Let f : Y → X be a morphism of real C ∞ manifolds and let F , K be sheaves on X (more precisely objects of the derived category D b (X)).

Andrea D’Agnolo

0. Introduction.

Let f : Y → X be a morphism of real C ^∞ manifolds and let F , K be sheaves on X (more precisely objects of the derived category D ^b (X)).

In this paper we study the microlocal inverse images of sheaves.

In particular we recall the construction of the functors f _µ,p ⁻¹ , f _µ,p ^! of [K-S 4] (which makes use of the categories of ind-objects and pro-objects on the microlocalization of D ^b (X)) and study some of their properties.

Then we give a theorem, namely Theorem 2.2.3 below, which asserts that the natural morphism:

(0.1) µhom(f _µ,p ⁻¹ K, f _µ,p ^! F ⊗ ω _Y ^⊗−1 ) _p

−→ µhom(K, F ⊗ ω ^⊗−1 _X ) _p

,

is an isomorphism as soon as a very natural hypothesis similar to that of “mi- crohyperbolicity” for microdifferential systems, is satisfied (here ω X denotes the dualizing complex on X and µhom the microlocalization bifunctor of [K-S 4]).

In fact, one could say that this theorem is a statement of the microlocal well poseness for the Cauchy problem.

As an application, we then state and prove a theorem, namely Theorem 3.1.1, on the well poseness for the Cauchy problem, in a sheaf theoretical frame.

This theorem generalize the results obtained in [D’A-S] and will allow us not only to recover the classical results on the ramified Cauchy problem (cf. [H-L-W], [K-S 1], [Sc]), but also the result of [K-S 2] on the hyperbolic Cauchy problem.

The author likes to express his gratitude to P. Schapira for frequent and fruitful discussions.

1. Review on sheaves.

In this chapter we collect the notations that will be used throughout this paper.

We also give some basic results on ind-objects and pro-objects that are necessary for the proof of the main theorem.

The frame is that of the microlocal study of sheaves as developed in [K-S 3] and [K-S 4].

Until chapter 4 all manifolds and morphisms of manifolds will be real and of class C ^∞ .

Appeared in: Publ. Res. Inst. Math. Sci. 27 (1991), no. 3, 509–532.

1

1.1. Geometry. To a manifold X one associates its tangent and cotangent bun- dles noted τ X : T X → X and π X : T ^∗ X → X respectively. We note ˙ T ^∗ X the cotangent bundle with the zero-section removed and denote by ˙π _X the projection T ˙ ^∗ X → X.

If M is a closed submanifold of X, one denotes by T _M ^∗ X the conormal bundle to M in X. If A is a subset of X, one denotes by N ^∗ (A) the strict conormal cone to A, a closed, proper, convex conic subset of T ^∗ X.

If f : Y → X is a morphism of manifolds, one denotes by ^t f ⁰ and f _π the natural mappings associated to f :

T ^∗ Y

t

f

←− Y × _X T ^∗ X −→ T ^f

^∗ X.

One sets: T _Y ^∗ X = ^t f ⁰⁻¹ (T _Y ^∗ Y ).

If N (resp. M ) is a closed submanifold of Y (resp. X) with f (N ) ⊂ M , one denotes by ^t f _N ⁰ and f _{N π} the natural mappings associated to f :

T _N ^∗ Y

t

f

←− N × _M T _M ^∗ X −→ T ^f

_M ^∗ X.

If A is a closed conic subset of T ^∗ X, one says that f is non-characteristic for A iff ^t f ⁰⁻¹ (T _Y ^∗ Y ) ∩ f _π ⁻¹ (A) ⊂ Y × _X T _X ^∗ X. If V is a subset of T ^∗ Y , we refer to [K-S 3]

for the definition of f being non-characteristic for A on V .

1.2. The category D ^b (X). We fix a commutative ring A with finite global di- mension (e.g. A = Z).

Let X be a manifold. One denotes by D ^b (X) the derived category of the category of bounded complexes of sheaves of A-modules on X.

If F ∈ Ob(D ^b (X)), we note by SS(F ) the micro-support of F (cf [K-S 3]). This is a closed conic involutive subset of T ^∗ X that describes the directions of non propagation for the cohomology of F .

If M is a closed submanifold of X, we denote by µ _M (F ) the Sato’s microlocal- ization of F along M , an object of D ^b (T _M ^∗ X). If G is another object of D ^b (X), following [K-S 3], we define the microlocalization of F along G by:

µhom(G, F ) = µ _∆ RHom(q ⁻¹ ₂ G, q ^! ₁ F ),

where ∆ is the diagonal of X × X and q ₁ , q ₂ denote the projections from X × X to X. This is an object of D ^b (T ^∗ X) with the following properties:

Rπ _X∗ µhom(G, F ) = RHom(G, F ), (1.2.1)

µhom(A _M , F ) = µ _M (F ), (1.2.2)

supp µhom(G, F ) ⊂ SS(G) ∩ SS(F ).

(1.2.3)

Here, as general notation on sheaves, for Z a locally closed subset of X, one denotes by A _Z the sheaf which is 0 on X \ Z and the constant sheaf with stalk A on Z.

If Y is another manifold and F ∈ Ob(D ^b (X)), G ∈ Ob(D ^b (Y )), one defines the external product of F and G by:

F × G = q ^L ₁ ⁻¹ F ⊗ q ^L ₂ ⁻¹ G,

where q 1 (resp. q 2 ) is the projection from X × Y to X (resp. Y ). This is an object of D ^b (X × Y ).

Let f : Y → X be a morphism of manifolds. We denote by ω _{Y /X} the relative dualizing complex defined by ω _{Y /X} = f ^! A _X . One sets ω _X = a ^! _X A, where a _X : X → {pt}. If or _X is the orientation sheaf, one has an isomorphism ω X ∼ = or X [dim X], and hence, for local problems, ω _X plays essentially the role of a shift.

If F is an object of D ^b (X), one says that f is non-characteristic for F if f is non-characteristic for SS(F ).

1.3. The category D ^b (X; p X ). Let X be a manifold and let Ω be a subset of T ^∗ X. One denotes by D ^b (X; Ω) the localized category D ^b (X)/D ^b _Ω (X), where D ^b _Ω (X) is the null system: D ^b _Ω (X) = {F ∈ Ob(D ^b (X)); SS(F ) ∩ Ω = ∅}. Recall that the objects of D ^b (X; Ω) are the same as those of D ^b (X) and that a morphism u : F → G in D ^b (X) becomes an isomorphism in D ^b (X; Ω) if Ω ∩ SS(H) = ∅, H being the third term of a distinguished triangle: F → G → H ^{u +1} →. Such an u is called an isomorphism on Ω. If p X ∈ T ^∗ X one writes D ^b (X; p X ) instead of D ^b (X; {p _X }).

A question naturally arising is whether a functor, acting on derived categories of sheaves, still has a “microlocal” meaning, i.e. if it is well defined as a functor acting on these localized categories. In this section we will mainly be concerned in giving an answer to this problem for several well known functors.

Let Y be another manifold and denote by q ₁ (resp. q ₂ ) the projections from X ×Y to X (resp. Y ). Let M be a closed submanifold of X. Take a point p _X ∈ T ^∗ X (resp. p _Y ∈ T ^∗ Y ) and set p _X×Y = (p _X , p _Y ) ∈ T ^∗ (X × Y ).

Proposition 1.3.1. The functors:

· × · : D ^{L b} (X) × D ^b (Y ) −→ D ^b (X × Y ), RHom(q ₂ ⁻¹ (·), q ₁ ^! (·)) : D ^b (Y ) ^◦ × D ^b (X) −→ D ^b (X × Y ),

µ M (·) : D ^b (X) −→ D ^b (T _M ^∗ X),

µhom(·, ·) : D ^b (X) ^◦ × D ^b (X) −→ D ^b (T ^∗ X),

are microlocally well defined, i.e. extend naturally as functors (that we denote by the same names):

· × · :D ^{L b} (X; p X ) × D ^b (Y ; p Y ) −→ D ^b (X × Y ; p X×Y ), RHom(q ₂ ⁻¹ (·), q ₁ ^! (·)) :

D ^b (Y ; p Y ) ^◦ × D ^b (X; p X ) −→ D ^b (X × Y ; p X×Y ), µ _M (·) :D ^b (X; p _X ) −→ D ^b (T _M ^∗ X; p _X ), (p _X ∈ T _M ^∗ X) µhom(·, ·) :D ^b (X; p X ) ^◦ × D ^b (X; p X ) −→ D ^b (T ^∗ X; p X ).

Here D ^b (Y ) ^◦ denotes the opposite category to D ^b (Y ), i.e. the category whose

objects are the same as those of D ^b (Y ) and whose morphisms are reversed.

Proof. Let F, G ∈ Ob(D ^b (X)) and H ∈ Ob(D ^b (Y )). Recall the following estimates of the micro-support (cf [K-S 3, Prop. 4.2.1, 4.2.2, Th. 5.2.1]):

SS(F × H) ⊂ SS(F ) × SS(H), ^L SS(RHom(q ₂ ⁻¹ (H), q ^! ₁ (F )) ⊂ SS(F ) × SS(H) ^a ,

SS(µ _M (F )) ⊂ C _T

_X (SS(F )), SS(µhom(G, F )) ⊂ C(SS(F ), SS(G)).

Since the proofs are similar we will treat only the first functor. The hypothesis F ∈ D ^b _{p

_} (X) or H ∈ D ^b _{p

_} (X) means that p X ∈ SS(F ) or p / _Y ∈ SS(H). Then / it follows from the first estimate that p _X×Y ∈ SS(F / × H). q.e.d. ^L

1.4. Complements on ind-objects and pro-objects. Let f : Y → X be a morphism of manifolds. Take a point p ∈ Y × _X T ^∗ X and set p _X = f _π (p), p _Y = ^t f ⁰ (p). Contrarily to the case of the functors treated in Proposition 1.3.1, the functors Rf _∗ , Rf _! (resp. f ⁻¹ , f ^! ) are not microlocal, i.e. are not well defined as functors from D ^b (Y ; p _Y ) (resp. D ^b (X; p _X )) to D ^b (X; p _X ) (resp. D ^b (Y ; p _Y )). To give a microlocal meaning to these functors one must enlarge the category D ^b (X; p _X ) and work with ind-objects and pro-objects. In this section we recall the definition of ind-objects and pro-objects and, as a preparation for the next section, we give some of their basic properties.

Let us first recall some basic notions on ind-objects and pro-objects due to Grothendieck [G] (for an exposition e.g. cf [K-S 4, Chapter 1, §11]).

Let C be a category. Denote by C ^∧ (resp. C ^∨ ) the category of covariant (resp.

contravariant) functors from C to the category of sets. Notice first that C may be considered as a full subcategory of C ^∧ or C ^∨ via the fully faithful functor:

h ^∧ : C −→ C ^∧ h ^∨ : C −→ C ^∨

X 7→ Hom C (X, ·) X 7→ Hom C (·, X)

An object φ of C ^∧ in the image of h ^∧ is called representable. An object X of C such that φ = h ^∧ (X) is called a representative of φ. Representatives are defined up to an isomorphism.

A category I is called filtrant if for i, j ∈ Ob(I) there exist k ∈ Ob(I) and morphisms i → k, j → k and if for two morphisms f, g ∈ Hom _I (i, j) there exists a morphism h : j → k such that h ◦ f = h ◦ g.

Let φ be a covariant functor from a filtrant category I to C. Recall that the object

“lim”

−→

I

φ(i) of C ^∨ is defined by “lim”

−→

I

φ(i)(X) = lim −→

I

Hom _C (X, φ(i)) for X ∈ Ob(C).

Here lim −→

I

denotes the classical inductive limit in the category of sets. Similarly, if φ is a contravariant functor from I to C, “lim”

←−

I

φ(i) is the object of C ^∧ defined by

“lim”

←−

I

φ(i)(X) = lim −→

I

Hom C (φ(i), X). The category of ind-objects (resp. pro-objects) is the full subcategory of C ^∨ (resp. C ^∧ ) consisting of those objects isomorphic to

“lim”

−→

I

φ(i) (resp. “lim”

←−

I

φ(i)) for some covariant (resp. contravariant) functor φ from

I to C.

We will give now some results on ind-objects and pro-objects which will be useful in section 2.

Let I, I ⁰ be two filtrant categories and, for simplicity, assume Ob(I) and Ob(I ⁰ ) being sets. One defines the filtrant category I × I ⁰ in the obvious way. Let C, C ⁰ be two categories and let φ, φ ⁰ be two covariant functors from I to C and from I ⁰ to C ⁰ respectively. One can prove the following result as in [K-S 4, Corollary 1.11.8].

Proposition 1.4.1. Keeping the same notations as above, let T be a bifunctor from C × C ⁰ to a category C ⁰⁰ . If “lim”

−→

I

φ(i) and “lim”

−→

I

φ ⁰ (i ⁰ ) are representable then so is the ind-object “lim”

−→

I×I

T (φ(i), φ ⁰ (i ⁰ )), a representative being given by T (“lim”

−→

I

φ(i), “lim”

−→

I

φ ⁰ (i ⁰ )).

A similar result holds if φ or φ ⁰ or both of them are contravariant.

Let C be a category. Let I and I ⁰ be filtrant categories and let ι : I → I ⁰ be a functor. Let φ be a covariant (resp. contravariant) functor from I ⁰ to C.

Definition 1.4.2. One says that I and I ⁰ are cofinal with respect to φ by ι if the following properties hold:

(a) For any i ⁰ ∈ Ob(I ⁰ ) there exists i ∈ Ob(I) and a morphism φ(i ⁰ ) → φ(ι(i)) (resp. φ(ι(i)) → φ(i ⁰ )).

(b) For any i ∈ Ob(I), i ⁰ ∈ Ob(I ⁰ ) and a morphism f : ι(i) → i ⁰ , there exists a morphism g : i → i ₁ in I such that φ(ι(g)) factors trough φ(f ).

If I and I ⁰ are cofinal with respect to the identical functor of I ⁰ for ι, we will say that I and I ⁰ are cofinal (by ι). This is the classical definition (cf [K-S 4, Ex.

1.38]). Note that if I and I ⁰ are cofinal by ι then they are cofinal with respect to any φ : I ⁰ → C by ι.

Let us now state a proposition that extend to this more general definition a result of [K-S 4, Ex. 1.38].

Proposition 1.4.3. With the same notations as above, if I and I ⁰ are cofinal with respect to φ by ι, the natural morphism:

“lim”

−→

I

φ ◦ ι → “lim”

−→

I

φ

(resp.

“lim”

←−

I

φ → “lim”

←−

I

φ ◦ ι) is an isomorphism.

Proof. For X ∈ Ob(C) set A _X = lim −→

I

Hom _C (X, φ(ι(i))), B _X = lim −→

I

Hom _C (X, ι(i ⁰ )).

We have to show that A X ' B _X for every X. Let [u : X → φ(i ⁰ )] be an element

of B _X (here [u] denotes the equivalence class of u in B _X ). Due to (a) of Definition

1.4.2 we can find a morphism v : i ⁰ → ι(i) in I ⁰ with i ∈ Ob(I). We define a map

F : A _X → B _X by F ([u]) = [φ(v) ◦ u]. We then have to show that F is well defined,

injective and surjective. Since the proofs of these facts are similar, we will assume

that the definition of F does not depend on the choice of the representative v of

[v] and we will only prove that it does not depend on the choice of u neither. Let

[u : X → φ(i ⁰ )] = [u ⁰ : X → φ(j ⁰ )] in B _X and let be given morphisms i ⁰ → ι(i),

j ⁰ → ι(j) in I ⁰ . In what follows × −→ will denote a morphism induced by a morphism in I ⁰ and ◦ −→ a morphism induced by one of I. [u] = [u ⁰ ] means that there is a commutative diagram:

φ(i ⁰ )

% & +

X φ(k ⁰ ).

& % + φ(j ⁰ )

Due to (a) of Definition 1.4.2 and to the fact that I and I ⁰ are filtrant, it is then easy to get the following commutative diagram:

φ(i ⁰ ) −→ × φ(ι(i)) −→ × φ(ι(l))

% & + % + & +

X φ(k ⁰ ) φ(ι(n)),

& % + & + % +

φ(j ⁰ ) −→ φ(ι(j)) × × −→ φ(ι(m)) i.e. we have a commutative diagram:

φ(ι(i))

% & +

X φ(ι(n)).

& % +

φ(ι(j))

Using (b) of Definition 1.4.2 one then easily get the diagram:

φ(ι(i)) −→ ◦ φ(ι(o)) φ(ι(s))

% & + + % & ◦ % ◦ × ↑

X φ(ι(n)) c φ(ι(q)) × −→

ε φ(ι(r)),

& % + + & % ◦ φ(ι(j)) −→ ◦ φ(ι(p))

where all the diagrams, except c, are commutative. Nevertheless ε ◦ c is commuta- tive. Hence we have the commutative diagram:

φ(ι(i))

% & ◦

X φ(ι(s)),

& % ◦

φ(ι(j)) which means that F ([u]) = F ([u ⁰ ]). q.e.d.

2. Microlocal inverse image theorem.

2.1 Microhyperbolic theorem for sheaves. Let f : Y → X be a morphism of manifolds. Let M (resp. N ) be a closed submanifold of X (resp. Y ), with f (N ) ⊂ M .

In [K-S 4] (or [K-S 3]) the main result on the comparison between inverse image

and microlocalization is the following.

Theorem 2.1.1. (cf [K-S 4, Theorem 6.7.1] or [K-S 3, Theorem 5.4.1].) Let V be an open subset of T _N ^∗ Y and let F ∈ Ob(D ^b (X)). Assume:

(i) f is non-characteristic for F on V ,

(ii) f _{N π} is non-characteristic for C _T

_X (SS(F )) on ^t f ⁰⁻¹ _N (V ), (iii) ^t f ⁰⁻¹ (V ) ∩ f _π ⁻¹ (SS(F )) ⊂ Y × _X T _M ^∗ X.

Then the natural morphism:

(2.1.1) µ _N (f ^! F ) 

 V −→ R ^t f _{N ∗} ⁰ f _{N π} ^! µ _M (F )   V , is an isomorphism.

2.2. Inverse image for µhom. In this section we aim at giving our main result, i.e. Theorem 2.2.3 below, which is a variation of Theorem 2.1.1. To this end, let us define microlocal images.

Let f : Y → X be a morphism. Let p ∈ Y × _X T ^∗ X and set p _X = f _π (p), p Y = ^t f ⁰ (p).

Definition 2.2.1. Let F be an object of D ^b (X). We denote by Proj _F (p _X ) (resp.

Ind _F (p X )) the filtrant category whose objects consist of the morphisms u : F ⁰ → F (resp. u : F → F ⁰ ) in D ^b (X) which are isomorphisms at p _X . A morphism (u : F ⁰ → F ) → (u ⁰ : F ⁰⁰ → F ) of Proj _F (p X ) is defined by a morphism v : F ⁰⁰ → F ⁰ in D ^b (X) with u ⁰ = u ◦ v (and similarly for Ind _F (p _X )).

Definition 2.2.2. (cf [K-S 4, Definition 6.1.7].)

(i) Let F ∈ Ob(D ^b (X; p _X )). One denotes by f _µ,p ⁻¹ F (resp. f _µ,p ^! F ) the pro- object (resp. ind-object) “lim”

←−

Proj

(p

)

f ⁻¹ F ⁰ (resp. “lim”

−→

Ind

(p

)

f ^! F ⁰ ). Here f ⁻¹ is the functor from Proj _F (p _X ) to D ^b (Y ; p _Y ) which associates f ⁻¹ F ⁰ to F ⁰ → F (and similarly for f ^! ). One calls f _µ,p ⁻¹ F the microlocal inverse image of F at p.

(ii) Let G ∈ Ob(D ^b (Y ; p _Y )). One denotes by f _! ^µ,p G (resp. f ∗ ^µ,p G) the pro- object (resp. ind-object) “lim”

←−

Proj

(p

)

Rf _! G ⁰ (resp. “lim”

−→

Ind

(p

)

Rf _∗ G ⁰ ). Here Rf _! is the functor from Proj _G (p _Y ) to D ^b (X; p _X ) which associates Rf _! G ⁰ to G ⁰ → G (and similarly for Rf _∗ ). One calls f _∗ ^µ,p G the microlocal direct image of G at p.

From now on, for a given p ∈ Y × _X T ^∗ X we will set p _X = f _π (p) and p _Y = ^t f ⁰ (p).

We shall now give a variation of Theorem 2.1.1.

For F and K objects of D ^b (X), there is a natural morphism:

(2.2.1) µhom(f ⁻¹ K, f ^! F ) −→ R ^t f _∗ ⁰ f _π ^! µhom(K, F ).

Theorem 2.2.3. Let F and K be objects of D ^b (X) and take p ∈ Y × _X T ^∗ X. Let V be an open neighborhood of p Y and assume:

(i) p / ∈ T _Y ^∗ X,

(ii) f _µ,p ⁻¹ K and f _µ,p ^! F are representable in D ^b (Y ; p _Y ),

(iii) f _π is non-characteristic for C(SS(F ), SS(K)) on ^t f ⁰⁻¹ (V ).

Then the morphism (2.2.1) induces an isomorphism:

(2.2.2) µhom(f _µ,p ⁻¹ K, f _µ,p ^! F ⊗ ω _Y ^⊗−1 ) p

−→ µhom(K, F ⊗ ω ∼ _X ^⊗−1 ) p

In the left hand side of (2.2.2) we consider µhom acting microlocally as remarked in Proposition 1.3.1. Taking the germ at p Y we get a bifunctor µhom : D ^b (Y ; p Y ) ^◦ × D ^b (Y ; p _Y ) → D ^b (Mod(A)). Hence the isomorphism in (2.2.2) holds in D ^b (Mod(A)), the derived category of the category of A-modules.

Let us explain how the morphism (2.2.2) is deduced from (2.2.1). Consider the maps:

T ^∗ Y

t

f

←−−−− Y × _X T ^∗ X −−−−→ T ^f

^∗ X

π





y ^π





y ^π



 y

Y ←−−−− ^∼ Y −−−−→ X. ^f

By adjunction, the morphism (2.2.1) induces the morphism:

(2.2.3) ^t f ⁰⁻¹ µhom(f ⁻¹ K, f ^! F ) −→ f _π ^! µhom(K, F ).

By (iii), the natural morphism: f _π ⁻¹ µhom(K, F )⊗π ⁻¹ ω _{Y /X} → f _π ^! µhom(K, F ) is an isomorphism on ^t f ⁰⁻¹ (V ) (cf [K-S 3, Proposition 5.3.2]). Composing (2.2.3) with the inverse of this last morphism and recalling that π ⁻¹ ω _{Y /X} ∼ = π ⁻¹ ω Y ⊗ π ⁻¹ f ⁻¹ ω ^⊗−1 _X we then get the morphism:

t f ⁰⁻¹ µhom(f ⁻¹ K, f ^! F ⊗ ω _Y ^⊗−1 ) −→ f _π ⁻¹ µhom(K, F ⊗ ω _X ^⊗−1 ).

Taking the fiber at p and via the natural morphisms f _µ,p ^! F → f ^! F and f _µ,p ⁻¹ K ← f ⁻¹ K, we obtain the morphism of (2.2.2).

In order to prove Theorem 2.2.3, we shall need Theorem 2.2.4 below.

Let M (resp. N ) be a closed submanifold of X (resp. Y ) such that f (M ) ⊂ N . For p ∈ N × _M T _M ^∗ X we will denote p _X = f _{N π} (p), p _Y = ^t f _N ⁰ (p), coherently with the previous notations. Recall that for F ∈ Ob(D ^b (X)) there is a natural morphism corresponding to (2.2.1):

(2.2.4) µ _N (f ^! F ) −→ R ^t f _{N ∗} ⁰ f _{N π} ^! µ _M (F ).

Theorem 2.2.4. Let F ∈ Ob(D ^b (X)) and take p ∈ N × M T _M ^∗ X. Let V be an open neighborhood of p _Y in T _N ^∗ Y and assume:

(i) p / ∈ T _Y ^∗ X,

(ii) f _µ,p ^! F is representable,

(iii) f _{N π} is non-characteristic for C _T

_X (SS(F )) on ^t f ⁰⁻¹ _N (V ).

Then the morphism (2.2.4) induces an isomorphism:

(2.2.5) µ N (f _µ,p ^! F ⊗ ω _Y ^⊗−1 ) p

−→ µ ∼ _M (F ⊗ ω _X ^⊗−1 ) p

.

The isomorphism holds once more in D ^b (Mod(A)), and (2.2.5) is deduced from

(2.2.4) similarly as (2.2.2) was deduced from (2.2.1).

2.3 A particular case. Let us first recall some results of [K-S 4] on microlocal images. Let f : Y → X be a morphism of manifolds and take p ∈ Y × _X T ^∗ X. Set p X = f π (p) and p Y = ^t f ⁰ (p).

Proposition 2.3.1. (cf [K-S 4, Proposition 6.1.8].) Let F ∈ Ob(D ^b (X; p _X )) and G ∈ Ob(D ^b (Y ; p _Y )). The following equalities hold:

Hom _D

_(X;p

₎

(f _! ^µ,p G, F ) = Hom _D

_{(Y ;p}

₎

(G, f _µ,p ^! F ), (2.3.1)

Hom _D

_(X;p

₎

(G, f _∗ ^µ,p F ) = Hom _D

_{(Y ;p}

₎

(f _µ,p ⁻¹ G, F ).

(2.3.2)

Moreover there are canonical morphisms:

f _! ^µ,p G −→ f _∗ ^µ,p G, (2.3.3)

f _µ,p ⁻¹ F ⊗ ω _{Y /X} −→ f _µ,p ^! F.

(2.3.4)

Proposition 2.3.2. (cf [K-S 4, Proposition 6.1.10].) Let G ∈ Ob(D ^b (Y )). If supp(G) is proper over X and if:

(2.3.5) f _π ⁻¹ (p _X ) ∩ ^t f ⁰⁻¹ (SS(G)) ⊂ {p},

then f _! ^µ,p G and f _∗ ^µ,p G are representable and one has the isomorphisms:

f _! ^µ,p G ∼ = f _∗ ^µ,p G ∼ = Rf ∗ G, in D ^b (X; p X ).

We are now ready to prove a particular case of Theorem 2.2.4.

Proposition 2.3.3. Let f : Y → X be a closed embedding and set M = f (N ).

Take a point p ∈ N × _M T _M ^∗ X ∼ = T _M ^∗ X and set p _X = f _{N π} (p), p _Y = ^t f _N ⁰ (p). Let F ∈ Ob(D ^b (X)) and assume that f _µ,p ^! F is representable. Then the natural morphism (2.2.4) induces an isomorphism:

µ _N (f _µ,p ^! F ) _p

−→ µ ^∼ _M (F ) _p

.

Proof. It is enough to prove the isomorphism for the cohomology groups. One has:

H ^j µ N (f _µ,p ^! F ) p

∼ = Hom D

(Y ;p

) (A N , f _µ,p ^! F [j])

∼ = Hom _D

_{(Y ;p}

₎

(A _N , f _µ,p ^! F [j])

∼ = Hom _D

_(X;p

₎

(f _! ^µ,p A _N , F [j])

∼ = Hom _D

_(X;p

₎ (A _M , F [j])

∼ = H ^j µ _M (F ) _p

.

Here the first isomorphism follows from [K-S 4, Th. 6.1.2], the second expresses

the fact that D ^b (Y ; p Y ) is a full subcategory of D ^b (Y ; p Y ) ^∨ , the third follows from

Proposition 2.3.1 and the forth from the fact that, since f _π is injective, we can

apply Proposition 2.3.2 and get: f _! ^µ,p A N = Rf ∗ A N = A M . q.e.d.

2.4. The microlocal cut-off lemma. First let us recall the definition of cutting functors as it has been given in [K-S 4, chap. 6].

Since we are concerned with problems of a local nature, we will assume X being a vector space. In what follows we will often identify X with T ₀ X.

Let γ be a (not necessarily proper) closed convex cone of T 0 X. Let ω be an open neighborhood of 0 in X with smooth boundary. We shall denote by q ₁ and q ₂ the projections from X × X to X and by s the map s(x ₁ , x ₂ ) = x ₁ − x ₂ . The following definition is a slight modification of that of [K-S 4, Prop. 6.1.4, 6.1.8].

Definition 2.4.1. Let γ and ω be as above and let F be an object of D ^b (X). We set:

Φ X (γ, ω; F ) = Rq 2∗ (s ⁻¹ A γ

⊗ q L ⁻¹ ₁ F ω ), Ψ _X (γ, ω; F ) = Rq _2! RΓ _s

_(γ

₎ (q ₁ ^! RΓ _ω (F )).

Notice that for γ ⁰ ⊂ γ, ω ⁰ ⊃ ω, one has the following natural morphisms in D ^b (X):

Φ X (γ, ω; F ) −→ Φ X (γ ⁰ , ω ⁰ ; F ), Ψ X (γ ⁰ , ω ⁰ ; F ) −→ Ψ X (γ, ω; F ).

(2.4.1)

In particular, recalling the isomorphisms

Rq _2∗ (s ⁻¹ A _{0} ⊗ q ^L ₁ ⁻¹ F ) → F, ^∼ F → ^∼ Rq 2! RΓ _s

₍₀₎ (q ₁ ^! F ), we get natural morphisms:

Φ _X (γ, ω; F ) −→ F,

F −→ Ψ X (γ, ω; F ).

(2.4.2)

One has the following result.

Proposition 2.4.2. (cf [K-S 4, Th. 5.2.3] or [K-S 3, Prop. 3.2.2]) With the same notations as above:

a) SS(F ) is contained in ω × γ ^◦a if and only if the morphism Φ _X (γ, ω; F ) → F (resp. F → Ψ _X (γ, ω; F )) is an isomorphism.

b) Φ X (γ, ω; F ) → F (resp. F → Ψ X (γ, ω; F )) is an isomorphism on ω × Int γ ^◦a .

In particular one has the following estimates:

SS(Φ _X (γ, ω; F )) ⊂ ω × γ ^◦a , SS(Ψ _X (γ, ω; F )) ⊂ ω × γ ^◦a .

In order to give a sharper result on the cutting of the micro-support one should

take care of the relation between γ and ω. Refining [K-S 4, Prop. 6.1.4], we give

the following definition:

Definition 2.4.3. Take ξ 0 ∈ ˙ T ₀ ^∗ X. Let γ ⊂ T 0 X and ω ⊂ X be such that:

(i) γ is a closed proper convex cone, (ii) ∂γ \ {0} is C ¹ ,

(iii) ξ ₀ ∈ Int γ ^◦a ,

(iv) ω is an open neighborhood of 0, (v) ∂ω is C ¹ ,

(vi) ω ⊂ {x; |x| < ε} for some ε > 0, (vii) ∀x ∈ ∂ω ∩ ∂γ, N _x ^∗ (ω) ^a = N _x ^∗ (γ).

We will call a pair (γ, ω) satisfying (i)–(vii) a refined cutting pair on X at (0; ξ ₀ ).

Note that since ∂ω and ∂γ are smooth, condition (vii) means that ∂ω and ∂γ are tangent at their intersection. More precisely, if g(x) < 0 (resp. h(x) < 0) is a local equation for ω (resp. γ) at x ∈ ∂ω ∩ ∂γ, this means that −d g(x) ∈ R ⁺ d h(x).

Let S be a vector space and take p _S ∈ T ₀ ^∗ S. If (γ, ω) is a refined cutting pair on X at (0; ξ 0 ), and if ω is defined by ω = {x; g(x) < 0} for a C ¹ function g with d g 6= 0, we can find an open neighborhood ω _S of 0 in X × S with smooth boundary such that:

(2.4.3)

 ω _S = {(x, s) ∈ X × S; hs, p _S i + g(x) < 0} near X × {0}, ω _S ⊂ {(x, s); |(x, s)| < ε}

The following proposition is an extension of Proposition 6.1.4 of [K-S 4].

Proposition 2.4.4. Let H ∈ Ob(D ^b (X × S)) and let (γ, ω) be a refined cutting pair on X at (0; ξ ₀ ), ξ ₀ 6= 0. Take p _S ∈ T ₀ ^∗ S and set H ⁰ = Φ _X×S (γ × {0}, ω _S ; H) (resp. H ⁰ = Ψ _X×S (γ × {0}, ω _S ; H)) for ω _S defined as in (2.4.3). The following estimate holds:

SS(H ⁰ ) ∩ (π _X ⁻¹ (0) × {p _S }) ⊂

({ξ ∈ γ ^◦a \ {0}; ∃x ∈ ω : ((x; ξ), p _S ) ∈ SS(H)} ∪ {0}) × {p S }.

We will give a proof based on the same line as the one of [K-S 4, Prop. 6.1.4].

Proof. By Proposition 2.4.2 we know that H ∼ = H ⁰ on ω _S × Int ((γ × {0}) ^◦a ) and that SS(H ⁰ ) ⊂ ω _S × (γ × {0}) ^◦a . It then remains to show that

(2.4.4)

ξ ∈ ∂((γ × {0}) ^◦a ) \ {0}, ((0; ξ), p _S ) ∈ SS(H ⁰ )

⇓

∃(x; ξ); x ∈ ω, ((x; ξ), p _S ) ∈ SS(H).

The map q ₂ : supp(s ⁻¹ A _γ×{0} ⊗ q ^L ₁ ⁻¹ H _ω

) → X × S is proper due to (2.4.3) and (i) of Definition 2.4.3. One may then apply Propositions 5.4.4, 5.4.5 and 5.4.14 of [K-S 4] and get the estimate:

(2.4.5)

((0; ξ), p S ) ∈ SS(H ⁰ )

⇓

∃x : ((x; ξ), p _S ) ∈ SS(A _γ×{0} ) ^a ∩ SS(H _ω

).

Let us then prove (2.4.4) using (2.4.5). Since ξ 6= 0 and ((x; ξ), p _S ) ∈ SS(A _γ×{0} ) ^a ,

we have x ∈ ∂γ.

If x ∈ X \ ω then SS(H ω

) ∩ π _X×S ⁻¹ ((x, 0)) = ∅.

If x ∈ ω then H ω

∼ = H at (x, 0).

If x ∈ ∂ω, by (vii) of Definition 2.4.3 we get: N _(x,0) ^∗ (ω _S ) ∩ N _(x,0) ^∗ (γ × {0}) ^a = R _≥0 (ξ, p S ). Assume ((x; ξ), p S ) / ∈ SS(H), then one may estimate SS(H _ω

) as

SS(H _ω

) ∩ π _X×S ⁻¹ ((x, 0)) ⊂ −R _≥0 (ξ, p _S ) + (SS(H) ∩ π _X×S ⁻¹ ((x, 0)))

which implies ((x; ξ), p _S ) ∈ SS(H). This is a contradiction and this complete the proof. q.e.d.

Corollary 2.4.5. (cf [K-S 4, Prop. 6.1.4, 6.1.8]) Keep the same notations as above.

Let K be a proper closed convex cone of T ₀ ^∗ X and let U ⊂ K be an open cone. Let F ∈ Ob(D ^b (X)) and let W be a conic neighborhood of K ∩ (SS(F ) \ {0}). Then:

a) (Refined microlocal cut-off lemma). There exists F ⁰ ∈ Ob(D ^b (X)) and a morphism u : F ⁰ → F satisfying:

(1) u is an isomorphism on U ; (2) π ⁻¹ _X (0) ∩ SS(F ⁰ ) ⊂ W ∪ {0}.

b) (Dual refined microlocal cut-off lemma). Same as a) with u : F → F ⁰ .

Proof. It is not restrictive to assume U ⊂ {0} ∪ Int K. Take ξ ₀ ∈ U and choose a refined cutting pair (γ, ω) on X at (0; ξ ₀ ) with K ^◦a ⊂ γ ⊂ U ^◦a . It then remains to apply Proposition 2.4.4 to the case S = {pt}. q.e.d.

2.5 Complements on the microlocal inverse image. As a preparation to the proof of the theorems of §2.2 we need to give some results concerning microlocal inverse images.

Let f : Y → X be a morphism of manifolds. Take p ∈ Y × _X T ^∗ X and set p _X = f _π (p), p _Y = ^t f ⁰ (p). Assume p _X ∈ T / _X ^∗ X. Set x ₀ = π _X (p _X ), y ₀ = π _Y (p _Y ). Fix a local system of coordinates (x) ∈ X in a neighborhood of x ₀ = π _X (p _X ) and let (x; ξ) be the associated symplectic coordinates in T ^∗ X. Since all statement in what follows are of a local nature, we may assume X is a vector space. Let p _X = (x ₀ ; ξ ₀ ) and recall that we assumed ξ ₀ 6= 0. Let γ ⊂ T _x

X be a cone and let ω ⊂ X be an open set such that:

(2.5.1)

 ξ ₀ ∈ Int γ ^◦a , x ₀ ∈ ω.

Let Cut X (p X ) be the category whose objects are the pairs (γ, ω) satisfying (2.5.1) and whose morphisms are defined as:

Hom _Cut

_(p

₎ ((γ, ω), (γ ⁰ , ω ⁰ )) =

 {>} if γ ⊃ γ ⁰ , ω ⊂ ω ⁰ ,

∅ otherwise.

Let φ _F (p _X ) : Cut _X (p _X ) → Proj _F (p _X ) be the functor associating to an object (γ, ω) of

Cut _X (p _X ) the morphism u : Φ _X (γ, ω; F ) → F defined in (2.4.2) (note that u belongs

to Ob(Proj _F (p _X )) due to Proposition 2.4.2) and to a morphism (γ, ω) > (γ ⁰ , ω ⁰ ) the

morphism defined in (2.4.1). Similarly, let ψ _F (p _X ) : Cut _X (p _X ) → Ind _F (p _X ) be

defined by ψ _F (p _X )((γ, ω)) = (F → Ψ _X (γ, ω; F )).

Proposition 2.5.1. Let F ∈ Ob(D ^b (X)) and take p ∈ Y × X T ^∗ X \ T _Y ^∗ X. The following isomorphisms hold:

f _µ,p ⁻¹ F ∼ = “lim”

←−

Cut

(p

)

f ⁻¹ Φ _X (γ, ω; F ) (i)

f _µ,p ^! F ∼ = “lim”

−→

Cut

(p

)

f ^! Ψ _X (γ, ω; F ) (ii)

Here f ⁻¹ Φ X (γ, ω; F ) is the functor from Cut X (p X ) to D ^b (Y ; p Y ) which associates the object f ⁻¹ Φ _X (γ, ω; F ) to (γ, ω) ∈ Ob(Cut _X (p _X )).

Proof. Since the proofs of (i) and (ii) are similar we will treat only the case (i).

Denote by f ⁻¹ F ⁰ the functor from Proj _F (p _X ) to D ^b (Y ; p _Y ) which associates f ⁻¹ F ⁰ to F ⁰ → F . Due to Proposition 1.4.3 we have to show that Cut _X (p X ) and Proj _F (p X ) are cofinal with respect to f ⁻¹ F ⁰ by φ _F (p _X ). Let u : F ⁰ → F be an object of Proj _F (p X ). In order to prove that (a) of Definition 1.4.2 holds, we have to find a pair (γ, ω) satisfying (2.5.1) and a morphism f ⁻¹ Φ _X (γ, ω; F ) → f ⁻¹ F ⁰ in D ^b (Y ; p Y ). To this end, embed u in a distinguished triangle F ⁰ → F → F ^u ₀ ⁺¹ → . Since u ∈ Proj _F (p _X ), we have p _X ∈ SS(F / ₀ ). Take a proper closed convex cone K and an open convex cone U such that p X ∈ U ⊂ K and SS(F ₀ )∩K ⊂ {0}. Following the proof of Corollary 2.4.5 we can find a refined cutting pair (γ, ω) on X at p _X such that f is non-characteristic for Φ X (γ, ω; F 0 ) at x 0 and f _π ⁻¹ SS(Φ X (γ, ω; F 0 )) ∩

t f ⁰⁻¹ (p _Y ) = ∅. Hence p _Y ∈ SS(f / ⁻¹ Φ _X (γ, ω; F ₀ )) and this means that the morphism v : f ⁻¹ Φ X (γ, ω; F ⁰ ) → f ⁻¹ Φ X (γ, ω; F ), obtained by applying f ⁻¹ Φ X (γ, ω; ·) to u, is an isomorphism at p _Y . Composing, in D ^b (Y ; p _Y ), v ⁻¹ with the natural morphism f ⁻¹ Φ X (γ, ω; F ⁰ ) → f ⁻¹ F ⁰ , we get the desired morphism f ⁻¹ Φ X (γ, ω; F ) → f ⁻¹ F ⁰ . As for (b) of Definition 1.4.2 we have to show that for any (γ, ω) as in (2.5.1), any (F → F ⁰ ) ∈ Ob(Proj _F (p X )) and any morphism u : F ⁰ → Φ _X (γ, ω; F ), there exists (γ ⁰ , ω ⁰ ) ∈ Ob(Cut _X (p _X )) such that the natural morphism Φ _X (γ ⁰ , ω ⁰ ; F ) → Φ X (γ, ω; F ) obtained from (2.4.1) factors as:

f ⁻¹ Φ X (γ ⁰ , ω ⁰ ; F ) −→ f ⁻¹ Φ X (γ, ω; F )

& %

f ⁻¹ F ⁰

Reasoning as for part (a), one can find a refined cutting pair (γ ⁰ , ω ⁰ ) > (γ, ω) so that the natural morphisms:

f ⁻¹ Φ X (γ ⁰ , ω ⁰ ; F ⁰ ) −→ f ⁻¹ Φ X (γ ⁰ , ω ⁰ ; Φ X (γ, ω; F ))

−→ f ⁻¹ Φ X (γ ⁰ , ω ⁰ ; F ),

are isomorphisms at p _Y . Composing, in D ^b (Y ; p _Y ), the inverse of this composite with the natural arrow f ⁻¹ Φ _X (γ ⁰ , ω ⁰ ; F ⁰ ) → f ⁻¹ F ⁰ , we get the claim. q.e.d.

Let S be another manifold and consider the map:

f = f × id ˆ _S : Y × S −→ X × S.

We will identify T ^∗ (X × S) with T ^∗ X × T ^∗ S. For p ∈ Y × _X T ^∗ X and p _S ∈ T ^∗ S, set ˆ p = (p, p _S ) and define p _X = f _π (p), p _Y = ^t f ⁰ (p), ˆ p _X = ˆ f _π (p) and ˆ p _Y = ^t f ˆ ⁰ (p).

Set x ₀ = π _X (p _X ), y ₀ = π _Y (p _Y ), s ₀ = π _S (p _S ). Fix a local system of coordinates (s)

on S and denote by (s; σ) the associated symplectic coordinates on T ^∗ S.

Proposition 2.5.2. Let F ∈ Ob(D ^b (X)), G ∈ Ob(D ^b (S)) and take p ∈ Y × X

T ^∗ X \ T _Y ^∗ X, p _S ∈ T ^∗ S. Assume that f _µ,p ⁻¹ F (resp. f _µ,p ^! F ) is representable. Then the following isomorphisms hold in D ^b (Y × S; ˆ p _Y ):

f ˆ _{µ, ˆ} ⁻¹ _p (F × G) ∼ ^L = (f _µ,p ⁻¹ F ) × G, ^L (i)

f ˆ _{µ, ˆ} ^! _p RHom(q ₁ ⁻¹ F, q ₂ ^! G) ∼ = RHom(q ⁻¹ ₁ f _µ,p ⁻¹ F, q ^! ₂ G) (ii)

(resp.

(iii) f ˆ _{µ, ˆ} ^! _p RHom(q ⁻¹ ₂ G, q ^! ₁ F ) ∼ = RHom(q ⁻¹ ₂ G, q ^! ₁ f _µ,p ^! F )).

Here q ₁ and q ₂ denote the projections from X × S to X and S respectively and we remark that (i)–(iii) make sense due to Proposition 1.3.1.

Proof. Since the proofs are similar we will treat only the case (i). For a pair (γ, ω) ∈ Ob(Cut _X (p _X )) and an open subset ω ⁰ ⊂ S, it is easy to check that

f ⁻¹ Φ X (γ, ω; F ) × G ^L _ω

∼ = ˆ f ⁻¹ Φ X×S (γ × {0}, ω × ω ⁰ ; F × G). ^L We then have the isomorphisms in D ^b (Y × S; ˆ p Y ):

(f _µ,p ⁻¹ F ) × G ∼ ^L = ( “lim”

←−

Cut

(p

)

f ⁻¹ Φ X (γ, ω; F )) × (“lim” ^L

←−

ω

G ω

)

∼ = “lim”

←−

Cut

(p

)×ω

(f ⁻¹ Φ _X (γ, ω; F ) × G ^L _ω

)

∼ = “lim”

←−

Cut

(p

)×ω

f ˆ ⁻¹ Φ _X×S (γ × {0}, ω × ω ⁰ ; F × G). ^L

Here ω ⁰ ranges over an open neighborhood system of s ₀ . Notice that the first isomorphism follows from Proposition 2.5.1 and the second one from Proposition 1.4.1.

We need now a lemma.

Lemma 2.5.3. Keeping the same notations as above and for ω _S as in (2.4.3), the following isomorphism holds:

f ˆ _{µ, ˆ} ⁻¹ _p (H) ∼ = “lim”

←−

Cut

(p

)

f ˆ ⁻¹ Φ X×S (γ × {0}, ω S ; H).

Proof. Let be given a morphism in D ^b (X) H ⁰ → H which is an isomorphism at p _X . Let H ₀ be the third term of a distinguished triangle: H ⁰ → H → H ₀ ⁺¹ →.

By the same proof as in Proposition 2.5.1 it is enough to show that there exists a pair (γ, ω) ∈ Ob(Cut X (p X )) such that ˆ p Y ∈ ˆ / f ⁻¹ Φ X×S (γ × {0}, ω S ; H 0 ). For that purpose it is enough to prove that SS(Φ _X×S (γ × {0}, ω _S ); H ₀ )) ∩ ˆ f _π ^t f ˆ ⁻¹ (ˆ p _Y ) ⊂ {0}.

Since ˆ f _π ^t f ˆ ⁻¹ (ˆ p _Y ) = f _π ^t f ⁰⁻¹ (p _Y ) × {p _S }, this follows from Proposition 2.4.4. q.e.d.

End of the proof of Proposition 2.5.2. The only thing which is left to prove is the isomorphism

“lim”

←−

Cut

(p

)

f ˆ ⁻¹ Φ X×S (γ × {0}, ω S ; F × G) ∼ ^L =

∼ = “lim”

←−

Cut

(p

)×ω

f ˆ ⁻¹ Φ _X×S (γ × {0}, ω × ω ⁰ ; F × G), ^L

but this follows from the fact that both ω S and ω × ω ⁰ describe a fundamental neighborhood system of (x ₀ , s ₀ ). q.e.d.

Let g : Y → X × S be a morphism of manifolds. Consider the composite:

f : Y −→ X × S ^g −→ X. ^q

For p ∈ Y × _X T ^∗ X we will set p _Y = ^t f ⁰ (p), p _X = f _π (p), y ₀ = π _Y (p _Y ) ∈ Y , (x ₀ , s ₀ ) = g(y ₀ ), ˆ p _X = ^t q ₁ ⁰ ((x ₀ , s ₀ ), p _X ) ∈ T ^∗ (X × S) and ˆ p = (y ₀ , ˆ p _X ) ∈ Y × _(X×S) T ^∗ (X × S).

Proposition 2.5.4. Let F ∈ Ob(D ^b (X)) and take p ∈ Y × X T ^∗ X \ T _Y ^∗ X. The following isomorphisms hold:

f _µ,p ⁻¹ F ∼ = g _{µ, ˆ} ⁻¹ _p (q ₁ ⁻¹ F ), (i)

f _µ,p ^! F ∼ = g _{µ, ˆ} ^! _p (q ₁ ^! F ).

(ii)

Proof. Since the proofs are similar we will treat only the case (i). Due to Proposition 2.5.1 we have to prove the isomorphism:

“lim”

←−

Cut

(p

)

g ⁻¹ q ₁ ⁻¹ Φ X (γ, ω; F ) ∼ = “lim”

←−

Cut

( ˆ p

)

g ⁻¹ Φ X×S (Γ, Ω; q ₁ ⁻¹ F ).

Let ˆ  : Cut X (p X ) → Cut X×S (ˆ p X ) be the functor of filtrant categories defined by ˆ

((γ, ω)) = (γ × {0}, ω × S) for (γ, ω) ∈ Ob(Cut _X (p _X )). One has the following evident isomorphism:

Φ _X×S (γ × {0}, ω × S; F × A ^L _S ) ∼ = q ⁻¹ ₁ Φ _X (γ, ω; F ),

and hence the proposition is proven if we show that Cut _X (p _X ) and Cut _X×S (ˆ p _X ) are cofinal with respect to g ⁻¹ Φ _X×S (Γ, Ω; q ₁ ⁻¹ F ) by ˆ . To this end it is enough to prove that they are cofinal by ˆ . In order to prove that (a) of Definition 1.4.3 holds, for a given (Γ, Ω) ∈ Ob(Cut _X×S (ˆ p _X )), we have to find (γ, ω) ∈ Ob(Cut _X (p _X )) and a morphism Φ X×S (γ ×{0}, ω ×S; F × A ^L _S ) → Φ X×S (Γ, Ω; F × A ^L _S ) in D ^b (X × S; (x ₀ , s ₀ )). It is not restrictive to assume (Γ, Ω) being a refined cutting pair on X × S at ˆ p _X . Consider a distinguished triangle: Φ _X×S (Γ, Ω; F × A ^L _S ) → F × A ^L _S → H ⁺¹ →. Choose a refined cutting pair (γ, ω) on X at p _X such that

- {(x, s ₀ )} × (γ × {0}) ^◦a ∩ SS(H) ⊂ {0} for x ∈ ω,

- N _x ^∗ (ω) ⊂ (γ × {0}) ^◦a ∀x ∈ ω ∩ γ.

Set H ⁰ = Φ X×S (γ × {0}, ω × S; H). Due to [K-S 4, Proposition 5.4.8] we have the estimate: SS(H _ω×S ) ⊂ N ^∗ (ω × S) ^a + SS(H). Due to (vii) of Definition 2.4.3, we have: N _(x,s ^∗

₎ (ω × S) = N _x ^∗ (γ) ^a × {0} for any x ∈ ∂γ ∩ ∂ω. and hence we get the estimate:

SS(H _ω×S ) ∩ (π ⁻¹ _X (x ₀ ) × {s ₀ }) ∩ (γ × {0}) ^◦a ⊂ {0} ∀x ∈ ∂γ ∩ ∂ω.

From the estimate:

SS(H ⁰ ) ∩ (π _X ⁻¹ (x ₀ ) × {s ₀ }) ⊂ SS(A _γ×{0} ) ^a ∩ SS(H _ω×S ), we then get:

SS(H ⁰ ) ∩ (π ⁻¹ _X (x 0 ) × {s 0 }) ⊂ {0},

and hence H ⁰ is a complex of constant sheaves. Moreover, since the stalks at (x ₀ , s ₀ ) of both sides of the morphism:

Φ _X×S (γ × {0}, ω × S;Φ _X×S (Γ, Ω; F × A ^L _S )) −→

−→ Φ _X×S (γ × {0}, ω × S; F × A ^L S ), (2.5.2)

are isomorphic to the stalk of F × A ^L _S at (x ₀ , s ₀ ), then H ⁰ = 0 at (x ₀ , s ₀ ). This means that (2.5.2) is an isomorphism at (x ₀ , s ₀ ) and to conclude it is then enough to compose the inverse of this morphism with the natural morphism

Φ X×S (γ × {0}, ω × S; Φ X×S (Γ, Ω; F × A ^L _S )) → Φ X×S (Γ, Ω; F × A ^L _S ).

Part (b) of Definition 1.4.2 is similarly proven. q.e.d.

2.6 Proof of the theorems. We are now ready to prove the theorems stated in

§2.2.

Proof of Theorem 2.2.4. Let us decompose f as:

Y −−−−→ Y × X ^j −−−−→ X ^q x





x





x



 N −−−−→ ^∼ N −−−−→ M,

where j is the graph map, q denotes the second projection and we identified N and j(N ). We will divide the proof in several steps.

The first step will concern the map q for which we shall use Theorem 2.1.1.

Remark that f _{N π} = (j × _M id _T

_X )◦q _{N π} . Then one checks easily that the hypothesis (iii) of Theorem 2.2.4 implies the corresponding hypothesis:

(iii)’ there is an open neighborhood W of (y ₀ , p _X ) in T _N ^∗ (Y × X) such that q _{N π} is non-characteristic for C _T

M

X (SS(F )) on ^t q ⁰⁻¹ _N (W ).

Here y 0 is the projection of p Y on Y .

Since q is smooth the hypotheses of Theorem 2.1.1 are all satisfied. Applying this theorem we get:

µ _N (q ^! F ) _(y

,p

) ∼

−→ R ^t q _{N ∗} ⁰ q ^! _{N π} µ _M (F ) _(y

,p

) . Moreover, since ^t q _N ⁰ is a closed embedding one has the isomorphisms:

R ^t q _{N ∗} ⁰ q _{N π} ^! µ _M (F ) _(y

,p

) ∼ = (q _{N π} ^! µ _M (F )) _(y

,p

)

∼ = µ M (F ) _p

⊗ ω _N/M . One then gets:

(2.6.1) µ _N (q ^! F ) _(y

,p

) ∼

−→ µ _M (F ) _p

⊗ ω _N/M .

As for the second step let us apply Proposition 2.3.3 to the closed embedding j.

We get the isomorphism:

(2.6.2) µ N (j _{µ, ˆ} ^! _p q ^! F ) p

−→ µ ∼ _N (q ^! F ) _(y

,p

) ,

where ˆ p = (y ₀ , ^t q ⁰ ((y ₀ , f (y ₀ )), p _X )). Notice that in Y × _{(Y ×X)} (T ^∗ Y × T ^∗ X), ˆ p is written as ˆ p = (y ₀ , (y ₀ , p _X )). Finally remark that

(2.6.3) f _µ,p ^! F ∼ = j _{µ, ˆ} ^! _p q ^! F

due to Proposition 2.5.4. By combining (2.6.1), (2.6.2) and (2.6.3) the proof is complete. q.e.d.

Proof of Theorem 2.2.3. Decompose the map ˜ f = f × f as follows:

Y × Y

2

f

−−−−→ Y × X

1

f

−−−−→ X × X x





x





x





∆ Y −−−−→ ∆ −−−−→ ∆ X ,

where ² f = id _Y × f , ¹ f = f × id _X , ∆ _Y is the diagonal of Y × Y , and ∆ = ² f (∆ _Y ).

One has the chain of isomorphisms:

µhom(f _µ,p ⁻¹ K, f _µ,p ^! F ) _(p

_,p

₎ =

= (µ _∆

RHom(q ⁻¹ ₂ f _µ,p ⁻¹ K, q ₁ ^! f _µ,p ^! F )) _(p

_,p

₎

∼ = (µ ∆

2 f _µ,(p ^!

_,p) RHom(q ₂ ⁻¹ K, q ₁ ^! f _µ,p ^! F )) _(p

_,p

₎

∼ = (µ ∆ RHom(q ₂ ⁻¹ K, q ₁ ^! f _µ,p ^! F )) _(p

_,p

₎ ⊗ ω _∆/∆

∼ = (µ ∆ 1 f _µ,(p,p ^!

₎ RHom(q ₂ ⁻¹ K, q ₁ ^! F )) _(p

_,p

₎ ⊗ ω _∆/∆

∼ = (µ ∆

RHom(q ₂ ⁻¹ K, q ₁ ^! F )) _(p

_,p

₎ ⊗ ω _∆

_/∆

= µhom(K, F ) _p

⊗ ω _{Y /X} .

Here q ₁ and q ₂ denote the projections from Y × Y , Y × X or X × X to the

corresponding factor, the meaning being clear from the context. The second and the

forth isomorphisms follow from Proposition 2.5.2 applied to ² f and ¹ f respectively.

The third and the fifth one follow from Theorem 2.2.4. q.e.d.

3. The inverse image theorem for sheaves.

In [D’A-S] is given a theorem on the well poseness for the Cauchy problem in a sheaf theoretical frame that allows to recover classical results as those of [H-L-W], [K-S 1] or [Sc].

In the statement of this theorem, among the others, there are some hypotheses concerning microlocal inverse images. When dealing with microlocal images there are two ways that may be taken: to work with ind-objects and pro-objects or else to restrict the attention to a class of complexes with prescribed conditions on the micro-support. The first choice is the one of §2.2, while the second is the one of [D’A-S]. Using the results of section 2, we are then able to state here a sharper result then that of [D’A-S] that will allow us to recover also the result of [K-S 2]

on the hyperbolic Cauchy problem.

3.1 Cauchy problem in sheaf theory. Let X be a manifold. We say that K ∈ Ob(D ^b (X)) is weakly cohomologically constructible (w-c-c for short), if the following conditions are satisfied:

(i) For any x ∈ X, “lim”

−→

U 3x

RΓ(U ; F ) is represented by F _x , (ii) For any x ∈ X, “lim”

←−

U 3x

RΓ _c (U ; F ) is represented by RΓ _{x} F . Here U ranges over an open neighborhood system of x.

In particular, weakly R-constructible complexes on a real analytic manifold are w-c-c (cf [K-S 3, §8.4]).

Let f : Y → X be a morphism of manifolds. Let Z be a subset of Y (e.g.

Z = {y} for y ∈ Y ).

Theorem 3.1.1. Let F and K be objects of D ^b (X), let L be an object of D ^b (Y ).

Assume to be given a morphism ψ : L → f ⁻¹ K. Let V be an open neighborhood of

˙π _Y ⁻¹ (Z). Assume that:

(i) f is non-characteristic for F on V ,

(ii) f π is non-characteristic for C(SS(F ), SS(K)) on ^t f ⁰⁻¹ (V ).

Assume that for every p Y ∈ ˙π _Y ⁻¹ (Z) there exist p 1 , . . . , p r in ^t f ⁰⁻¹ (p Y ) with:

(iii) ^t f ⁰⁻¹ (p _Y ) ∩ f _π ⁻¹ (SS(F )) ⊂ {p ₁ , . . . , p _r }, (iv) f _µ,p ⁻¹

K is representable for j = 1, . . . , r,

(v) the morphism induced by ψ, L → f _µ,p ⁻¹

K, is an isomorphism in D ^b (Y ; p _Y ) for j = 1, . . . , r.

Finally assume:

(vi) K and L are w-c-c,

(vii) the morphism induced by ψ, RΓ _{y} (L ⊗ ω _Y ) → RΓ _{x} (K ⊗ ω _X ), is an isomorphism for every y ∈ Z, x = f (y).

Then the natural morphism induced by ψ:

(3.1.1) f ⁻¹ RHom(K, F ) 

 Z −→ RHom(L, f ⁻¹ F ) 

 Z ,

is an isomorphism.

The only difference between this statement and that of Theorem 2.1.1 of [D’A-S]

is the hypothesis (iv) which is actually weakened.

Proof. One has a morphism induced by ψ:

R ^t f _! ⁰ f _π ⁻¹ µhom(K, F ) → µhom(L, f ⁻¹ F ).

As in [D’A-S], following an idea of [K-S 1], we consider the commutative diagram:

Rπ Y ! A −−−−→ Rπ Y ∗ A −−−−→ R ˙π Y ∗ A −−−−→ ⁺¹



 y



 y



 y

Rπ _{Y !} B −−−−→ Rπ _{Y ∗} B −−−−→ R ˙π _{Y ∗} B −−−−→ , ⁺¹ where A = R ^t f _! ⁰ f _π ⁻¹ µhom(K, F ) and B = µhom(L, f ⁻¹ F ).

Due to (1.2.1), we are easily reduced to prove that the first and the third vertical arrows are isomorphisms on Z.

The proof of the first vertical arrow being an isomorphism follows from hypothe- ses (vi) and (vii) and is given in [D’A-S].

Let us consider the third vertical arrow.

We have to prove that the natural morphism:

R ^t f _! ⁰ f _π ⁻¹ µhom(K, F ) _p

→ µhom(L, f ⁻¹ F ) _p

,

is an isomorphism for every p Y ∈ ˙π _Y (Z). Due to the assumption (iii) we can find refined cutting pairs (γ _j , ω _j ) on X at p _X,j (where p _X,j = f _π (p _j )) such that:

f _π ⁻¹ SS(Ψ _X (γ _j , ω _j ; F )) ∩ ^t f ⁰⁻¹ (p _Y ) ⊂ {p _j }.

Of course, Ψ _X (γ _j , ω _j ; F ) is isomorphic to F in D ^b (X; p _X,j ), and hence, due to Proposition 2.3.2:

f _µ,p ^!

F = f ^! Ψ _X (γ _j , ω _j ; F ).

Set F _j = Ψ _X (γ _j , ω _j ; F ). One has the isomorphism F ∼ = ⊕ j F _j in D ^b (X; f _π ^t f ⁰⁻¹ (p _Y )).

Since f is non-characteristic for F one also has the isomorphism f ⁻¹ F ∼ = ⊕ j f ⁻¹ F _j in D ^b (Y ; p _Y ) and hence we get the following chain of isomorphisms:

R ^t f _! ⁰ f _π ⁻¹ µhom(K, F ) _p

∼ = (R ^t f _! ⁰ f _π ⁻¹ ⊕ ^r _j=1 µhom(K, F _j )) _p

∼ = ⊕ ^r _j=1 (f _π ⁻¹ µhom(K, F _j )) _p

∼ = ⊕ ^r _j=1 µhom(K, F j ) p

∼ = ⊕ ^r _j=1 µhom(f _µ,p ⁻¹

K, f _µ,p ^!

F _j ) _p

⊗ ω _{Y |X} ^⊗

∼ = ⊕ ^r _j=1 µhom(f _µ,p ⁻¹

K, f _µ,p ⁻¹

F _j ) _p

∼ = ⊕ ^r _j=1 µhom(L, f ⁻¹ F j ) p

∼ = µhom(L, f ⁻¹ F ) _p

.

Here the first isomorphism is due to the fact that f is non-characteristic for F and that µhom is a microlocal functor, the forth to Theorem 2.2.3 and assumptions (ii), (iv), the fifth to assumption (i) and the sixth to assumption (v). q.e.d.

4. Applications to the Cauchy problem.

We said that Theorem 3.1.1 generalizes the corresponding result of [D’A-S]. As it was for [D’A-S], we are then able to recover (and even extend to general systems) the classical results of [H-L-W] (cf. also [K-S 1]) on the initial value problem for a linear partial differential operator when the data are ramified along the characteristic hypersurfaces as well as a result of [Sc] that shows how the holomorphic solution for the Cauchy problem can be expressed as a sum of functions which are holomorphic in domains whose boundary is given by the real characteristic hypersurfaces issued from the boundary of a strictly pseudoconvex domain where the data are defined.

Moreover we get the following results.

4.1. Other applications.

a) Our aim here is to recover the results of [K-S 2] concerning hyperbolic systems (cf [B-S] for the case of a single differential operator).

Let N and M be two real analytic manifolds, and let f be a real analytic map from N to M , which extends to a holomorphic map from Y to X. Here Y and X are complexifications of N and M respectively. Let M be a left coherent D _X -module.

Definition 4.1.1. One says that M is hyperbolic with respect to f if the following conditions are satisfied.

(i) f is non-characteristic for M,

(ii) ^t f ⁰⁻¹ (T _N ^∗ Y ) ∩ f _π ⁻¹ (char(M)) ⊂ f _π ⁻¹ (T _M ^∗ X), (iii) f _π is non-characteristic for C(T _M ^∗ X, char(M)).

Recall that the sheaf of Sato’s hyperfunctions on M is defined by:

B _M = RΓ M (O X ) ⊗ ω _M/X ^⊗−1 .

We can now state the well-poseness for the hyperbolic Cauchy problem in the hyperfunction frame (cf [K-S 2, Corollary 2.1.2]).

Proposition 4.1.2. Let M be a hyperbolic system with respect to f . Then the natural morphism:

f ⁻¹ RHom D

(M, B M ) −→ RHom D

(M Y , B N ), is an isomorphism.

Proof. One has the isomorphisms:

RHom _D

(M, B _M ) ∼ = RHom(ω _M/X , RHom _D

(M, O _X )) RHom _D

(M _Y , B _N ) ∼ = RHom(ω N/Y , RHom _D

(M _Y , O _Y ))

(in the second isomorphism we used the hypothesis (i) of Definition 4.1.1 and the

Cauchy-Kowalevski-Kashiwara’s theorem). We then have to show that we can apply

Theorem 3.1.1, for the choice F = RHom D

(M, O X ), K = ω _M/X , L = ω _N/Y .

Hypotheses (i)–(iii) as well as (vi) are easily verified, hypotheses (iv) and (v) follow

from the next Lemma 4.1.3, while hypothesis (vii) follows from Lemma 4.1.4. q.e.d.