Microlocal study of ind-sheaves I: micro-support and regularity

Masaki Kashiwara and Pierre Schapira June 2001

Abstract

We define the notions of micro-support and regularity for indsheaves, and prove their invariance by contact transformations. We apply the results to the ind-sheaves of temperate holomorphic solutions of D-modules. We prove that the micro-support of such an ind-sheaf is the characteristic variety of the corresponding D-module and that the ind-sheaf is regular if the D-module is regular holonomic.

Contents

1 Introduction 2 2 Notations and review 3 3 Complements of homological algebra 10 4 Micro-support and regularity 12 5 Invariance by contact transformations 20 6 Ind-sheaves and D-modules 22 7 An example 24

Mathematics Subject Classification: 35A27, 32C38 The first named author benefits by 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”.

1 Introduction

Recall that a system of linear partial differential equations 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 ℝ-constructible 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

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 [4] 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 analytic 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 growth conditions. So far this difficulty was overcome in two cases, the temperate case including Schwartz’s distributions and meromorphic functions with poles on Z and the dual case including C^{∞}- functions and the formal completion of O_{X} along Z. The method was two construct specific functors, the functor T Hom of [2] and the functor ⊗^{w} of [5].

There is a more radical method, which consists in replacing the too narrow framework of sheaves by that of ind-sheaves, as explained in [6]. For example, the presheaf of holomorphic temperate functions on a complex manifold X (which, to a subanalytic open subset of X, associates the space of holomorphic functions with temperate growth at the boundary) is clearly not a sheaf. However it makes sense as an object (denoted by O_{X}^{t}) of the derived category of ind-sheaves on X. Then it is natural to ask if the microlocal theory of sheaves, in particular the theory of micro-support, applies in this general setting.

In this paper we give the definition and the elementary properties of the micro-support of ind-sheaves as well as the notion of regularity.

We prove in particular that the micro-support SS(·) and the regular micro-support SS_{reg}(·) of ind-sheaves behave naturally with respect to distinguished triangles and that these micro-supports are invariant by “quantized contact transformations” (in the framework of sheaf theory, as explained in [4]).

When X is a complex manifold and M is a coherent D_{X}-module, we study the ind-sheaf Sol^{t}(M) := RHom_{D_{X}} (M, O_{X}^{t}). We prove that

(i) SS(Sol^{t}(M)) = char(M),

(ii) if M is holonomic, Sol^{t}(M) is regular if M is regular holonomic.

Finally, we treat an example: we calculate the ind-sheaf of the temperate holomorphic solutions of an irregular differential equation.

This paper is the first one of a series. In Part II, we shall introduce the microlocalization functor for ind-sheaves, and in Part III we shall study the functorial behavior of micro-supports.

2 Notations and review

We will mainly follow the notations in [4] and [6].

Geometry. In this paper, all manifolds will be real analytic (sometimes, complex analytic). Let X be a manifold. One denotes by τ: T X ⟶ X the tangent bundle to X and by π: T^{∗}X ⟶ X the cotangent bundle. One denotes by a: T^{∗}X ⟶ T^{∗}X the antipodal map. If S ⊂ T^{∗}X, one denotes by Ṡ the set S \ T_{X}^{∗}X, and one denotes by S^{a} the image of S by the antipodal map. In particular, Ṫ^{∗}X = T^{∗}X \ X, the set T^{∗}X with the zero-section removed. One denotes by π̇: Ṫ^{∗}X ⟶ X the projection.

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 Whitney normal cone to S along Y, a conic subset of T_{Y} X.

If S is a locally closed subset of T^{∗}X, we say that S is ℝ^{+}-conic (or simply “conic”, for short) if it is locally invariant under the action of ℝ^{+}. If S is smooth, this is equivalent to saying that the Euler vector field on T^{∗}X is tangent to S.

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

T^{∗}X ⟵_{f_{d}} X ×_{Y} T^{∗}Y ⟶_{f_{π}} T^{∗}Y (2.1)

(In [4], f_{d} is denoted by ^{t}f′.) We denote by q_{1} and q_{2} the first and second projections defined on X × Y.

Sheaves. Let k be a field. We denote by Mod(k_{X}) the abelian category of sheaves of k-vector spaces and by D^{b}(k_{X}) its bounded derived category.

We denote by ℝ-C(k_{X}) the abelian category of ℝ-constructible sheaves of k-vector spaces on X, and by D_{ℝ−c}^{b}(k_{X}) (resp. D_{w−ℝ−c}^{b}(k_{X})) the full triangulated subcategory of D^{b}(k_{X}) consisting of objects with ℝ-constructible (resp. weakly ℝ-constructible) cohomology. On a complex manifold, one defines similarly the categories D_{ℂ−c}^{b}(k_{X}) and D_{w−ℂ−c}^{b}(k_{X}) of ℂ-constructible and weakly ℂ-constructible sheaves.

If Z is a locally closed subset of X and if F is a sheaf on X, recall that F_{Z} is a sheaf on X such that F_{Z}|_{Z} ≃ F |_{Z} and F_{Z}|_{X\Z} ≃ 0. One writes k_{XZ} instead of (k_{X})_{Z} and one sometimes writes k_{Z} instead of k_{XZ}.

If f: X ⟶ Y is a morphism of manifolds, one denotes by ω_{X/Y} the relative dualizing complex on X and if Y = {pt} one simply denotes it by ω_{X}. Recall that

ω_{X} ≃ or_{X}[dim_{ℝ} X]

where or_{X} is the orientation sheaf and dim_{R} X is the dimension of X as a real manifold. We denote by D′_{X} and D_{X} the duality functors on D^{b}(k_{X}), defined by

D′_{X}(F) = RHom (F, k_{X}), D_{X}(F) = RHom (F, ω_{X}).

If F is an object of D^{b}(k_{X}), SS(F) denotes its micro-support, a closed conic involutive subset of T^{∗}X. For an open subset U of T^{∗}X, one denotes by D^{b}(k_{X}; U) the localization of the category D^{b}(k_{X}) with respect to the triangulated subcategory consisting of sheaves F such that SS(F) ∩ U = ∅.

We shall also use the functor μhom as well as the operation ˆ+ and refer to loc. cit. for details.

O and D On a complex manifold X we consider the structural sheaf O_{X} of holomorphic functions and the sheaf D_{X} of linear holomorphic differential operators of finite order.

We denote by Mod_{coh}(D_{X}) the abelian category of coherent D_{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}) (resp. D_{hol}^{b}(D_{X})), D_{rh}^{b}(D_{X})) its full triangulated category consisting of objects with coherent cohomologies (resp. holonomic cohomologies, regular holonomic cohomologies).

Categories. In this paper, we shall work in a given universe U, and a category means a U-category. If C is a category, C^{∧} denotes the category of functors from C^{op} to Set. The category C^{∧} admits inductive limits, however, in case C also admits inductive limits, the Yoneda functor h^{∧}: C ⟶ C^{∧} does not commute with such limits. Hence, one denotes by \underrightarrow{lim} the inductive limit

in C and by “\underrightarrow{lim}” the inductive limit in C^{∧}.

One denotes by Ind(C) the category of ind-objects of C, that is the full subcategory of C^{∧} consisting of objects F such that there exist a small filtrant category I and a functor α: I ⟶ C, with

F ≃ “\underrightarrow{lim}” α, i.e., F ≃ “\underrightarrow{lim}”_{i∈I} F_{i}, with F_{i} ∈ C.

The category C is considered as a full subcategory of Ind(C).

If φ: C ⟶ C′ is a functor, it defines a functor Iφ: Ind(C) ⟶ Ind(C′) which commutes with “\underrightarrow{lim}”.

If C is an additive category, we denote by C(C) the category of complexes in C and by K(C) the associated homotopy category. If C is abelian, one denotes by D(C) its derived category. One defines as usual the full subcategories C^{∗}(C), K^{∗}(C), D^{∗}(C), with ∗ = +, −, b. One denotes by Q the localization functor:

Q: K^{∗}(C) ⟶ D^{∗}(C).

We keep the same notation Q to denote the composition C^{∗}(C) ⟶ K^{∗}(C) ⟶ D^{∗}(C).

One denotes by C^{[a,b]}(C) the full subcategory of C(C) consisting of objects F^{•} satisfying F^{i} = 0 for i ∉ [a, b]. If a, b ∈ ℤ with a ≤ b, there is a natural isomorphism

Ind(C^{[a,b]}(C)) ⟶^{∼} C^{[a,b]}(Ind(C)).

Ind-sheaves. Here, X is a Hausdorff locally compact space with a countable base of open sets and k is a field. One denotes by I(k_{X}) the abelian category of ind-sheaves of k-vector spaces on X, that is, I(k_{X}) = Ind(Mod^{c}(k_{X})), the category of ind-objects of the category Mod^{c}(k_{X}) of sheaves with compact support on X. We denote by D^{b}(I(k_{X})) the bounded derived category of I(k_{X}).

There is a natural fully faithful exact functor

ι_{X}: Mod(k_{X}) ⟶ I(k_{X}),

F ↦ “\underrightarrow{lim}”_{U⊂⊂X} F_{U} (U open).

Most of the time, we shall not write this functor and identify Mod(k_{X}) with a full abelian subcategory of I(k_{X}) and D^{b}(k_{X}) with a full triangulated subcategory of D^{b}(I(k_{X})).

The category I(k_{X}) admits an internal hom denoted by Ihom and this functor admits a left adjoint, denoted by ⊗. If F ≃ “\underrightarrow{lim}”_{i} F_{i} and G ≃

“\underrightarrow{lim}”_{j} G_{j}, then

Ihom (G, F) ≃ \underleftarrow{lim}_{j} “\underrightarrow{lim}”_{i} Hom (G_{j}, F_{i})
G ⊗ F ≃ “\underrightarrow{lim}”_{i} “\underrightarrow{lim}”_{j}(G_{j} ⊗ F_{i}).

The functor ι_{X} admits a left adjoint

α_{X}: I(k_{X}) ⟶ Mod(k_{X}),

To F = “\underrightarrow{lim}”_{i∈I} F_{i}, this functor associates α_{X}(F) = \underrightarrow{lim}_{i∈I} F_{i}. This functor also

admits a left adjoint

β_{X}: Mod(k_{X}) ⟶ I(k_{X}),

and both functors α_{X} and β_{X} are exact. The functor β_{X} is not so easy to describe. For example, for an open subset U and a closed subset Z, one has;

β_{X}(k_{XU}) ≃ “\underrightarrow{lim}”_{V ⊂⊂U} k_{XV} (V open),

β_{X}(k_{XZ}) ≃ “\underrightarrow{lim}”_{Z⊂V} k_{X\overline{V}} (V open).

One sets

Hom (G, F) = α_{X}Ihom (G, F) ∈ Mod(k_{X}).

One has

Hom_{I(k_{X})}(G, F) = Γ(X; Hom (G, F)).

The functors Ihom and Hom are left exact and admit right derived functors RIhom and RHom.

Let f: X ⟶ Y be a morphism of topological spaces (Y satisfies the same assumptions as X). There are natural functors

f^{−1} : I(k_{Y})) ⟶ I(k_{X})
f_{∗}: I(k_{X}) ⟶ I(k_{Y})
f_{!!} : I(k_{X}) ⟶ I(k_{Y}).

The proper direct image functor is denoted by f_{!!} instead of f_{!} because it does not commute with ι, that is ι_{Y} f_{!} ≠ f_{!!}ι_{X} in general..

These functors induce derived functors, and moreover the functor Rf_{!!} admits a right adjoint denoted by f^{!}:

f^{−1} : D^{b}I(k_{Y})) ⟶ D^{b}(I(k_{X})),
Rf_{∗}: D^{b}(I(k_{X})) ⟶ D^{b}(I(k_{Y})),
Rf_{!!} : D^{b}(I(k_{X})) ⟶ D^{b}(I(k_{Y})),
f^{!} : D^{b}(I(k_{Y})) ⟶ D^{b}(I(k_{X})).

Let a_{X}: X ⟶ {pt} denote the canonical map. We also introduce a notation. We set

IΓ(X; ·) = a_{X}_{∗}(·),
RIΓ(X; ·) = Ra_{X}_{∗}(·).

Ind-sheaves on real manifolds. Let X be a real analytic manifold. Among all ind-sheaves, there are those which are ind-objects of the category of ℝ-constructible sheaves, and we shall encounter them in our applications.

We denote by ℝ-C^{c}(k_{X}) the full abelian subcategory of ℝ-C(k_{X}) consisting of ℝ-constructible sheaves with compact support. We set

Iℝ−c(k_{X}) = Ind(ℝ-C^{c}(k_{X}))

and denote by D_{I ℝ−c}^{b}(I(k_{X})) the full subcategory of D^{b}(I(k_{X})) consisting of objects with cohomology in Iℝ−c(k_{X}). (Note that in [6], Iℝ−c(k_{X}) was denoted by I_{ℝ−c}(k_{X}).)

Theorem 2.1. The natural functor D^{b}(Iℝ−c(k_{X})) ⟶ D_{I ℝ−c}^{b}(I(k_{X})) is an equivalence.

There is an alternative construction of Iℝ−c(k_{X}), using Grothendieck topologies. Denote by Op_{X} the category of open subsets of X (the morphisms U ⟶ V are the inclusions), and by by Op_{X_{sa}} its full subcategory consisting of open subanalytic subsets of X. One endows this category with a Grothendieck topology by deciding that a family {U_{i}}_{i} in Op_{X_{sa}} is a covering of U ∈ Op_{X_{sa}} if for any compact subset K of X, there exists a finite subfamily which covers U ∩ K. In other words, we consider families which are locally finite in X. One denotes by X_{sa} the site defined by this topology.

Sheaves on X_{sa} are easy to construct. Indeed, consider a presheaf F of k-vector spaces defined on the subcategory Op_{X_{sa}}^{c} of relatively compact open subanalytic subsets of X and assume that the sequence

0 ⟶ F (U ∪ V) ⟶ F (U) ⊕ F (V) ⟶ F (U ∩ V)

is exact for any U and V in Op_{X_{sa}}^{c}. Then there exists a unique sheaf F̃ on X_{sa} such that F̃(U) ≃ F (U) for all U ∈ Op_{X_{sa}}^{c}. Sheaves on X_{sa} define naturally ind-sheaves on X. Indeed:

Theorem 2.2. There is a natural equivalence of abelian categories

Iℝ−c(k_{X}) ⟶^{∼} Mod(k_{X_{sa}}),

given by

Iℝ−c(k_{X}) ∋ F ↦ (Op_{X_{sa}}^{c} ∋ U ↦ Hom_{Iℝ−c(k_{X})}(k_{U}, F)).

As usual, we denote by C_{X}^{∞} the sheaf of complex-valued functions of class C^{∞}, by Db_{X} (resp. B_{X}) the sheaf of Schwartz’s distributions (resp. Sato’s hyperfunctions), and by D_{X} the sheaf of analytic finite-order differential operators.

Let U be an open subset of X. One sets C_{X}^{∞}(U) = Γ(U; C_{X}^{∞}).

Definition 2.3. Let f ∈ C_{X}^{∞}(U). One says that f has polynomial growth at p ∈ X if it satisfies the following condition. For a local coordinate system (x_{1}, …, x_{n}) around p, there exist a sufficiently small compact neighborhood K of p and a positive integer N such that

sup_{x∈K∩U} (dist(x, K \ U))^{N}|f(x)| < ∞ . (2.2)

It is obvious that f has polynomial growth at any point of U. We say that f is tempered at p if all its derivatives have polynomial growth at p. We say that f is tempered if it is tempered at any point.

For an open subanalytic set U in X, denote by C_{X}^{∞,t}(U) the subspace of C_{X}^{∞}(U) consisting of tempered functions. Denote by Db_{X}^{t}(U) the space of tempered distributions on U, defined by the exact sequence

0 ⟶ Γ_{X\U}(X; Db_{X}) ⟶ Γ(X; Db_{X}) ⟶ Db_{X}^{t}(U) ⟶ 0.

It follows from the results of Lojasiewicz [8] that U ↦ C_{X}^{∞}(U) and U ↦ Db_{X}^{t}(U) are sheaves on the subanalytic site X_{sa}, hence define ind-sheaves.

Definition 2.4. We call C_{X}^{∞,t} (resp. Db_{X}^{t}) the ind-sheaf of tempered C^{∞}- functions (resp. tempered distributions).

One can also define the ind-sheaf of Whitney C^{∞}-functions, but we shall not recall here its construction. These ind-sheaves are well-defined in the category Mod(β_{X}D_{X}). Roughly speaking, it means that if P is a differential operator defined on the closure Ū of an open subset U, then it acts on C_{X}^{∞,t}(U) and Db_{X}^{t}(U).

Let now X be a complex manifold. We denote by \overline{X} the complex conjugate manifold and by X^{ℝ} the underlying real analytic manifold, identified with the diagonal of X × \overline{X}. We denote by D_{X} the sheaf of rings of finiteorder holomorphic differential operators, not to be confused with D_{X^{ℝ}}. We set

O_{X}^{t} := RIhom_{βD\overline{X}} (βO\overline{X}, Db_{X^{ℝ}}^{t})

One can prove that the natural morphism

RIhom_{βD\overline{X}} (βO\overline{X}, C_{X^{ℝ}}^{∞,t}) ⟶ RIhom_{βD\overline{X}} (βO\overline{X}, Db_{X^{ℝ}}^{t})

is an isomorphism. One calls O_{X}^{t} the ind-sheaf of tempered holomorphic functions. One shall be aware that in fact, O_{X}^{t} is not an ind-sheaf but an object of the derived category D^{b}(I(ℂ_{X})), or better, of D^{b}(β_{X}D_{X}). It is not concentrated in degree 0 as soon as dim X > 1.

Let G ∈ D_{ℝ−c}^{b}(ℂ_{X}). It follows from the construction of O_{X}^{t} that:

RHom (G, O_{X}^{t}) ≃ T Hom (G, O_{X}),

where T Hom (·, O_{X}) denotes the functor of temperate cohomology of [2] (see also [5] for a detailed construction and [1] for its microlocalization).

3 Complements of homological algebra

The results of this section are extracted from [7]. Let C denote a small abelian category. We shall study some links between the derived category D^{b}(Ind(C)) and the category Ind(D^{b}(C)).

We define the functor J: D^{b}(Ind(C)) ⟶ (D^{b}(C))^{∧} by setting for F ∈ D^{b}(Ind(C)) and G ∈ D^{b}(C)

(3.1)  J(F)(G) = Hom_{D^{b}(Ind(C))}(G, F).

Theorem 3.1. (i) The functor J takes its values in Ind(D^{b}(C)). (ii) Consider a small and filtrant category I, integers a ≤ b and a functor

I ⟶ C^{[a,b]}(C), i ↦ F_{i}. If F ∈ D^{b}(Ind(C)), F ≃ Q(“\underrightarrow{lim}”_{i} F_{i}) and

G ∈ D^{b}(C), then:

(a) J(F) ≃ “\underrightarrow{lim}”_{i} Q(F_{i}),

(b) Hom_{D^{b}(Ind(C))}(G, F) ≃ \underrightarrow{lim}_{i} Hom_{D^{b}(C)}(G, F_{i}).

(iii) For each k ∈ ℤ, the diagram below commutes.

\xymatrix{D^{b}(Ind(C)) \ar[rr]_{J} \ar[dr]_{H^{k}} &  & Ind(D^{b}(C)) \ar[dl]^{IH^{k}} \\  & Ind(C)}

Lemma 3.2. Assume that C has finite homological dimension. Let φ: X ⟶ Y be a morphism in Ind(D^{b}(C)) and assume that φ induces an isomorphism IH^{k}(φ): IH^{k}(X) ⟶^{∼} IH^{k}(Y) for every k ∈ ℤ. Then φ is an isomorphism.

Theorem 3.3. Let ψ: D^{b}(Ind(C)) ⟶ D^{b}(Ind(C′)) be a triangulated functor which satisfies: if F ∈ D^{b}(Ind(C)), F ≃ Q(“\underrightarrow{lim}”_{i} F_{i}) with F_{i} ∈ C^{[a,b]}(C), then

H^{k}ψ(F) ≃ “\underrightarrow{lim}”_{i} H^{k}ψ(Q(F_{i})). Assume moreover that the homological di-

mension of C′ is finite. Then there exists a unique functor Jψ: Ind(D^{b}(C)) ⟶ Ind(D^{b}(C′)) which commutes with “\underrightarrow{lim}” and such that the diagram below

commutes:

\xymatrix{D^{b}(Ind(C)) \ar[r]_{ψ} \ar[d]_{J} & D^{b}(Ind(C′)) \ar[d]_{J} \\ Ind(D^{b}(C)) \ar[r]^{Jψ} & Ind(D^{b}(C′)) .}

Remark 3.4. The functor J: D^{b}(Ind(C)) ⟶ Ind(D^{b}(C)) is neither full nor faithful. Indeed, let C = Mod^{c}(k_{X}) and let F ∈ Mod(k_{X}) considered as a full subcategory of I(k_{X}). Then

Hom_{D^{b}(I(k_{X}))}(k_{X}, F [n]) ≃ H^{n}(X; F).

On the other hand,

Hom_{Ind(D^{b}(Mod^{c}(k_{X})))}(J(k_{X}), J(F [n])) ≃ \underleftarrow{lim}_{U⊂⊂X} H^{n}(U; F).

Let T be a full triangulated subcategory D^{b}(C). One identifies Ind(T ) with a full subcategory of Ind(D^{b}(C)).

Let F ∈ D^{b}(Ind(C)). Let us denote by T_{F} the category of arrows G ⟶ F in D^{b}(Ind(C)) with G ∈ T . The category T_{F} is filtrant.

Lemma 3.5. For F ∈ D^{b}(Ind(C)), the conditions below are equivalent.

(i) J(F) ∈ Ind(T ), (ii) for each k ∈ ℤ, one has H^{k}(F) ≃ “\underrightarrow{lim}”_{G⟶F ∈T_{F}} H^{k}(G).

Definition 3.6. Let T be a full triangulated subcategory of D^{b}(C). One denotes by J^{−1}Ind(T ) the full subcategory of D^{b}(Ind(C)) consisting of objects F ∈ D^{b}(Ind(C)) such that J(F) ∈ Ind(T ).

Proposition 3.7. The category J^{−1}Ind(T ) is a triangulated subcategory of D^{b}(Ind(C)).

We will apply these results to the category I(k_{X}) = Ind(Mod^{c}(k_{X})). Hence J is the functor:

J: D^{b}(I(k_{X}))⟶Ind(D^{b}(Mod^{c}(k_{X}))).

By the definition one has

J(F) ≃ “\underrightarrow{lim}”_{U⊂⊂X} J(F_{U}) for any F ∈ D^{b}(I(k_{X})).

As a corollary of Theorem 3.3, one gets:

Proposition 3.8. For G ∈ D^{b}(k_{X}) and F ∈ D^{b}(I(k_{X})), assume that J(F) ≃ “\underrightarrow{lim}”_{i} J(F_{i}) with F_{i} ∈ D^{b}(k_{X}). Then there are natural isomorphisms:

(3.2)  J(G ⊗ F) ≃ “\underrightarrow{lim}”_{i} J(G ⊗ F_{i}),
(3.3)  J(RIhom (G, F)) ≃ “\underrightarrow{lim}”_{i} J(RIhom (G, F_{i})).

4 Micro-support and regularity

Let γ be a closed convex proper cone in an affine space X. One denotes by γ^{∘} its polar cone,

γ^{∘} = {ξ ∈ X^{∗}; ⟨x, ξ⟩ ≥ 0 for all x ∈ γ}.

Let W ⊂ X be an open subset. We introduce the functor Φ_{γ,W} : D^{b}(I(k_{X})) ⟶ D^{b}(I(k_{X})) as follows. Denote by q_{1}, q_{2} : X × X ⟶ X the first and second projections and denote by s: X × X ⟶ X the map (x, y) ↦ x − y. One sets

Φ_{γ,W} (F) = Rq_{1}_{!!}(k_{s^{−1}γ∩q_{1}^{−1}W ∩q_{2}^{−1}W} ⊗ q_{2}^{−1}F).

One writes Φ_{γ} instead of Φ_{γ,X}. Define the functor Φ_{γ,W}^{−} by replacing the kernel k_{s^{−1}γ∩q_{1}^{−1}W ∩q_{2}^{−1}W} with the complex k_{s^{−1}γ∩q_{1}^{−1}W ∩q_{2}^{−1}W} ⟶ k_{s^{−1}(0)} in which k_{s^{−1}(0)} is situated in degree 0. We have a distinguished triangle in D^{b}(I(k_{X}))

Φ_{γ,W} (F) ⟶ F ⟶ Φ_{γ,W}^{−} (F) ⟶^{+1} .

Note that if F ∈ D^{b}(k_{X}), then

supp(Φ_{γ,W} (F)) ⊂ \overline{W} ,
Φ_{γ}(F) ⟶ F is an isomorphism on X × Intγ^{∘},
SS(Φ_{γ}(F)) ⊂ X × γ^{∘}.
SS(Φ_{γ,W}^{−} (F)) ⋂ W × Intγ^{∘} = ∅

Lemma 4.1. Let F ∈ D^{b}(I(k_{X})) and let p ∈ T^{∗}X. The conditions (1a)–(4b) below are all equivalent. Moreover, if F ∈ D_{I ℝ−c}^{b}(I(k_{X})), these conditions are equivalent to (5a).

(1a) Assume that for a small and filtrant category I, integers a ≤ b and a functor I ⟶ C^{[a,b]}(Mod(k_{X})), i ↦ F_{i} one has F ≃ Q(“\underrightarrow{lim}”_{i∈I} F_{i}).

Then there exists a conic open neighborhood U of p in T^{∗}X such that for any i ∈ I there exists a morphism i ⟶ j in I which induces the zero-morphism 0 : F_{i} ⟶ F_{j} in D^{b}(k_{X}; U).

(1b) There exist a conic open neighborhood U of p in T^{∗}X, a small and filtrant category I, integers a ≤ b and a functor I ⟶ C^{[a,b]}(Mod(k_{X})), i ↦ F_{i}, such that SS(F_{i}) ∩ U = ∅ and F ≃ Q(“\underrightarrow{lim}”_{i} F_{i}) in a neighbor-

hood of π(p).

(2a) Assume that for a small and filtrant category I, integers a ≤ b and a functor I ⟶ D^{[a,b]}(k_{X}), i ↦ F_{i} one has J(F) ≃ “\underrightarrow{lim}”_{i∈I} J(F_{i}). Then

there exists a conic open neighborhood U of p in T^{∗}X such that for any i ∈ I there exists a morphism i ⟶ j in I which induces the zeromorphism 0 : F_{i} ⟶ F_{j} in D^{b}(k_{X}; U).

(2b) There exist a conic open neighborhood U of p in T^{∗}X, a small and filtrant category I, integers a ≤ b, a functor I ⟶ D^{b}(k_{X}), i ↦ F_{i} and F′ isomorphic to F in neighborhood of π(p) such that SS(F_{i}) ∩ U = ∅

and J(F′) ≃ “\underrightarrow{lim}”_{i} J(F_{i}).

(3a) There exists a conic open neighborhood U of p in T^{∗}X such that for any G ∈ D^{b}(k_{X}) with supp(G) ⊂⊂ π(U), SS(G) ⊂ U ∪ T_{X}^{∗}X, one has Hom_{D^{b}(I(k_{X}))}(G, F) = 0.

(3b) There exists a conic open neighborhood U of p in T^{∗}X such that for any G ∈ D^{b}(k_{X}) with supp(G) ⊂⊂ π(U), SS(G) ⊂ U^{a} ∪ T_{X}^{∗}X, one has RIΓ(X; G ⊗ F) = 0.

Assume now that X is an affine space and let p = (x_{0}; ξ_{0}).

(4a) There exist a relatively compact open neighborhood W of x_{0} and a

closed convex proper cone γ with ξ_{0} ∈ Intγ^{∘} such that Φ_{γ,W} (F) ≃ 0. (4b) There exist F′ ∈ D^{b}(I(k_{X})) with F′ ≃ F in a neighborhood of x_{0} and

F′ has compact support, and a closed convex proper cone γ as in (4a) such that Φ_{γ}(F′) ≃ 0 in a neighborhood of x_{0}.

(5a) Same condition as (3a) with G ∈ D_{ℝ−c}^{b}(k_{X}).

Proof. The plan of the proof is as follows:

[figure: (2a); (3a); (2b); (1a); (5a); (1b); (3b); (4a); (4b)]

(2a) ⇒ (1a) follows from F ≃ Q(“\underrightarrow{lim}”_{i} F_{i}) ⇒ J(F) ≃ “\underrightarrow{lim}”_{i} J(Q(F_{i})).

(1a) ⇒ (3b). Let F ≃ Q(“\underrightarrow{lim}”_{i} F_{i}) and let i ∈ I. There exists i ⟶ j such

that the morphism F_{i} ⟶ F_{j} in D^{b}(k_{X}) is zero in D^{b}(k_{X}; U). Hence, there exists a morphism F_{j} ⟶ F′_{ij} in D^{b}(k_{X}) which is an isomorphism on U and such that the composition F_{i} ⟶ F_{j} ⟶ F′_{ij} is the zero-morphism in D^{b}(k_{X}). Consider the commutative diagram in which the row on the bottom is a distinguished triangle in D^{b}(k_{X}) and SS(F_{ij}) ∩ U = ∅:

[figure: F_{i}; 0; F_{ij}; F_{j}; F′_{ij}; +1]

Since the arrow F_{i} ⟶ F′_{ij} is zero, the dotted arrow may be completed, making the diagram commutative. Hence, we may assume from the beginning that for any i ∈ I there exists i ⟶ j such that the morphism F_{i} ⟶ F_{j} factorizes as F_{i} ⟶ F_{ij} ⟶ F_{j} with SS(F_{ij}) ∩ U = ∅.

We may assume X is affine and U = W × λ where W is open and relatively compact and λ is an open convex cone. Then SS(G⊗F_{ij})∩U = ∅, and the sheaf G ⊗ F_{ij} has compact support. Hence, RΓ(X; G ⊗ F_{ij}) ≃ 0 which implies H^{j}RIΓ(X; G ⊗ F) ≃ “\underrightarrow{lim}”_{i} H^{j}RΓ(X; G ⊗ F_{i}) ≃ 0 for all j.

We conclude therefore RIΓ(X; G ⊗ F) ≃ 0.

(3b) ⇒ (4a). Let F = Q(“\underrightarrow{lim}”_{i} F_{i}), with F_{i} ∈ C^{[a,b]}(Mod(k_{X})). Set

H_{ε} = {x; ⟨x − x_{0}; ξ_{0}⟩ > −ε}

and let K ⊂⊂ π(U) be a compact neighborhood of x_{0}. Then there exist an open convex cone γ and an open neighborhood W of x_{0} satisfying the following conditions:

\{ \begin{matrix}W ⊂ H_{ε} ∩ K, \\ (x + γ) ∩ H_{ε} ⊂ W for all x ∈ W, \\ \overline{W} × γ^{∘} ⊂ U ∪ T_{X}^{∗}X. \end{matrix}

Set

G_{x} = k_{(x+γ^{a})∩H_{ε}}, G = ⨁_{x∈W} G_{x}.

Since supp(G) ⊂⊂ π(U) and SS(G) ⊂ \overline{W} × γ^{∘}^{a}, we get by the hypothesis:

“\underrightarrow{lim}”_{i} H^{k}RΓ(X; G ⊗ F_{i}) ≃ 0.

Hence,

“\underrightarrow{lim}”_{i}( ⨁_{x∈W} H^{k}RΓ(X; G_{x} ⊗ F_{i})) ≃ 0.

Hence one obtains:

{for any i ∈ I, there exists i ⟶ j such that H^{k}RΓ(X; G_{x} ⊗F_{i}) ⟶ H^{k}RΓ(X; G_{x} ⊗ F_{j}) is zero for any x ∈ W and any k ∈ ℤ.

On the other-hand,

H^{k}(Φ_{γ,W} (F_{i}))_{x} ≃ H^{k}RΓ(X; G_{x} ⊗ F_{i}).

Therefore, for any i ∈ I there exists i ⟶ j such that for any k ∈ ℤ, the morphism H^{k}(Φ_{γ,W} (F_{i})) ⟶ H^{k}(Φ_{γ,W} (F_{j})) is the zero morphism, and this implies

H^{k}(Φ_{γ,W} (F)) ≃ “\underrightarrow{lim}”_{i} H^{k}Φ_{γ,W} (F_{i}) ≃ 0.

This gives the desired result: Φ_{γ,W} (F) = 0.

(4a) ⇒ (4b) is obvious by taking F_{W} as F′.

(4b) ⇒ (1b). Let W be an open relatively compact neighborhood of x_{0} such that F |_{W} ≃ F′|_{W} and Φ_{γ}(F′)|_{W} ≃ 0.

Then one has a distinguished triangle:

Rq_{1}_{!!}(k_{s^{−1}(γ\{0})∩q_{1}^{−1}W} ⊗ q_{2}^{−1}F′) ⟶ Φ_{γ}(F′)_{W} ⟶ F′_{W} ⟶^{+1},

and hence one obtains Rq_{1}_{!!}(k_{s^{−1}(γ\{0})∩q_{1}^{−1}W} [1] ⊗ q_{2}^{−1}F′) ≃ F′_{W}. Let F′ = Q(“\underrightarrow{lim}”_{i} F_{i}) with F_{i} ∈ C^{[a,b]}(Mod(k_{X})), and take a finite injective resolution

I of k_{s^{−1}(γ\{0})∩q_{1}^{−1}W} [1]. Since I ⊗ F_{i} is a finite complex of soft sheaves, Rq_{1}_{!}(k_{s^{−1}(γ\{0})∩q_{1}^{−1}W} [1] ⊗ q_{2}^{−1}F_{i}) is represented by F′_{i} := q_{1}_{!}(I ⊗ q_{2}^{−1}F_{i}). Hence one has

Rq_{1}_{!!}(k_{s^{−1}(γ\\{0\})∩q_{1}^{−1}W} ⊗ q_{2}^{−1}F′) ≃ Q(“\underrightarrow{lim}”_{i} F′_{i}).

Since SS(F′_{i}) ∩ W × Intγ^{∘} = ∅, we obtain the desired result.

(1b) ⇒ (2b) is obvious.

(2b) ⇒ (3a). Let J(F) ≃ “\underrightarrow{lim}”_{i} J(F_{i}). If G ∈ D^{b}(k_{X}), we get the isomor-

phism:

Hom_{D^{b}(I(k_{X}))}(G, F) ≃ \underrightarrow{lim}_{i} Hom_{D^{b}(k_{X})}(G, F_{i}).

We may assume that X is affine and U = W × λ where W is open and λ is an open convex cone. Then the micro-support of RHom (G, F_{i}) is contained in SS(F_{i}) + λ̄^{a} and this set does not intersect X × λ. Since RHom (G, F_{i}) has compact support, Hom (G, F_{i}) is zero.

(3a) ⇒ (2a). We may assume that X is affine, p = (x_{0}; ξ_{0}) and U = X′ × Intγ^{∘}, with ξ_{0} ∈ Intγ^{∘} for a neighborhood X′ of x_{0}. Let V be an open neighborhood of x_{0} and let W = {x; ⟨x − x_{0}; ξ_{0}⟩ > −ε}. Then by taking V and ε small enough, the sheaf Φ_{γ}(H_{W})_{V} satisfies the condition in (3a) for any H ∈ D^{b}(k_{X}). Let J(F) = “\underrightarrow{lim}”_{i} J(F_{i}). Then \underrightarrow{lim}_{i} Hom_{D^{b}(k_{X})}(G, F_{i}) ≃ 0

for any G = Φ_{γ}(H_{W})_{V}. Let i ∈ I and choose H = F_{i}. There exists i ⟶ j such that the composition (Φ_{γ}(F_{iW}))_{V} ⟶ F_{i} ⟶ F_{j} is zero. The morphism (Φ_{γ}((F_{iW}))_{V} ⟶ F_{i} is an isomorphism on U′ := (V ∩ W) × Intγ^{∘}. Therefore, F_{i} ⟶ F_{j} is zero in D^{b}(k_{X}; U′).

(3a) ⇒ (5a) is obvious.

(5a) ⇒ (3b). (Assuming F ∈ D_{I ℝ−c}^{b}(I(k_{X})).) Let (2a-rc) denote the condition (2a) in which one asks moreover that F_{i} ∈ D_{ℝ−c}^{[a,b]}(k_{X}). Define similarly (1a-rc). Then the same proof of (3a) ⇒ (2a) ⇒ (1a) ⇒ (3b) can be applied to show (5a) ⇒ (2a-rc) ⇒ (1a-rc) ⇒ (3b).

q.e.d.

Definition 4.2. Let F ∈ D^{b}(I(k_{X})). The micro-support of F, denoted by SS(F), is the closed conic subset of T^{∗}X whose complementary is the set of points p ∈ T^{∗}X such that one of the equivalent conditions in Lemma 4.1 is satisfied.

Proposition 4.3. (i) For F ∈ D^{b}(I(k_{X})), one has SS(F) ∩ T_{X}^{∗}X = supp(F).

(ii) Let F ∈ D^{b}(k_{X}). Then SS(ι_{X}F) = SS(F).

(iii) Let F ∈ D^{b}(I(k_{X})). Then SS(α_{X}F) ⊂ SS(F).

(iv) Let F_{1} ⟶ F_{2} ⟶ F_{3} ⟶^{+1} be a distinguished triangle in D^{b}(I(k_{X})). Then SS(F_{i}) ⊂ SS(F_{j}) ∪ SS(F_{k}) if {i, j, k} = {1, 2, 3}.

Proof. (i) supp(F) ⊂ SS(F) follows for example from (1b) of Lemma 4.1. The other inclusion is obvious.

(ii) The inclusion SS(F) ⊂ SS(ι_{X}F) follows from (2a) since J(F) is “\underrightarrow{lim}” F. The converse inclusion follows from (1b).

(iii) is obvious, using condition (3b).

(iv) is obvious by (3b).

q.e.d.

Definition 4.4. Let Λ_{i}, i ∈ I be a family of closed conic subsets of T^{∗}X, indexed by the objects of a small and filtrant category I. One sets

lim_{i} Λ_{i} = ⋂_{J⊂I} \overline{⋃_{j∈J} Λ_{j}}

where J ranges over the family of cofinal subcategories of I.

In other words, p ∈ T^{∗}X does not belong to lim_{i} Λ_{i} if there exists an open neighborhood U of p and a cofinal subset J of I such that Λ_{j} ∩ U = ∅ for every j ∈ J.

It follows immediately from the definition that if J(F) ≃ “\underrightarrow{lim}”_{i} J(F_{i}),

then

SS(F) ⊂ lim_{i} SS(F_{i}). (4.1)

It follows from Proposition 3.8 that if G ∈ D^{b}(k_{X}), one has the inclusions

\begin{aligned}SS(G ⊗ F) & ⊂ & lim_{i} (SS(G)ˆ+SS(F_{i})), \\ SS(RIhom (G, F)) & ⊂ & lim_{i} (SS(G)^{a} ˆ+SS(F_{i})). \end{aligned} (4.2)

Example 4.5. Let X = ℝ^{2} endowed with coordinates (x, y) and denote by (x, y; ξ, η) the associated coordinates on T^{∗}X. Let

Y = {(x, y); y = 0},
U = {(x, y); x^{2} < y},
Z_{ε} = {(x, y); x^{2} < y ≤ ε^{2}}.

Set F_{ε} = k_{Z_{ε}} and F = k_{U} ⊗ β_{X}(k_{\{0\}}) ≃ “\underrightarrow{lim}”_{ε} F_{ε}. Then

SS(k_{Y}) = T_{Y}^{∗} X = {(x, y; ξ, η) ; y = ξ = 0},
SS(F_{ε}) = {(x, y; 0, 0) ; x^{2} ≤ y ≤ ε^{2}}
⋃{(x, y; ξ, η) ; y = x^{2}, |x| ≤ ε, ξ = −2xη, η ≤ 0}
⋃{(x, y; ξ, η) ; y = ε^{2}, |x| ≤ ε, ξ = 0, η ≤ 0}
⋃{(±ε, ε^{2}; ξ, η) ; 0 ≤ ±ξ ≤ −2εη, η ≤ 0},
SS(F) = {(x, y; ξ, η); x = y = ξ = 0, η ≤ 0}.

On the other-hand, one has

SS(F) = lim_{ε} SS(F_{ε}),

RHom (k_{Y} , F) ≃ k_{{0}} [−2] ,

lim_{ε} (T_{Y}^{∗} X ˆ+SS(F_{ε})) = T_{{0}}^{∗}X,

T_{Y}^{∗} X ˆ+SS(F) = {(x, y; ξ, η); x = y = ξ = 0}

⫋ SS(RHom (k_{Y} , F)).

Note that SS(F) is not involutive.

Recall that subanalytic isotropic subsets of T^{∗}X are defined in [4]. Let us say for short that a conic locally closed subset Λ of T^{∗}X is isotropic if Λ is contained in a conic locally closed subanalytic isotropic subset.

Definition 4.6. (i) We denote by D_{w−ℝ−c}^{b}(I(k_{X})) the full triangulated subcategory of D_{I ℝ−c}^{b}(I(k_{X})) consisting of objects F such that SS(F) is isotropic. We call an object of this category a weakly ℝ-constructible ind-sheaf.

(ii) Let us denote by D_{ℝ−c}^{b}(I(k_{X})) the full triangulated subcategory of D_{w−ℝ−c}^{b}(I(k_{X})) consisting of objects F such that one has RHom (G, F) ∈ D_{ℝ−c}^{b}(k_{X}) for any G ∈ D_{ℝ−c}^{b}(k_{X}). We call an object of this category an ℝ-constructible ind-sheaf.

Note that the functor α_{X} induces functors

α_{X}: D_{w−ℝ−c}^{b}(I(k_{X})) ⟶ D_{w−ℝ−c}^{b}(k_{X}),
α_{X}: D_{ℝ−c}^{b}(I(k_{X})) ⟶ D_{ℝ−c}^{b}(k_{X}).

The last property follows from α_{X}(F) = RHom (ℂ_{X}, F).

Conjecture 4.7. Let F ∈ D_{w−ℝ−c}^{b}(I(k_{X})) and let G ∈ D_{w−ℝ−c}^{b}(k_{X}). Then RIhom (G, F) and G ⊗ F belong to D_{w−ℝ−c}^{b}(I(k_{X})).

Example 4.5 shows that the knowledge of SS(F) and SS(G) does not allows us to estimate the micro-support of RHom (F, G) by the one for sheaves, and that is one reason for the definition below.

Definition 4.8. Let F ∈ D^{b}(I(k_{X})).

(i) Let S ⊂ T^{∗}X be a locally closed conic subset and let p ∈ T^{∗}X. We say that F is regular along S at p if there exist F′ isomorphic to F in a neighborhood of π(p), an open neighborhood U of p with S∩U closed in U, a small and filtrant category I and a functor I ⟶ D^{[a,b]}(k_{X}), i ↦ F_{i} such that J(F′) ≃ “\underrightarrow{lim}”_{i} J(F_{i}) and SS(F_{i}) ∩ U ⊂ S.

(ii) If U is an open subset of T^{∗}X and F is regular along S at each p ∈ U, we say that F is regular along S on U.

(iii) Let p ∈ T^{∗}X. We say that F is regular at p if F is regular along SS(F) at p.

If F is regular at each p ∈ SS(F), we say that F is regular.

(iv) We denote by SS_{reg}(F) the conic open subset of SS(F) consisting of points p such that F is regular at p, and we set

SS_{irr}(F) = SS(F) \ SS_{reg}(F).

Note that SS_{irr}(F) = SS(F) for F in Example 4.5.

Proposition 4.9. (i) Let F ∈ D^{b}(I(k_{X})). Then F is regular along any locally closed set S at each p ∉ SS(F).

(ii) Let F_{1} ⟶ F_{2} ⟶ F_{3} ⟶^{+1} be a distinguished triangle in D^{b}(I(k_{X})). If F_{j} and F_{k} are regular along S, so is F_{i} for i, j, k ∈ {1, 2, 3}, j ≠ k.

(iii) Let F ∈ D^{b}(k_{X}). Then ι_{X}F is regular.

Proof. (i) and (iii) are obvious and the proof of (ii) is similar to that of Proposition 4.3 (iv). q.e.d.

It is possible to localize the category D^{b}(I(k_{X})) with respect to the microsupport, exactly as for usual sheaves.

Let V be a subset of T^{∗}X and let Ω = T^{∗}X \ V. We shall denote by D_{V}^{b} (k_{X}) the full triangulated subcategory of D^{b}(k_{X}) consisting of objects F such that SS(F) ⊂ V, and by D^{b}(k_{X}; Ω) the localization of D^{b}(k_{X}) by D_{V}^{b} (k_{X}).

Similarly, we denote by D_{V}^{b} (I(k_{X})) the full triangulated subcategory of D^{b}(I(k_{X})) consisting of objects F such that SS(F) ⊂ V.

Definition 4.10. One sets

D^{b}(I(k_{X}; Ω)) = D^{b}(I(k_{X}))/D_{V}^{b} (I(k_{X})),

the localization of D^{b}(I(k_{X})) by D_{V}^{b} (I(k_{X})).

Let F_{1} and F_{2} are two objects of D^{b}(I(k_{X})) whose images in D^{b}(I(k_{X}; Ω)) are isomorphic. There exist a third object F_{3} ∈ D^{b}(I(k_{X}; Ω)) and distinguished triangles in D^{b}(I(k_{X})): F_{i} ⟶ F_{3} ⟶ G_{i} ⟶^{+1} (i = 1, 2) such that SS(G_{i}) ∩ Ω = ∅. It follows that SS(F_{1}) ∩ Ω = SS(F_{3}) ∩ Ω = SS(F_{2}) ∩ Ω.

Therefore if F ∈ D^{b}(I(k_{X}; Ω)), the subsets SS(F) and SS_{irr}(F) of Ω are well-defined.

5 Invariance by contact transformations

It is possible to define contact transformations on ind-sheaves. We shall follow the notations in [4] Chapter VII.

We denote by p_{1} and p_{2} the first and second projections defined on T^{∗}(X × Y) ≃ T^{∗}X × T^{∗}Y, and we denote by p_{2}^{a} the composition of p_{2} with the antipodal map on T^{∗}Y.

We denote by r: X × Y ⟶ Y × X the canonical map and we keep the same notation to denote its inverse.

By a kernel K on X × Y we mean an object of D^{b}(k_{X×Y}). To a kernel K one associates the kernel on Y × X

K^{∗} := r_{∗}RHom (K, ω_{X×Y/Y}).

One defines the functor

(5.1)  Φ_{K}: D^{b}(k_{Y}) ⟶ D^{b}(k_{X})
G ↦ Rq_{1}_{!}(K ⊗ q_{2}^{−1}G).

Consider another manifold Z and a kernel L on Y × Z. One defines the projection q_{12} from X × Y × Z to X × Y, and similarly with q_{23}, q_{13}.

One sets

K ∘ L = Rq_{13}_{!}(q_{12}^{−1}K ⊗ q_{23}^{−1}L). (5.2)

Choosing Z = {pt}, one has Φ_{K}(G) = K ∘ G for G ∈ D^{b}(k_{Y}).

Let Ω_{X} and Ω_{Y} be two conic open subsets of T^{∗}X and T^{∗}Y, respectively. One denotes by N(Ω_{X}, Ω_{Y}) the full subcategory of D^{b}(k_{X×Y}; Ω_{X} × T^{∗}Y) of objects K satisfying;

\{ \begin{matrix}SS(K) ∩ (Ω_{X} × T^{∗}Y) ⊂ Ω_{X} × Ω_{Y}^{a} , \\ p_{1} : SS(K) ∩ (Ω_{X} × T^{∗}Y) ⟶ Ω_{X} is proper. \end{matrix} (5.3)

Let us recall some results of loc. cit.

(i) Let K ∈ N(Ω_{X}, Ω_{Y}). Then the functor Φ_{K} induces a well-defined functor: Φ_{K}^{μ}: D^{b}(k_{Y}; Ω_{Y}) ⟶ D^{b}(k_{X}; Ω_{X}).

(ii) Let L ∈ N(Ω_{Y} , Ω_{Z}). Then K ∘ L ∈ N(Ω_{X}, Ω_{Z}). Moreover, the two functors Φ_{K∘L}^{μ} and Φ_{K}^{μ} ∘ Φ_{L}^{μ} from D^{b}(k_{Z}; Ω_{Z}) to D^{b}(k_{X}; Ω_{X}) are isomorphic.

We construct the functor analogous to the functor Φ_{K} for ind-sheaves by defining

(5.4)  Φ̃_{K}: D^{b}(I(k_{Y})) ⟶ D^{b}(I(k_{X}))
G ↦ Rq_{1}_{!!}(K ⊗ q_{2}^{−1}G).

Applying Theorem 3.3, we get:

Lemma 5.1. Let G ∈ D^{b}(I(k_{Y})) and assume that J(G) ≃ “\underrightarrow{lim}”_{i} J(G_{i}), with

I small and filtrant and G_{i} ∈ D^{b}(k_{Y}). Then J(Φ̃_{K}(G)) ≃ “\underrightarrow{lim}”_{i} J(Φ_{K}(G_{i})).

Now assume that dim X = dim Y and that there exists a smooth conic Lagrangian submanifold Λ ⊂ Ω_{X} × Ω_{Y}^{a} such that p_{1} : Λ ⟶ Ω_{X} and p_{2}^{a}: Λ ⟶ Ω_{Y} are isomorphisms. In other words, Λ is the graph of a homogeneous symplectic isomorphism χ: Ω_{Y} ⟶^{∼} Ω_{X}.

Let K be a kernel satisfying the assumptions of Theorem 7.2.1 of loc. cit., that is:

K is cohomologically constructible,

(p_{1}^{−1}(Ω_{X}) ∪ p_{2}^{a}^{−1}(Ω_{Y})) ∩ SS(K) ⊂ Λ,

k_{Λ} ⟶^{∼} μhom(K, K) on Ω_{X} × Ω_{Y}^{a}.

(5.5)

Theorem 5.2. Assume (5.5).

(i) The functor Φ̃_{K} induces a well-defined functor: Φ̃_{K}^{μ}: D^{b}(I(k_{Y}; Ω_{Y})) ⟶ D^{b}(I(k_{X}; Ω_{X})). Similarly, the functor Φ̃_{K^{∗}} induces a well-defined functor: Φ̃_{K^{∗}}^{μ}: D^{b}(I(k_{X}; Ω_{X})) ⟶ D^{b}(I(k_{Y}; Ω_{Y})).

(ii) The functor Φ̃_{K}^{μ}: D^{b}(I(k_{Y}; Ω_{Y})) ⟶ D^{b}(I(k_{X}; Ω_{X})) and the functor Φ̃_{K^{∗}}^{μ}: D^{b}(I(k_{X}; Ω_{X})) ⟶ D^{b}(I(k_{Y}; Ω_{Y})) are equivalences of categories inverse one to each other.

(iii) If G ∈ D^{b}(I(k_{Y})), then SS(Φ̃_{K}(G)) ∩ Ω_{X} = χ(SS(G) ∩ Ω_{Y}).

(iv) If G is regular at p ∈ Ω_{Y}, then Φ̃_{K}(G) is regular at χ(p) ∈ Ω_{X}. In other words, SS_{irr}(Φ_{K}(G)) ∩ Ω_{X} = χ(SS_{irr}(G) ∩ Ω_{Y}).

Proof. (i) Let G ∈ D^{b}(I(k_{Y})) and assume that SS(G)∩Ω_{Y} = ∅. Let us prove that SS(Φ̃_{K}(G)) ∩ Ω_{X} = ∅. Let p_{X} ∈ Ω_{X} and let p_{Y} = χ^{−1}(p_{X}). There exist an open neighborhood U_{Y} of p_{Y} in Ω_{Y} and an inductive system such that J(G) ≃ “\underrightarrow{lim}”_{i∈I} J(G_{i}), and for any i ∈ I there exists i ⟶ j such that the

morphism G_{i} ⟶ G_{j} is zero in D^{b}(k_{Y}; U_{Y}). Applying Lemma 5.1 we find that J(Φ̃_{K}(G)) ≃ “\underrightarrow{lim}”_{i} J(Φ_{K}(G_{i})). Since the morphism Φ_{K}(G_{i}) ⟶ Φ_{K}(G_{j}) is

zero in D^{b}(k_{X}; U_{X}), the result follows. (ii) One has the isomorphism K ∘ K^{∗} ≃ k_{Δ_{X}} in N(Ω_{X}, Ω_{X}) and the isomorphism K^{∗} ∘ K ≃ k_{Δ_{Y}} in N(Ω_{Y} , Ω_{Y}). Hence, it is enough to remark that

Φ̃_{K}^{μ} ∘ Φ̃_{K^{∗}}^{μ} ≃ Φ̃_{K∘K^{∗}}^{μ}, (5.6)

which follows from the fact that the two functors Φ̃_{K} ∘ Φ̃_{K^{∗}} and Φ̃_{K∘K^{∗}}, from D^{b}(I(k_{X})) to D^{b}(I(k_{X})) are isomorphic. (iii) For an open subset U_{Y} ⊂ Ω_{Y}, set U_{X} = χ(U_{Y}). Then K ∈ N(U_{X}, U_{Y}) and K satisfies (5.5) with Ω replaced with U. Let G ∈ D^{b}(I(k_{Y})) with SS(G) = ∅ in a neighborhood of p_{Y} ∈ Ω_{Y}. By the proof of (i), SS(Φ̃_{K}(G)) = ∅ in a neighborhood of χ(p_{Y}). (iv) The proof is similar to that of (iii). q.e.d.

6 Ind-sheaves and D-modules

Let now X be a complex manifold and let M be a coherent D_{X}-module. We set for short

Sol(M) = RHom_{D_{X}} (M, O_{X}),
Sol^{t}(M) = RIhom_{β_{X}D_{X}} (β_{X}M, O_{X}^{t}).

Theorem 6.1. One has

SS(Sol^{t}(M)) = char(M).

Proof. (i) The inclusion char(M) ⊂ SS(Sol^{t}(M)) follows from

SS(Sol(M)) = char(M), α_{X}(Sol^{t}(M)) ≃ Sol(M).

and Proposition 4.3 (ii).

(ii) Let us prove the converse inclusion using condition (5a) of Lemma 4.1. Assume that G ∈ D_{ℝ−c}^{b}(ℂ_{X}) satisfy SS(G) ∩ char(M) ⊂ T_{X}^{∗}X. One has the morphisms

RHom (G, RIhom_{β_{X}D_{X}} (β_{X}M, O_{X}^{t})) ≃ RHom_{D_{X}} (M, T Hom (G, O_{X}))
⟶ RHom_{D_{X}} (M, RHom (G, O_{X})).

It follows from [1, Corollary 4.2.5] that the second morphism is an isomorphism. Hence the result follows from SS(Sol(M)) = char(M) and Lemma 4.1 (5a). q.e.d.

The following conjecture is a consequence of Conjecture 4.7.

Conjecture 6.2. If M is a holonomic D_{X}-module, then Sol^{t}(M) belongs to D_{ℝ−c}^{b}(I(ℂ_{X})).

Theorem 6.3. If M is a regular holonomic D_{X}-module, then Sol^{t}(M) ⟶ Sol(M) is an isomorphism.

Proof. This is a reformulation of a result of [2] which asserts that for any G ∈ D_{ℝ−c}^{b}(ℂ_{X}), the natural morphism

RHom_{D_{X}} (M, T Hom (G, O_{X})) ⟶ RHom_{D_{X}} (M, RHom (G, O_{X}))

is an isomorphism. q.e.d.

We conjecture the following statement in which “only if ” part is a consequence of the theorem above.

Conjecture 6.4. Let M be a holonomic D_{X}-module. Then M is regular holonomic if and only if Sol^{t}(M) is regular.

7 An example

In this section X = ℂ endowed with the holomorphic coordinate z, and we shall study the ind-sheaf of temperate holomorphic solutions of the D_{X}- module M := D_{X} exp(1/z) = D_{X}/D_{X}(z^{2}∂_{z} + 1). We set for short

\begin{aligned}S^{t}:= & H^{0}(Sol^{t}(M)) & ≃ Ihom_{β_{X}D_{X}} (β_{X}M, O_{X}^{t}), \\ S:= & H^{0}(Sol(M)) & ≃ Hom_{D_{X}} (M, O_{X}). \end{aligned}

Notice first that O_{X}^{t} is concentrated in degree 0 (since dim X = 1), and it is a sub-ind-sheaf of O_{X}. It follows that the morphism S^{t} ⟶ S is a monomorphism.

Moreover,

S ≃ ℂ_{X,X\{0}} · exp(1/z).

Lemma 7.1. Let V ⊂ X be a connected open subset. Then Γ(V; S^{t}) ≠ 0 if and only if V ⊂ X \ {0} and exp(1/z)|_{V} is tempered.

Proof. The space Γ(V; S) has dimension one and is generated by the function exp(1/z). Hence, the subspace Γ(V; S^{t}) ≃ Γ(V; S) ∩ Γ(V; O^{t}) is not zero if and only if exp(1/z) ∈ Γ(V; O_{X}^{t}), that is, if and only if exp(1/z)|_{V} is tempered. q.e.d.

Let us set z = x + iy.

Lemma 7.2. Let W be an open subanalytic subset of ℙ^{1}(ℂ) with ∞ ∉ W. Assume that there exist positive constants C and A such that

exp(x) ≤ C(1 + x^{2} + y^{2})^{N} on W. (7.1)

Then there exists a constant B such that x ≤ B on W.

Proof. If x is not bounded on W, then there exists a real analytic curve γ: [0, ε[⟶ ℙ^{1}(ℂ) such that Re γ(0) = ∞ and γ(t) ∈ W for t > 0. Writing γ(t) = (x(t), y(t)), one has

y(t) = cx(t)^{q} + O(x(t)^{q−ε}).

for some q ∈ ℚ, c ∈ ℝ and ε > 0. Then (7.1) implies that exp(x) has a polynomial growth when x ⟶ ∞, which is a contradiction. q.e.d.

Let B̄_{ε} denote the closed ball with center (ε, 0) and radius ε and set U_{ε} = X \ B̄_{ε}.

Proposition 7.3. One has the isomorphism

“\underrightarrow{lim}”_{ε>0} ℂ_{XU_{ε}} ⟶^{∼} Ihom_{β_{X}D_{X}} (β_{X}M, O_{X}^{t}). (7.2)

Proof. It follows from Lemma 7.2 that exp(1/z) is temperate (in a neighborhood of 0) on an open subanalytic subset V ⊂ X \ {0} if and only if Re(1/z) is bounded on V, that is, if and only if V ⊂ U_{ε} for some ε > 0.

Let V be a connected relatively compact subanalytic open subset of X \ {0}. Then a morphism ℂ_{V} ⟶ ℂ_{X\{0}} · exp(1/z) factorizes through a morphism ℂ_{V} ⟶ S^{t} if and only if it factorizes through ℂ_{U_{ε}}. Hence we get the isomorphism (7.2) by Theorem 2.2. q.e.d.

Remark 7.4. In fact one can show

H^{1}(Sol^{t}(M)) ⟶^{∼} H^{1}(Sol(M)) ≃ ℂ_{0}.

The isomorphism H^{1}(Sol(M)) = O_{X}/(z^{2}∂_{z} + 1)O_{X} ⟶^{∼} ℂ_{0} is given by

(O_{X})_{0} ∋ v(z) ↦ ∮ v(z)z^{−2} exp(−1/z) dz.

Note that φ(z):=z^{−2} exp(−1/z) is a solution to the adjoint equation

(−∂_{z}z^{2} + 1)φ(z) = 0.

The distinguished triangle

S^{t} ⟶ Sol^{t}(M) ⟶ H^{1}(Sol^{t}(M))[−1] ⟶^{+1}

gives a non-zero element of Ext^{2}(ℂ_{0}, S^{t}) ⟶^{∼} Ext^{2}(ℂ_{0}, ℂ_{X}) ≃ ℂ.

References

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[2] M. Kashiwara, The Riemann-Hilbert problem for holonomic systems, Publ. RIMS, Kyoto Univ. 20 (1984) pp. 319–365.

[3] M. Kashiwara and T. Oshima, Systems of microdifferential equations with regular singularities and their boundary values problems, Annals of Math. 106 (1977) pp. 145–200.

[4] M. Kashiwara and P. Schapira, Sheaves on Manifolds, Grundlehren der Math. Wiss. 292 Springer (1990).

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[6] M. Kashiwara and P. Schapira, Ind-sheaves, Astérisque, Soc. Math. France 271 (2001).

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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, Pl. Jussieu 75252 Paris Cedex 05 France schapira@math.jussieu.fr http://www.math.jussieu.fr/˜ schapira/