N = E u is regular along Λ. We consider on
X
N the canonical bifiltration, that is W N = (W E )u. By definition 3.1 of regularity,
kj kj X
ChΛ(∞,1)(N) is contained in SΛ hence grWM is annihilated by a power of ϕ. As the
operator ϑ is of bi-order (0,1), there is an integer q such that
ϑqW N ⊂ W N +W N
k,j k−1,j+q k,j+q−1
Hence there is an operator P ∈ W E and an operator Q ∈ W E such that
−1,q X 0,q−1 X
ϑqu= Pu+Qu
10 YVESLAURENT
We have (ϑq −P)2u = Q(ϑq − P)u+ [ϑq − P,Q]u = Q2u+[ϑq − P,Q]u. The com-
mutator [ϑq − P,Q] belongs to W E as well as Q2 so (ϑq − P)2u = Q u with
0,2(q−1) X 2
Q ∈ W E . By induction we find that (ϑq −P)ru= Q u with Q ∈ W E .
2 0,2(q−1) X r r 0,r(q−1) X
Let b be the b-function of u. If p is the degree of b, there is some integer r such that
b(ϑ)W N ⊂ W N.
0,j −1,j+p+r
We get
b(ϑ)(ϑq −A)ru⊂ b(ϑ)W N ⊂ W N
0,rq−r −1,p+rq
The operator (ϑq − A)r is equal to ϑrq − A where A belongs to W E . hence
r r −1,rq X
b(ϑ)ϑrqu ⊂ W N and as p+rq is the degree of b(ϑ)ϑrq, this is a regular b-function
−1,p+rq
for u. (cid:3)
Remark 3.5. It is clear in the proof that θdb(θ) may be replaced by (θ−α)db(θ) for any
complex number α.
Corollary 3.6. Let M be holonomic E -module regular along the conic lagrangian man-
X
ifold Λ. Any good bifiltration WM relative to Λ admits locally a regular b-function.
If b(θ) is the b-function associated to this bifiltration, there exits an integer d such that
θdb(θ) is a regular b-function.
Proof. Let (u ,...,u ) be local generators of the bifiltration. By definition, there are
1 N
integers (λ ,...,λ ) and (ν ,...,ν ) such that
1 N 1 N
N
WkjM= XWk−λi,j−νiEXui
i=1
If b(θ) is the b-function of the bifiltration, b(θ−λ ) is a b-function for the section u . By
i i
theorem 3.4 and remark 3.5, there exists an integer di such that (θ −λi)dib(θ −λi) is a
regular b-function for d . Let d be the maximum of d for i= 1,...,N.
i i
As remarked in section 2.2, [θ,P] = +kP on grVE hence if P is an operator of
k X
W E we have ϑP = P(ϑ+k−λ )+R with R in W E .
k−λi,j−νi X i k−λi−1,j−νi X
Ifuisasection of W Mwehaveu= P u withP inW E and(θ−k)db(θ−
kj P i i i k−λi,j−νi X
k) is a regular b-function for u. (cid:3)
3.3. ApplicationtoD -modules. PreviousresultshavebeenestablishedforE -modules
X X
on the whole of T∗X including the zero section that is also for D -modules.
X
If x is a point of the zero section X of T∗X, a conic lagrangian submanifold of T∗X
defined in a neighborhood of x is the conormal T∗X to a submanifold Y of X. Let
Y
B∞ = H1(O ) be the cohomology of O and B = H1 (O ) the corresponding
Y|X Y X X Y|X [Y] X
algebraic cohomology.
If Y has codimension 1 we have:
C∞ /C = π−1(B∞ /B ) with π :T∗X → Y
Y|X Y|X Y|X Y|X Y
So if M is a holonomic D -module condition (iv) of definition 3.1 is equivalent to
X
(iv)” ∀j ≥ 0, Extj (M,B )= Extj (M,B∞ )
DX Y|X DX Y|X
and by [11] this is equivalent to formula 1.2.
If Y has codimension greater than 1, the situation is different. By a canonical trans-
formation, the situation is microlocally equivalent to the situation of codimension 1. But
in the definition 3.1 we have to consider points of T˙∗X = T∗X −Y the complementary
Y Y