Micro-support and Cauchy problem for temperate solutions of regular D-Modules

Masaki Kashiwara ^{1} , Teresa Monteiro Fernandes ^{2} , and Pierre Schapira

December 2000

Abstract

Let X be a complex manifold, V a smooth involutive submanifold of T^{∗}X, M a microdifferential system regular along V, and F an ℝ-constructible sheaf on X. We study the complex of temperate microfunction solutions of M associated with F, that is, the complex RHom_{D_{X}} (M, T μhom(F, O_{X})). We give a bound to its micro-support and solve the Cauchy problem under a suitable hyperbolicity assumption.

1 Introduction

The Cauchy problem for solutions of linear differential operators as well as the problem of propagation of singularities, are closely related subjects which have been intensively studied in the 80’th. In the analytic case, it is shown in [8] that these problems may be reduced to purely geometric ones, using sheaf theory, the only analytic tool being the Cauchy-Kowalevski theorem.

To be more precise, recall that a system of linear partial differential operators on a complex manifold X is the data of a coherent module M over the sheaf of rings D_{X} of holomorphic differential operators. Let F be a complex of sheaves on X with ℝ-constructible cohomologies (one says an ℝconstructible sheaf, for short). The complex of “generalized functions” associated with F is described by the complex RHom (F, O_{X}), and the complex of solutions of M with values in this complex is described by the complex

1The first named author benefits from a “Chaire Internationale de Recherche Blaise Pascal de l’Etat et de la Région d’Ile-de-France, gérée par la Fondation de l’Ecole Normale Supérieure”.

2Research supported by FCT and Programa Ciência, Tecnologia e Inovação do Quadro Comunitário de Apoio

RHom_{D_{X}} (M, RHom (F, O_{X})).

One may also microlocalize the problem by replacing RHom (F, O_{X}) with μhom(F, O_{X}). In (loc cit) one shows that most of the properties of this complex, especially those related to propagation or Cauchy problem, are encoded in two geometric objects, both living in the cotangent bundle T^{∗}X, the characteristic variety of the system M, denoted by char(M), and the micro-support of F, denoted by SS(F).

The complex RHom (F, O_{X}) allows us to treat various situations. For example if M is a real manifold and X is a complexification of M, by taking as F the dual D′(ℂ_{M}) of the constant sheaf on M, one obtains the sheaf B_{M} of Sato’s hyperfunctions. If Z is a complex analytic hypersurface of X and F = ℂ_{Z}[−1] is the (shifted) constant sheaf on Z, one obtains the sheaf of holomorphic functions with singularities on Z.

However, the complex RHom (F, O_{X}) does not allow us to treat sheaves associated with holomorphic functions with temperate growth such as Schwartz’s distributions or meromorphic functions with poles on Z. To consider such cases, one has to replace it by the complex T hom(F, O_{X}) of temperate cohomology, introduced in [4] or its microlocalization T μhom(F, O_{X}) constructed by Andronikof [1]. At this stage, a serious difficulty appears: the geometric methods of [8] do not apply any more, and indeed, it is a well known fact that to solve for example the Cauchy problem for distributions requires more informations than the data of the characteristic variety of the system.

In fact, very little is known concerning the problems of propagation of singularities and the Cauchy problem in the space of distributions, apart the case where M has real simple characteristics (see [2, 10] for a formulation in the language of sheaves and D-modules) and some very specific situation (e.g., operators with constant coefficients on ℝ^{n}). We refer to [3] for historical and bibliographical comments.

In this paper we give an estimate for the microsupport of the sheaf of temperate microfunction solutions associated with an ℝ-constructible object F, when M has regular singularities along an involutive manifold V in the sense of [7]. More precisely, we prove the estimate

SS(RHom_{D_{X}} (M, T μhom(F, O_{X}))) ⊂ ρ_{V}^{−1}(C_{V} (SS(F))^{a}), (1)

where ρ_{V} : V ×_{T^{∗}X} T^{∗}(T^{∗}X) ⟶^{∼} V ×_{T^{∗}X} T (T^{∗}X) ⟶ T_{V} (T^{∗}X) is the pro-

jection and a is the antipodal map of T_{V} (T^{∗}X) as a vector bundle over V.

One can translate this result as follows. For a bicharacteristic leaf Σ of V, one has Σ ×_{T^{∗}X} T_{V} (T^{∗}X) ≃ T^{∗}Σ, and Σ ×_{V} C_{V} (SS(F))^{a} may be regarded as a subset of T^{∗}Σ. Then for any Σ, (1) implies

SS(RHom_{D_{X}} (M, T μhom(F, O_{X}))|_{Σ}) ⊂ Σ ×_{V} C_{V} (SS(F))^{a}. (2)

What makes this paper original, in our opinion, is that we treat general ℝ-constructible sheaves F. Let us illustrate our results by an example.

We consider a smooth morphism f: X ⟶ Y, we set V = X ×_{Y} T^{∗}Y, and we assume that M is a coherent module regular along V. Let M be a real analytic manifold with complexification X, S a closed subanalytic subset of M. We obtain the estimate

SS(RHom_{D_{X}} (M, Γ_{S}(Db_{M}))) ⊂ V ˆ+(SS(ℂ_{S}))^{a},

where Db_{M} denote the sheaf of distributions on M, and the operation ˆ+ is defined in [8] and recalled in Section 2.

2 Notations and main results

We will mainly follow the notations in [8].

Let X be a real analytic manifold. We shall denote by τ: T X ⟶ X the tangent bundle to X and by π: T^{∗}X ⟶ X the cotangent bundle. Set Ṫ^{∗}X = T^{∗}X \ X and π̇: Ṫ^{∗}X ⟶ X the projection Ṫ^{∗}X ⟶ X. For a smooth submanifold Y of X, T_{Y} X denotes the normal bundle to Y and T_{Y}^{∗} X the conormal bundle. In particular, T_{X}^{∗}X is identified with X, the zero section.

For a submanifold Y of X and a subset S of X, we denote by C_{Y} (S) the normal cone to S along Y, a conic subset of T_{Y} X.

If A and B are two conic subsets of T^{∗}X, the operation Aˆ+B is defined in (loc cit) and will be recalled below. The set A^{a} denotes the image of A by the antipodal map, (x; ξ) ↦ (x; −ξ).

For a cone γ ⊂ T X, the polar cone γ^{∘} to γ is the closed convex cone of T^{∗}X defined by

γ^{∘} = {(x; ξ); x ∈ π(γ), ⟨v, ξ⟩ ≥ 0 for any (x; v) ∈ γ}.

Let f: X ⟶ Y be a morphism of complex manifolds. One has two natural morphisms

T^{∗}X ⟵_{f_{d}} X ×_{Y} T^{∗}Y ⟶_{f_{π}} T^{∗}Y

(In [7], f_{d} is denoted by ^{t}f′.)

We denote by D(ℂ_{X}) (respectively D^{b}(ℂ_{X}), D_{ℝ−c}^{b}(ℂ_{X})) the derived category of complexes of sheaves of ℂ-vector spaces (respectively with bounded cohomologies, with bounded and ℝ-constructible cohomologies).

We denote by D′_{X} the duality functor on D^{b}(ℂ_{X}), defined by

D′_{X}(F) = RHom (F, ℂ_{X}).

If F is an object of D^{b}(ℂ_{X}), SS(F) denotes its micro-support, a closed ℝ^{+}-conic involutive subset of T^{∗}X.

On a complex manifold X we consider the sheaf O_{X} of holomorphic functions, the sheaf Θ_{X} of holomorphic vector fields, the sheaf D_{X} of linear holomorphic differential operators of finite order, and its subsheaves D_{X}(m) of operators of order at most m. We shall also consider the sheaf E_{X} on T^{∗}X of microdifferential operators of finite order ([14] and [13] for an exposition) and its subsheaves E_{X}(m) of operators of order at most m. We denote by Mod_{coh}(D_{X}) (respectively by Mod_{coh}(E_{X})) the abelian category of coherent D_{X}-modules (respectively coherent E_{X}-modules). We denote by D^{b}(D_{X}) the bounded derived category of left D_{X}-modules and by D_{coh}^{b}(D_{X}) its full triangulated category consisting of objects with coherent cohomologies. We define similarly D^{b}(E_{X}) and D_{coh}^{b}(E_{X}).

The notion of regularity of an E_{X}-module along an involutive submanifold V of T^{∗}X will be recalled in Section 3.

The main purpose of this paper is to prove the three following results. Let V be an involutive vector subbundle of T^{∗}X. Let M be a coherent

D_{X}-module regular along V and let F ∈ D_{ℝ−c}^{b}(ℂ_{X}),

Theorem 2.1. We have the estimate:

(3)  SS(RHom_{D_{X}} (M, T hom(F, O_{X}))) ⊂ V ˆ+SS(F)^{a}.

Note that V ˆ+SS(F)^{a} coincides with the closure of V +SS(F)^{a} by Lemma 4.1.

As a particular case of Theorem 2.1, assume that X is the complexification of a real analytic manifold M and let S be a closed subanalytic subset of M. We obtain the estimate:

SSRHom_{D_{X}} (M, Γ_{S}(Db_{M})) ⊂ (SS(ℂ_{S}))^{a} ˆ+V.

For a complex submanifold Z of X, we denote by M_{Z} the induced system of M on Z.

Theorem 2.2. Assume T_{Z}^{∗}X ∩ ((SSF)^{a} ˆ+V) ⊂ T_{X}^{∗}X. Then

RHom_{D_{X}} (M, T hom(F, O_{X}))|_{Z} ≃ RHom_{D_{Z}}(M_{Z}, T hom(F_{Z}, O_{Z})).

Let U be a conic open subset of Ṫ^{∗}X, and let V be a closed smooth conic regular involutive submanifold in U. We denote by

ρ_{V} : V ×_{T^{∗}X} T^{∗}(T^{∗}X) ⟶^{∼} V ×_{T^{∗}X} T (T^{∗}X) ⟶ T_{V} T^{∗}X

the canonical projection.

For an ℝ-constructible sheaf F, the cohomology of T μhom(F, O_{X}) is provided with an action of E_{X} as proved in [1], therefore, when T μhom(F, O_{X}) is concentrated in a single degree, we regard it as an E_{X}-module.

Theorem 2.3. Let U and V be as above and let M be a coherent E_{X}- module defined on U and regular along V. Assume that F ∈ D_{ℝ−c}^{b}(ℂ_{X}) and T μhom(F, O_{X})|_{U} is concentrated in a single degree. Then

(4)  SS(RHom_{E_{X}} (M, T μhom(F, O_{X}))) ⊂ ρ_{V} ^{−1}(C_{V} (SS(F))^{a}).

For a bicharacteristic leaf Σ of V, one has Σ ×_{T^{∗}X} T_{V} (T^{∗}X) ≃ T^{∗}Σ, and Σ ×_{V} C_{V} (SS(F))^{a} may be regarded as a subset of T^{∗}Σ.

Corollary 2.4. Let Σ be a bicharacteristic leaf of V. Then one has

SS(RHom_{D_{X}} (M, T μhom(F, O_{X}))|_{Σ}) ⊂ Σ ×_{V} C_{V} (SS(F))^{a}.

Proof. The map ρ_{V} decomposes as V ×_{T^{∗}X} T^{∗}(T^{∗}X) ⟶_{j_{d}} T^{∗}V ⟶_{h} T_{V} T^{∗}X.

Here j : V ↪ T^{∗}X is the embedding. Set S = RHom_{D_{X}} (M, T μhom(F, O_{X})). Then the support of S is contained in V, and

SS(S|_{V}) = j_{d}(SS(S)) ⊂ h^{−1}(C_{V} (SS(F))^{a}).

Hence the corollary follows from the following lemma. q.e.d.

Lemma 2.5. Let f: X ⟶ Y be a smooth morphism of real analytic manifolds. Let T^{∗}(X/Y) be the relative cotangent bundle and h: T^{∗}X ⟶ T^{∗}(X/Y) be the canonical projection. Let S be a closed conic subset of T^{∗}(X/Y), and let F ∈ D^{b}(ℂ_{X}). Then the following two conditions are equivalent.

(i) SS(F) ⊂ h^{−1}(S).

(ii) for any y ∈ Y, identifying f^{−1}(y) ×_{X} T^{∗}(X/Y) with T^{∗}(f^{−1}(y)),

SS(F |_{f^{−1}(y)}) ⊂ f^{−1}(y) ×_{X} S.

Proof. (i)⇒(ii) Since SS(F ⊗ ℂ_{f^{−1}(y)}) = h^{−1}(SS(F |_{f^{−1}(y)})), it is enough to show that

SS(F ⊗ ℂ_{f^{−1}(y)}) ⊂ h^{−1}S.

Since SS(F ⊗ ℂ_{f^{−1}(y)}) ⊂ SS(F)ˆ+T_{f^{−1}(y)}^{∗}X, we may reduce the assertion to

h^{−1}S ˆ+T_{f^{−1}(y)}^{∗}X) ⊂ h^{−1}S. (5)

Let N = X ×_{Y} T^{∗}Y ⊂ T^{∗}X. Then T^{∗}(X/Y) = T^{∗}X/N, and T_{f^{−1}(y)}^{∗}X ⊂ N. Hence (5) is a consequence of h^{−1}S ˆ+N = \overline{h^{−1}S + N} (Lemma 4.1 (i)). (ii)⇒(i) Let us take a coordinate system x = (x_{1}, x_{2}) on X such that f is given by x ↦ x_{1}. Assume that (x_{0}, ξ_{0}) ∈ T^{∗}X \ h^{−1}S. Set L(x, δ, ε) = {z ∈ X; ε > ⟨z − ξ_{0}, ξ_{0}⟩ ≥ δ|z − ξ_{0}|}. It is enough to show that

RΓ(X; F ⊗ ℂ_{L(x,δ,ε)}) = 0

for x sufficiently close to x_{0} and 0 < ε, δ ≪ 1. For any y ∈ Y, RΓ(f^{−1}(y); F ⊗ ℂ_{L(x,δ,ε)}|_{f^{−1}(y)}) = 0 by the assumption. Hence Rf_{∗}(F ⊗ℂ_{L(x,δ,ε)}) = 0, which implies RΓ(X; F ⊗ ℂ_{L(x,δ,ε)}) = 0. q.e.d.

3 Regularity for D-Modules

The results contained in this section are extracted or adapted from [2] and [5].

Recall that a good filtration on a D_{X}-module M is a sequence of coherent O_{X}-submodules {M_{k}}_{k∈ℤ} satisfying:

(i) D_{X}(l)M_{k} ⊂ M_{k+l} for any l, k ∈ ℤ,

(ii) M = ⋃_{k} M_{k},

(iii) locally on X, M_{k} = 0 for k ≪ 0,

(iv) locally on X, D_{X}(l)M_{k} = M_{l+k} for k ≫ 0 and any l ≥ 0.

We shall use the following notations. For a coherent D_{X}-module M, we denote by E_{X}M the coherent E_{X}-module E_{X} ⊗_{π^{−1}D_{X}} π^{−1}M. For a coherent

E_{X}-module M and a coherent E_{X}(0)-submodule N_{0} of M, we set

N_{0}(m) := E_{X}(m)N_{0} for m ∈ ℤ.

Lemma 3.1 ([6]). Let M be a coherent D_{X}-module and let N_{0} be a coherent E_{X}(0)-submodule of E_{X}M|_{Ṫ^{∗}X} such that N_{0} generates E_{X}M on Ṫ^{∗}X. Set

M_{k} = {u ∈ M; (1 ⊗ u)|_{Ṫ^{∗}X} ∈ N_{0}(k)} for k ≥ 0,
M_{k} = 0 for k < 0.

Then M_{k} defines a good filtration on M.

We shall use the notion of regularity along a closed analytic subset V of Ṫ^{∗}X due to [7]. Let us denote by J_{V} the subsheaf of E_{X} of microdifferential operators of order at most 1 whose symbol of order 1 vanishes on V. In particular J_{V} contains E_{X}(0). Then E_{V} denotes the sub-sheaf of rings of E_{X} generated by J_{V}. More precisely,

E_{V} = ∪_{m≥0}J_{V}^{m}.

Definition 3.2. [7] Let M be a coherent E_{X}-module. An E_{V} -lattice in M is an E_{V} -submodule N_{0} of M such that N_{0} is E_{X}(0)-coherent and generates M over E_{X}.

Definition 3.3. [7] A coherent E_{X}-module M is called regular along V if M has an E_{V} -lattice locally on Ṫ^{∗}X.

Note that if a coherent E_{X}-module M is regular along V, its support is contained in V.

If M ∈ D_{coh}^{b}(E_{X}), one says that M is regular along V if H^{k}(M) is regular along V for every k ∈ ℤ.

Definition 3.4. Let V be a closed analytic subset of T^{∗}X. A coherent D_{X}-module M is called regular along V if the characteristic variety of M is contained in V and if E_{X}M is regular along V̇ := V \ T_{X}^{∗}X.

One extends this definition to D_{coh}^{b}(D_{X}) as in the E_{X}-module case. The next result follows from [7].

Lemma 3.5. Consider a distinguished triangle M′ ⟶ M ⟶ M′′ ⟶_{1} in

D_{coh}^{b}(D_{X}) and assume that two of these three objects are regular along V. Then so is the third one.

The next lemma gives a characterization of the D_{X}-modules which are regular along V when V has a special form. If V is an involutive vector subbundle of T^{∗}X, then one can find locally on X a smooth morphism f: X ⟶ Y such that V = X ×_{Y} T^{∗}Y Lemma 3.6. Let f: X ⟶ Y be a smooth morphism and let V := X ×_{Y} T^{∗}Y. Let M be a coherent D_{X}-module regular along V. Then, locally on X, M is the pull-back of a coherent D_{Y} -module by f. In particular M admits a finite resolution locally on X

0 ⟶ L^{N} ⟶ L^{N−1} ⟶ ⋯ ⟶ L^{0} ⟶ M ⟶ 0,

each D_{X}-module L^{j} being isomorphic to a finite direct sum of the D_{X}-module D_{X⟶Y}.

Proof. Since the assertion is local on X we may assume that X = Z ×Y and f is the projection Z × Y ⟶ Y. Hence V̇ = Z × Ṫ^{∗}Y. Let Θ_{f} ⊂ Θ_{X} denote the O_{X}-module of vector fields tangent to the fibers of f. With the above definition, an EV̇ -lattice for E_{X}M|_{Ṫ^{∗}X} is a coherent E_{X}(0)-submodule N_{0} of E_{X}M such that N_{0} generates E_{X}M on Ṫ^{∗}X and such that Θ_{f}N_{0} ⊂ N_{0}.

Locally on X, there exists a coherent O_{X}-submodule M_{0} of M such that M ≃ D_{X}M_{0}. Let us prove that the coherent EV̇ -module N_{0} := EV̇ M_{0} is an EV̇ -lattice in E_{X}M|_{Ṫ^{∗}X}. Since M_{0} generates E_{X}M it is sufficient to prove that N_{0} is E_{X}(0)-coherent.

Locally on X, there exists a finite covering of Ṫ^{∗}X by ℂ^{×}-conic open subsets U_{j} and EV̇ -modules N_{j} such that N_{j} is an EV̇ -lattice in M on U_{j}. Hence, for each j, there exists m_{j} ∈ ℤ such that N_{0} is contained in N_{j}(m_{j}) on U_{j}.

Consider the increasing sequence {J_{V}^{k}M_{0}}_{k≥0} of coherent E_{X}(0)-submodules of N_{0}. For each j, the restriction of this sequence to U_{j} is contained in N_{j}(m_{j}), which is E_{X}(0)-coherent, hence it is locally stationary. Therefore N_{0} is E_{X}(0)-coherent.

To summarize, we have constructed an EV̇ -module N_{0}, coherent over E_{X}(0)-module and which generates M. We may now apply Lemma 3.1 and consider the good filtration M_{k} associated to N_{0}.

By the construction, Θ_{f}M_{k} ⊂ M_{k}. Setting L = M_{k} for k ≫ 0, we find a coherent O_{X}-submodule L of M such that Θ_{f}L ⊂ L and M = D_{X}L. Hence, locally on X, there is a D_{X}-linear epimorphism φ_{0} : (D_{X⟶Y})^{N_{0}} ⟶ M. Repeating the same construction with M replaced by ker(φ_{0}), we construct an exact sequence

D_{X⟶Y}^{N_{1}} ⟶_{φ_{1}} D_{X⟶Y}^{N_{0}} ⟶_{φ_{0}} M ⟶ 0.

Since

Hom_{D_{X}} (D_{X⟶Y} , D_{X⟶Y}) ≃ f^{−1}D_{Y} ,

there is a D_{Y} -linear morphism ψ: D_{Y}^{N_{1}} ⟶ D_{Y}^{N_{0}} such that φ_{1} = f^{−1}ψ. Set N = coker ψ. This is a coherent D_{Y} -module and we have an isomorphism

M ≃ D_{X⟶Y} ⊗_{f^{−1}D_{Y}} f^{−1}N .

To conclude, we choose a finite resolution of N by finitely free D_{Y} -modules, and tensorize over f^{−1}D_{Y} by the flat f^{−1}D_{Y} -module D_{X⟶Y}.

q.e.d.

4 Review on normal cones

We shall now recall some constructions of [8] which will be useful for the next steps. To start with, we shall assume that X is a real manifold. Let S_{1} and S_{2} be two subsets of X. The normal cone C(S_{1}, S_{2}) is a closed conic subset of T X which can be described as follows:

Let (x) be a system of local coordinates on X. Then (x_{0}; v_{0}) ∈ C(S_{1}, S_{2}) if and only if there exists a sequence {(x_{n}, y_{n}, c_{n})} in S_{1} ×S_{2} ×ℝ^{+} such that

(6)  x_{n} ⟶_{n} x_{0}, y_{n} ⟶_{n} x_{0}, c_{n}(x_{n} − y_{n}) ⟶_{n} v_{0}

If A is a conic subset of T^{∗}X, we denote by A^{∘} its polar which is a conic subset in T X. Let A and B be two conic subsets of T^{∗}X. One defines the sum

A + B = {(x; ξ) ∈ T^{∗}X; ξ = ξ_{1} + ξ_{2} for some (x; ξ_{1}) ∈ A and (x; ξ_{2}) ∈ B}.

If A and B are two closed conic subsets of T^{∗}X, one also defines Aˆ+B, a closed conic set containing A + B, which may be described in a local canonical coordinate system (x; ξ) as follows: (x_{0}; ξ_{0}) belongs to Aˆ+B if and only if there exist a sequence {(x_{n}; ξ_{n})}_{n} in A, a sequence {(y_{n}; η_{n})}_{n} in B, such that

(7)  x_{n} ⟶_{n} x_{0}, y_{n} ⟶_{n} x_{0}, (ξ_{n} + η_{n}) ⟶_{n} ξ_{0}, |x_{n} − y_{n}||ξ_{n}| ⟶_{n} 0.

If A ∩ B^{a} ⊂ T_{X}^{∗}X, we have

Aˆ+B = A + B.

Let now Y be a closed submanifold of X and let i : Y ⟶ X be the inclusion morphism. Using the Hamiltonian isomorphism, we get an embedding of T^{∗}Y into T_{T_{Y}^{∗} X}(T^{∗}X). Let A be a conic subset of T^{∗}X. One sets

i^{♯}(A) := T^{∗}Y ∩ C_{T_{Y}^{∗} X}(A).

This set can be described explicitly by local coordinate systems as follows. Let (x, y) be a local coordinate system on X such that Y = {x = 0}. Then (y_{0}; η_{0}) ∈ i^{♯}(A) ⇔ there exists a sequence {(x_{n}, y_{n}; ξ_{n}, η_{n})}_{n} in A such that

(8)  x_{n} ⟶_{n} 0, y_{n} ⟶_{n} y_{0}, η_{n} ⟶_{n} η_{0} and |x_{n}||ξ_{n}| ⟶_{n} 0.

Lemma 4.1. Let X be an open subset in a finite dimensional real vector space E with 0 ∈ E . Let Λ be closed conic subset of T^{∗}X. Let L be a vector subspace of the dual vector space E^{∗} and let V = X × L. Then we have

(i) V ˆ+Λ = \overline{V + Λ}.

(ii) For any θ ∈ E^{∗} we have

\overline{V + (Λˆ+X × ℝ^{≤0}θ)} ∩ X × ℝ^{≥0}θ ⊂ \overline{V + Λ}.

(iii) Let i: E^{∗} ⟶ T^{∗}X be the map θ ↦ (0; θ) and assume (0; 0) ∈ Λ. Then, for any vector subspace N of E^{∗},

({0} × N) ∩ (V ˆ+Λ) ⊂ T_{X}^{∗}X

if and only if

(N + i^{−1}(Λ)) ∩ L = {0}.

Proof. (i) The inclusion V ˆ+Λ ⊂ \overline{V + Λ} is clear. Conversely, assume that there are sequences {(x_{n}; ζ_{n})}_{n} ⊂ V and {(y_{n}; η_{n})}_{n} ⊂ Λ such that

x_{n} ⟶_{n} x, y_{n} ⟶_{n} x, ζ_{n} + η_{n} ⟶_{n} ξ.

Then the sequence {(y_{n}; ζ_{n})}_{n} is contained in V. Therefore, {(y_{n}; ζ_{n})}_{n} and {(y_{n}; η_{n})}_{n} satisfy (2). (ii) Suppose that (z; θ) ∈ \overline{V + (Λˆ+X × ℝ^{≤0}θ)}. Then by (i) there exist sequences (ζ_{n})_{n} in L, {(y_{n}; η_{n})}_{n} in Λ, λ_{n} ≥ 0, such that y_{n} ⟶_{n} z and

(ζ_{n} + η_{n} − λ_{n}θ) ⟶_{n} θ. Therefore

ζ_{n} + η_{n} − (λ_{n} + 1)θ ⟶_{n} 0.

Since λ_{n} ≥ 0, we get

\frac{ζ_{n} + η_{n}}{λ_{n} + 1} ⟶_{n} θ,

hence (z; θ) ∈ \overline{V + Λ}.

(iii) The condition is obviously necessary. Let us now assume that there exists θ ∈ N, θ ≠ 0, such that (0; θ) ∈ V ˆ+Λ. Then there exist sequences (ζ_{n})_{n} in L and (y_{n}; η_{n})_{n} in Λ such that y_{n} ⟶_{n} 0 and ζ_{n} + η_{n} ⟶_{n} θ. Taking suitable subsequences, we may assume that ζ_{n}/|ζ_{n}| converges to l ∈ L, l ≠ 0. Suppose that ζ_{n} ⟶_{n} 0. Then (0; θ) ∈ Λ, a contradiction. If |ζ_{n}| is unbounded

we get ζ_{n}/|ζ_{n}| + η_{n}/|ζ_{n}| ⟶_{n} 0 hence (0; −l) ∈ L ∩ i^{−1}(Λ), a contradiction. In the other case, we may assume that ζ_{n} ⟶_{n} l and setting (0; η) = (0; θ − l) ∈

(N + L) ∩ i^{−1}(Λ), we get η + l = θ, a contradiction. q.e.d.

5 Proof of Theorem 2.1

Let X be an open subset of a finite-dimensional real vector space E. For (x_{0}; ξ_{0}) ∈ T^{∗}X, ε > 0, δ > 0, an open convex proper cone γ of E and v ∈ γ, we introduce the following notation:

L_{ε,ξ_{0}} = {y ∈ E; ⟨y − x_{0}, ξ_{0}⟩ > −ε},
Z(x, γ, ε, ξ_{0}) = (x + \overline{γ}) ∩ L_{ε,ξ_{0}}.

Here \overline{γ} denotes the closure of γ, The following result is proved in [8].

Lemma 5.1. Let F ∈ D^{b}(ℂ_{X}) and let p = (x_{0}; ξ_{0}) ∈ T^{∗}X. The conditions below are equivalent:

(i) p ∉ SS(F).

(ii) There exists an open neighborhood U of x_{0} and an open convex proper subanalytic cone γ ⊂ E such that ξ_{0} ∈ Int(γ^{∘})^{a}, satisfying: for any x ∈ U and sufficiently small ε > 0, Z(x, γ, ε, ξ_{0}) is contained in X and

(9)  RΓ_{c}(Z(x, γ, ε, ξ_{0}); F) = 0.

We can now embark into the proof of Theorem 2.1

We may assume X = Z × Y, f is the projection, and X, Z, Y are open subsets of affine complex spaces. Moreover, using the results of § 3, we may assume that M = D_{X⟶Y}. We shall set for short:

H(F) := RHom_{D_{X}} (M, T hom(F, O_{X})).

Let θ = (x_{0}; ξ_{0}) ∉ V ˆ+(SS(F))^{a}. Let us take an open convex proper cone γ and an open neighborhood U of x_{0} such that ξ_{0} ∈ Int(γ^{∘})^{a} and U × γ^{∘} ∩ (V ˆ+(SS(F))^{a}) ⊂ T_{X}^{∗}X. By Lemma 5.1 and keeping its notations, it is enough to prove RΓ_{c}(Z(x, γ, ε, ξ_{0}); H(F)) = 0 for any x ∈ U and sufficiently small ε > 0. Taking v ∈ γ, set Z_{δ} = Z(x − δv, γ, ε − δ, ξ_{0}) for 0 < δ ≪ ε. Then we have

ℂ_{Z(x,γ,ε,ξ_{0})} = \underrightarrow{lim}_{δ>0} ℂ_{Z_{δ}},

and hence we obtain

H_{c}^{j}(Z(x, γ, ε, ξ_{0}); H(F)) ≃ \underrightarrow{lim}_{δ>0} H_{c}^{j}(X; H(F) ⊗ ℂ_{Z_{δ}}).

Set

Z′_{δ} = (x − δv + γ) ∩ {y ∈ X; ⟨y − x_{0}, ξ_{0}⟩ ≥ −ε + δ}.

Then for 0 < δ′ < δ, there is a chain of morphisms

H(F) ⊗ ℂ_{Z_{δ}} ⟶ H(F ⊗ ℂ_{Z′_{δ}}) ⟶ H(F) ⊗ ℂ_{Z_{δ′}} ⟶ H(F ⊗ ℂ_{Z′_{δ′}}). Therefore we have

\underrightarrow{lim}_{δ>0} H_{c}^{j}(X; H(F) ⊗ ℂ_{Z_{δ}}) ≃ \underrightarrow{lim}_{δ>0} H_{c}^{j}(X; H(F ⊗ ℂ_{Z′_{δ}})).

Since f is proper over the support of ℂ_{Z′_{δ}}, we may apply Theorem 7.2 of [9] and obtain

H_{c}^{j}(Z(x, γ, ε, ξ_{0}); H(F)) ≃ \underrightarrow{lim}_{δ>0} H_{c}^{j}(Y; T hom(Rf_{!}(F ⊗ ℂ_{Z′_{δ}}), O_{Y} )).

Hence, we are reduced to prove Rf_{!}(F ⊗ ℂ_{Z′(x,γ,ε,ξ_{0})}) = 0, (10) where Z′(x, γ, ε, ξ_{0}) = (x+γ)∩{y ∈ X; ⟨y −x_{0}, ξ_{0}⟩ ≥ −ε}. In order to prove this, we shall apply [8, Proposition 5.4.17]. Set X_{t} = {y ∈ X; ⟨y − x_{0}, ξ_{0}⟩ ≥ t}. Then if we prove (y; −ξ_{0}) ∉ (SS(F ⊗ ℂ_{(x+γ)}) + V ), (11) for y ∈ U, we have

Rf_{!}(F ⊗ ℂ_{Z′(x,γ,ε,ξ_{0})}) = Rf_{!}(F ⊗ ℂ_{(x+γ)∩X_{t}}).

Hence taking t > 0, we obtain the desired result (10).

Thus the proof is reduced to (11). We have

SS(ℂ_{(x+γ)}) ⊂ X × γ^{∘}^{a}.

Hence we have

SS(F ⊗ ℂ_{(x+γ)}) ⊂ SS(F)ˆ+(X × γ^{∘}^{a}).

Since SS(F) ∩ (X × γ^{∘}) ⊂ T_{X}^{∗}X, we get

SS(F) + (X × γ^{∘}) = SS(F)ˆ+(X × γ^{∘})

and

SS(F ⊗ ℂ_{(x+γ)}) ⊂ SS(F)+(X × γ^{∘}^{a}).

On the other hand, by the choice of γ, we have

(X × γ^{∘}) ∩ (SS(F) + V) ⊂ T_{X}^{∗}X.

Hence

(X × γ^{∘}) ∩ (SS(F) + (X × γ^{∘}^{a}) + V ) ⊂ T_{X}^{∗}X

and we obtain

(X × Int(γ^{∘})) ∩ ((SS(F)ˆ+(X × γ^{∘}^{a})) + V ) = ∅.

Then the desired result follows from −ξ_{0} ∈ Int(γ^{∘}).

6 Proof of Theorem 2.2

We shall now embark in the proof of Theorem 2.2. Since the question is local on X, by Lemma 3.6 we may assume that M is isomorphic to D_{X⟶Y}. By (7.5) of [9], if d denotes the codimension of Z, we have a natural isomorphism (12)

RHom_{D_{Z}}(M_{Z}, T hom(F_{Z}, O_{Z})) ≃ RHom_{D_{X}} (M, T hom(F_{Z}, O_{X}))|_{Z}[2d]

With this isomorphism in hand, Theorem 2.2 will be a consequence of the next Lemma for a real submanifold.

If Z is a real submanifold of a complex manifold X, we still denote by T_{Z}^{∗}X the conormal bundle to Z of the real underlying manifold X^{ℝ} .

Lemma 6.1. Let Z be a real analytic closed submanifold of X of codimension d ≥ 1 and assume that

T_{Z}^{∗}X ∩ (V ˆ+SS(F)^{a}) ⊂ T_{X}^{∗}X.

Let M be a regular D_{X}-module along V. Then the following natural morphism is an isomorphism: (13)

RHom_{D_{X}} (M, T hom(F, O_{X}))|_{Z} ⟶ RHom_{D_{X}} (M, T hom(F_{Z}, O_{X}))|_{Z}[d]

Proof. Since T_{Z}^{∗}X ∩V ⊂ T_{X}^{∗}X, there exists a coordinate system (y_{1}, ⋯, y_{N}) on Y as a real analytic manifold and a coordinate system (f_{1}, ⋯, f_{m}, y_{1} ∘ f, ⋯, y_{N} ∘f) of X such that Z is defined by the equations f_{1} = ⋯ = f_{d} = 0. We shall argue by induction on d.

(i) Let us prove the result for d = 1. Assume that Z is a real analytic hypersurface defined by the equation f(z) = 0, and set

Z^{−} = {z ∈ X; f(z) ≤ 0}, Z^{+} = {z ∈ X; f(z) ≥ 0}.

Assume that df, −df ∉ V ˆ+SS(F)^{a}. We shall show that the morphism (14)

RHom_{D_{X}} (M, T hom(F, O_{X}))|_{Z} ⟶ RHom_{D_{X}} (M, T hom(F_{Z}, O_{X}))|_{Z}[1].

is an isomorphism. The morphism (14) is given by

F ⟶ F_{Z^{+}} ⊕ F_{Z^{−}} ⟶ F_{Z} ⟶^{+1}

Therefore to obtain (14) it is enough to prove that

RHom_{D_{X}} (M, T hom(F_{Z^{±}}, O_{X}))|_{Z} = 0.

Since

SS(F_{Z^{±}}) ⊂ SS(F)ˆ+(X × ℝ^{≥0}(±df)),

±df ∈ V ˆ+SS(F_{Z^{±}})^{a} implies ±df ∈ V ˆ+SS(F)^{a} by Lemma 4.1 (ii), which contradicts the assumption. Hence

±df ∉ V ˆ+SS(F_{Z^{±}})^{a}.

Denoting

S_{±} = Hom_{D_{X}} (M, T hom(F_{Z^{±}}, O_{X})),

then, by Theorem 2.1, ±df ∉ SS(S_{±}) hence S_{±}|_{Z} = RΓ_{Z^{±}}(S_{±}|_{Z}) = 0.

(ii) Let us set Z_{1} = {f_{1} = 0} and Z_{2} = {f_{2} = ⋯ = f_{d} = 0}. Since

T_{Z_{2}}^{∗}X ∩ (V ˆ+(SS(F)^{a})) ⊂ T_{X}^{∗}X

in a neighborhood of Z, the hypothesis of induction implies

RHom_{D_{X}} (M, T hom(F, O_{X}))|_{Z} ≃ RHom_{D_{X}} (M, T hom(F_{Z_{2}}, O_{X}))|_{Z}[d − 1].

On the other hand, SS(F_{Z_{2}}) ⊂ SS(F)ˆ+T_{Z_{2}}^{∗}X. Hence by (i)

RHom_{D_{X}} (M, T hom(F, O_{X}))|_{Z} ≃ RHom_{D_{X}} (M, T hom(F_{Z_{2}}, O_{X}))|_{Z}[d − 1]

≃ RHom_{D_{X}} (M, T hom(F_{Z}, O_{X}))|_{Z}[d].

This ends the proof of (13).

q.e.d.

7 Proof of Theorem 2.3

We start by recalling the functor of tempered microlocalization.

7.1 Review on T μhom

We shall recall the construction of the functor T νhom(·, O_{X}) of tempered specialization of ([1]).

Let X̃^{ℂ} be the complex normal deformation of X ×X along the diagonal Δ which we identify with X by the first projection p_{1}. We may then identify T X with the normal bundle T_{Δ}(X × X). Let t : X̃^{ℂ} ⟶ ℂ and p : X̃^{ℂ} ⟶ X × X be the canonical maps, let Ω̃ be t^{−1}(ℂ − {0}) and Ω = t^{−1}(ℝ^{+}) ⊂ Ω̃. Let p_{2} : X × X ⟶ X be the second projection.

Consider the following diagram of morphisms:

(15)  T X ≃ T_{Δ}(X × X) ↪^{i} X̃^{ℂ} ↩^{j} Ω = t^{−1}(ℝ^{+})

Let p̃ : Ω̃ ⟶ X × X, be the restriction of p. Finally denote by \overline{p}_{1} the composition p_{1} ∘ p and by \overline{p}_{2} the composition p_{2} ∘ p.

Under these notations, T νhom(F, O_{X}) is defined by

T νhom(F, O_{X}) = i^{−1}RHom_{DX̃^{ℂ}}(D_{X̃^{ℂ}⟶_{\overline{p}_{1}}X}, T hom(\overline{p}_{2}^{−1}F ⊗ ℂ_{Ω}, OX̃^{ℂ})).

Let D_{ℝ^{+}}^{b}(ℂ_{T X}) (resp. D_{ℝ^{+}}^{b}(ℂ_{T^{∗}X})) be the derived category of complexes of sheaves on T X (resp. T^{∗}X) with conic cohomologies. We denote by the symbol the Fourier-Sato Transform from D_{ℝ^{+}}^{b}(ℂ_{T X}) to D_{ℝ^{+}}^{b}(ℂ_{T^{∗}X}). Then by definition, T μhom(F, O_{X}) = T νhom(F, O_{X}) . Let us recall that under the identification of T^{∗}(T X) with T^{∗}(T^{∗}X) by the Hamiltonian isomorphism we have SS(F) = SS(F) for any F ∈ D_{ℝ^{+}}^{b}(ℂ_{T X}).

Remark that for any coherent D_{X}-module M, one has

RHom_{π^{−1}D_{X}} (π^{−1}M, T μhom(F, O_{X})) (16)

≃ RHom_{τ^{−1}D_{X}} (τ^{−1}M, T νhom(F, O_{X})) .

The proof of Theorem 2.3 will be performed in two steps.

7.2 First reduction

First of all remark that the statement of Theorem 2.3 is local on T^{∗}X and invariant by local canonical transformation as proved in [1], 5.5. Therefore, since V is regular involutive, locally in Ṫ^{∗}X, we may choose a canonical coordinate system (x; ξ), x = (x_{1}, ..., x_{n}), ξ = (ξ_{1}, ..., ξ_{n}), such that V = {(x; ξ) ∈ Ṫ^{∗}X; ξ_{1} = 0, ..., ξ_{d} = 0}, in other words, locally on X, we have X ≃ Z × Y where Y is an open subset of ℂ^{n−d} and V = X ×_{Y} T^{∗}Y, is

associated to the projection f : X ⟶ Y. By the results of [7], we have an exact sequence of coherent E_{X}-modules

(17)  0 ⟶ N ⟶ (E_{X}/(E_{X}D_{x_{1}} + E_{X}D_{x_{2}} + ... + E_{X}D_{x_{d}}))^{N} ⟶ M ⟶ 0

where N is still regular along V.

By ”devissage” thanks to (17), we may then assume that

M = E_{X}/(E_{X}D_{x_{1}} + E_{X}D_{x_{2}} + ... + E_{X}D_{x_{d}}) = E_{X⟶Y}.

Hence from now on we will assume M = E_{X⟶Y}. Of course, in that case, M ≃ E_{X} ⊗_{π^{−1}D_{X}} π^{−1}D_{X⟶Y}.

7.3 End of the proof

Having (16) in mind, we are bound to prove the analogue of (4) with T μhom(F, O_{X}) replaced by T νhom(F, O_{X}). We have

RHom_{τ^{−1}D_{X}} (τ^{−1}M, T νhom(F, O_{X}))

≃ i^{−1}RHom_{DX̃} (D_{X̃^{ℂ}→_{h}Y} , T hom(\overline{p}_{2}^{−1}F ⊗ ℂ_{Ω}, OX̃^{ℂ}))

where h = f ∘ \overline{p}_{1} is a smooth morphism since f and \overline{p}_{1} are smooth. By Proposition 6.6.2 of [8], we have an inclusion

SS( i^{−1}RHom_{Dℂ ; X̃} (D_{X̃^{ℂ}→_{h}Y} , T hom(\overline{p}_{2}^{−1}F ⊗ ℂ_{Ω}, OX̃^{ℂ}))) (18)

⊂ i^{♯}(SS(RHom_{DX̃^{ℂ}}(D_{X̃^{ℂ}→_{h}Y} , T hom(\overline{p}_{2}^{−1}F ⊗ ℂ_{Ω}, OX̃^{ℂ})))).

Therefore it is enough to consider the case of the partial De Rham system

D_{X̃^{ℂ}→_{h}Y}.

Let us take a local coordinate system (x) = (x_{1}, x_{2}) on X such that f: X ⟶ Y is by (x_{1}, x_{2}) ↦ x_{1}. Then V = X ×_{Y} T^{∗}Y is given by V = {(x_{1}, x_{2}; ξ_{1}, ξ_{2}); ξ_{2} = 0}. Endow X × X with the system of local coordinates (x, x′), so that Δ ⊂ X × X is defined by x = x′. Under the change of coordinates : x, y = x − x′, Δ will be defined by y = 0. Using (x, y), X̃^{ℂ} is endowed with the coordinates (t, x, y), and

(19)  p(t, x, y) = (x, x − ty), \overline{p}_{1}(t, x, y) = x, and \overline{p}_{2}(t, x, y) = x − ty

Let (t, x, y; τ, ξ, η) be the associated coordinates of T^{∗}(X̃^{ℂ}).

Let Ṽ be the submanifold \overline{p}_{1}_{d}(X̃^{ℂ} ×_{Y} T^{∗}Y) of T^{∗}X̃^{ℂ}, which is explicitly

given by

Ṽ = {(t, x, y; τ, ξ, η); τ = 0, η = 0, (x; ξ) ∈ V }.

By Theorem 2.1 we have the following estimate: (20)

SS(RHom_{DX̃^{ℂ}}(D_{X̃^{ℂ}⟶_{h}Y} , T hom(\overline{p}_{2}^{−1}F ⊗ℂ_{Ω}, OX̃^{ℂ}))) ⊂ Ṽ ˆ+SS( \overline{p}_{2}^{−1}F ⊗ ℂ_{Ω})^{a}.

By (18) and (20) we get (21)

SS(RHom_{τ^{−1}D_{X}} (τ^{−1}M, T νhom(F, O_{X}))) ⊂ i^{♯}(Ṽ ˆ+SS(\overline{p}_{2}^{−1}F ⊗ ℂ_{Ω})^{a}).

Therefore it is enough to prove the inclusion

i^{♯}(Ṽ ˆ+SS( \overline{p}_{2}^{−1}F ⊗ ℂ_{Ω})^{a}) ⊂ C(V, SS(F)).

More precisely, since SS(\overline{p}_{2}^{−1}F ⊗ ℂ_{Ω}) ⊂ SS(\overline{p}_{2}^{−1}F)ˆ+SS(ℂ_{Ω}) we shall prove the inclusion

i^{♯}(Ṽ ˆ+(SS( \overline{p}_{2}^{−1}F)^{a} ˆ+SS(ℂ_{Ω})^{a})) ⊂ C(V, SS(F)).

We have

(22) SS(ℂ_{Ω}) = {(t, x, y; τ, ξ, η); ξ = 0, η = 0, Im t = 0, Re t ≥ 0, Re τ = 0}
∪ {(t, x, y; τ, ξ, η); ξ = 0, η = 0, t = 0, Re τ ≤ 0},

and since \overline{p}_{2} is smooth,

SS(\overline{p}_{2}^{−1}F) = \overline{p}_{2,d}\overline{p}_{2,π}^{−1}(SS(F)) (23)
= {(t, x, y; τ, ξ, η); (x − ty; ξ) ∈ SS(F), η = −tξ, τ = −⟨ξ, y⟩}.

Hence SS(\overline{p}_{2}^{−1}F)^{a} ∩ SS(ℂ_{Ω}) ⊂ T_{X}^{∗}X, which implies

SS( \overline{p}_{2}^{−1}F)^{a} ˆ+SS(ℂ_{Ω})^{a} = SS( \overline{p}_{2}^{−1}F)^{a}+SS(ℂ_{Ω})^{a}.

Remark that the identification T^{∗}(T X) with T (T^{∗}X) is described by

T^{∗}(T X) ∋ (x, y; ξ, η) ⟷ (x, η; y, ξ) ∈ T (T^{∗}X).

Let (x_{0}, y_{0}; ξ_{0}, η_{0}) ∈ T^{∗}(T X) and assume that

(x_{0}, y_{0}; ξ_{0}, η_{0}) ∈ i^{♯}(Ṽ ˆ+(SS( \overline{p}_{2}^{−1}F)^{a}+SS(ℂ_{Ω})^{a})).

Then there exist sequences \{(t_{n}, x_{n}, y_{n}; 0, ξ_{n}, 0)\}_{n} in Ṽ, \{(t′_{n}, x′_{n}, y′_{n}; τ′_{n}, ξ′_{n}, η′_{n})\}_{n} in SS(\overline{p}_{2}^{−1}F)^{a} and {(t′′_{n}, x′′_{n}, y′′_{n}; τ′′_{n}, 0, 0)}_{n} in SS(ℂ_{Ω})^{a} such that

(i) t_{n} ⟶_{n} 0, t′_{n} ⟶_{n} 0, t′′_{n} ⟶_{n} 0, t′_{n}, t′′_{n} ≥ 0.

(ii) x_{n} ⟶_{n} x_{0}, x′_{n} ⟶_{n} x_{0}, x′′_{n} ⟶_{n} x_{0}.

(iii) y_{n} ⟶_{n} y_{0}, y′_{n} ⟶_{n} y_{0}, y′′_{n} ⟶_{n} y_{0}.

(iv) τ′_{n} + τ′′_{n} ⟶_{n} 0.

(v) ξ_{n} + ξ′_{n} ⟶_{n} ξ_{0}.

(vi) η′_{n} ⟶_{n} η_{0} (hence t′_{n}ξ′_{n} ⟶_{n} −η_{0}, and t′_{n}ξ_{n} ⟶_{n} η_{0} by (v)).

By (vi), there exists a sequence of positive numbers (a_{n})_{n} such that a_{n} ⟶_{n} 0 and a_{n}ξ_{n} ⟶_{n} η_{0}. Consider the sequence {(x′_{n} − t′_{n}y′_{n}; −a_{n}ξ′_{n})}_{n} in SS(F) and {(x′_{n} + (a_{n} − t′_{n})y′_{n}; a_{n}ξ_{n})}_{n} in V. Then

(x′_{n} − t′_{n}y′_{n}; −a_{n}ξ′_{n}) ⟶_{n} (x_{0}; η_{0}), {(x′_{n} + (a_{n} − t′_{n})y′_{n}; a_{n}ξ_{n}) ⟶_{n} (x_{0}; η_{0})

and

a_{n}^{−1}((x′_{n}+(a_{n}−t′_{n})y′_{n}, a_{n}ξ_{n})−(x′_{n}−t′_{n}y′_{n}, −a_{n}ξ′_{n})) = (y′_{n}, ξ_{n}+ξ′_{n}) ⟶_{n} (y_{0}, ξ_{0}).

Hence one has (x_{0}, η_{0}; y_{0}, ξ_{0}) ∈ C(V, SS(F)).

Since C(V, SS(F)) = ρ_{V} ^{−1}(C_{V} (SS(F))^{a}), we finally obtain SS( RHom_{π^{−1}D_{X}} (π^{−1}M, T μhom(F, O_{X}))) ⊂ ρ_{V} ^{−1}(C_{V} (SS(F))^{a})

as asserted.

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Teresa Monteiro Fernandes Centro de Álgebra da Universidade de Lisboa, Complexo 2, 2 Avenida Prof.Gama Pinto, 1699 Lisboa codex Portugal tmf@ptmat.lmc.fc.ul.pt

Masaki Kashiwara Research Institute for Mathematical Sciences, Kyoto University Kyoto 606-8502 Japan masaki@kurims.kyoto-u.ac.jp

Pierre Schapira Université Pierre et Marie Curie, case 82 Analyse Algébrique, UMR7586 4, place Jussieu,75252 Paris cedex 05 France schapira@math.jussieu.fr http://www.math.jussieu.fr/˜ schapira/