(i = 1, 2) be a pair of real Lagrangian submanifolds of T

SHEAVES WITH APPLICATIONS TO SYSTEMS WITH SIMPLE CHARACTERISTICS

Andrea D’Agnolo Giuseppe Zampieri

Abstract. In this paper we state a hypoellipticity result in the framework of mi- crolocal boundary value problems (which has to be compared with the analogous results of [S-K-K], [K-S 2] at the interior). More precisely, let M be a system of mi- crodifferential equations with simple characteristics on a complex manifold X, and let Λ

(i = 1, 2) be a pair of real Lagrangian submanifolds of T

X. Denote by C

the associated complexes of microfunctions. If the pair (Λ

, Λ

) is “positive”, we prove the injectivity of the natural “restriction” morphism

E xt

X

(M, C

) → Ext

X

(M, C

)

between solution sheaves, where j is the first possibly non-vanishing degree of the cohomology.

R´ esum´ e fran¸ cais: Dans ce papier, on ´ etabli un r´ esultat d’hypoellipticit´ e dans le cadre des probl` emes aux limites microlocaux (cela est ` a comparer aux r´ esultats analogues ` a l’int´ erieur de [S-K-K], [K-S 2]). Plus pr´ ecis´ ement, soit M un syst` eme d’´ equations microdiff´ erentiels ` a caract´ eristiques simples sur une vari´ et´ e complexe X, et soit Λ

(i = 1, 2) un couple de sous-vari´ et´ ees Lagrangiennes r´ eelles de T

X. On note par C

les complexes de microfonctions associ´ es. Si le couple (Λ

, Λ

) est

“positif”, nous prouvons l’injectivit´ e du morphisme naturel de “restriction”

E xt

X

(M, C

) → Ext

X

(M, C

)

entre les faisceaux de solutions, o` u j est le premier degr´ e de cohomologie ´ eventuelle- ment non nul.

1991 Mathematics Subject Classification: 58G17

Appeared in: Bull. Soc. Math. France 123 (1995), no. 4, 605–637.

1

Table of contents Introduction

1. Symplectic geometry and Levi forms 1.1. Generalized Levi forms

1.2. Homogeneous symplectic spaces 1.3. Levi forms

2. Review on sheaves 2.1. General notations

2.2. Functors in localized categories

2.3. Simple sheaves and contact transformations 3. Levi-simple sheaves

4. Positivity

5. Generalized microfunctions

6. Microlocal direct images of simple sheaves

7. Faithfulness of the microlocal direct image functor 8. Proof of the faithfulness

9. An analytical lemma

10. End of proof of Theorem 7.1

11. Applications to systems with simple characteristics 11.1. A general statement

11.2. Microlocal boundary value problems Bibliography

Introduction

Let M be a real analytic manifold, let X be a complexification of M , and let M be a system of microdifferential equations with simple characteristics along a germ of smooth regular involutive submanifold V of the cotangent bundle T ^∗ X to X.

A classical result of Sato, Kawai and Kashiwara [S-K-K] asserts that the complex RHom E

(M, C M ) of microfunction solutions to M has vanishing cohomology in degree smaller than s ⁻ (M, V ), where s ⁻ (M, V ) denotes the number of negative eigenvalues of the generalized Levi form of V with respect to T _M ^∗ X.

The aim of this paper is to prove a similar result “up to the boundary”. More precisely, let Ω ⊂ M be an open subset with real analytic boundary S, and con- sider the complex C Ω of microfunctions at the boundary introduced by Schapira [S 2] as a framework for the study of boundary value problems. Under some cleanness hypotheses, we get in this paper vanishing for the cohomology of the complex RHom _E

(M, C _Ω ) in degree smaller then s ⁻ (M, V ): this is a criterion of hypoellipticity for microlocal boundary value problems. Moreover, we give geo- metrical conditions that ensure the vanishing of the whole complex of solutions RHom _E

(M, C _Ω ).

Our method of proof is similar to that of Kashiwara and Schapira [K-S 2] where, assuming the intersection V ∩ T _M ^∗ X is clean and regular, they recover the above mentioned results of [S-K-K], and obtain new vanishing theorems. Let us examine their proof in some details.

First, they use a complex contact transformation χ which replaces T _M ^∗ X with

the set Λ of exterior conormals to a strictly pseudoconvex open subset U ⊂ X,

and which “straightens” V , i.e. which replaces V with X × _Y T ^∗ Y , where the

fiber product is taken over a smooth holomorphic map f : X → Y . A quanti- zation of χ reduces M to a partial de Rham system, and interchanges the complex RHom E

(M, C M ) with µ ∂U (f ⁻¹ O _Y ), where µ ∂U denotes the Sato microlocaliza- tion functor along ∂U . Finally, an adjunction formula reduces the study of the complex µ ∂U (f ⁻¹ O _Y ) at p ∈ (X × Y T ^∗ Y ) ∩ Λ to the study of µ ∂U

(O Y ) at f π (p) (with T _∂U ^∗

Y = f _π (T _∂U ^∗ X ∩ V )). One is then reduced to the analysis of the mi- crolocal direct image f _! ^µ,p A _∂U of the constant sheaf A _∂U . The correspondence of cotangent bundles T ^∗ X

t

f

← X × _Y T ^∗ Y → T ^f

^∗ Y associates to Λ a germ of Lagrangian manifold f _π ^t f ⁰⁻¹ (Λ) ⊂ T ^∗ Y at f _π (p). The sheaf A _∂U is simple along Λ with shift 1/2, and it is thus possible to apply the following result of [K-S 3, Ch. 7]: if F is simple along Λ with shift d, then the microlocal direct image f _! ^µ,p F is simple along f _π ^t f ⁰⁻¹ (Λ) with a shift which is calculated using generalized Levi forms. The vanishing theorems now easily follow.

Let Ω ⊂ M be an open subset with real analytic boundary S and consider the distinguished triangle A _Ω → A _M → A _S ⁺¹ →. In order to apply the above scheme of proof to boundary value problems, we have to deal simultaneously with the two Lagrangian manifolds T _M ^∗ X and T _S ^∗ X. To this end, we first introduce a notion of

“positive” pair of Lagrangian manifolds (to be compared with the one of [S 1]) which allows us to give a microlocal meaning to the restriction morphism A _M → A _S . In particular, if (Λ ₁ , Λ ₂ ) is a pair of positive Lagrangian submanifolds of T ^∗ X, there is a natural “restriction” morphism C _Λ

→ C _Λ

between the associated complexes of generalized Sato microfunctions. Then, we give a criterion of faithfulness for the functor f _! ^µ,p acting on simple sheaves. This allows us to state a very general result of injectivity for the the natural morphism

Ext ^j _E

X

(M, C _Λ

) → Ext ^j _E

(M, C _Λ

)

between solution sheaves to a simply characteristic system M, where j is the first possibly non vanishing degree of the cohomology. In particular, this implies the criterion of hypoellipticity for microlocal boundary value problems we were looking for.

1. Symplectic geometry and Levi form

For the notions reviewed in this section, we refer to [S-K-K], [S 1], [K-S 1], and [D’A-Z 2].

§ 1.1. Generalized Levi forms. Let (E, σ) be a complex symplectic vector space.

A complex subspace ρ ⊂ E is called isotropic (resp. Lagrangian, resp. involutive) if ρ ^⊥ ⊃ ρ (resp. ρ ^⊥ = ρ, resp. ρ ^⊥ ⊂ ρ), where (·) ^⊥ denotes the orthogonal with respect to σ. A real subspace λ ⊂ E is called R-Lagrangian if it is Lagrangian in the underlying real symplectic vector space (E ^R , σ ^R ), where σ ^R = 2 Re σ.

If ρ ⊂ E is isotropic, the form σ induces a symplectic structure on the space ρ ^⊥ /ρ, which is denoted by (E ^ρ , σ ^ρ ). For a subset λ ⊂ E, let

λ ^ρ = ((λ ∩ ρ ^⊥ ) + ρ)/ρ ⊂ E ^ρ

and recall that if λ is an R-Lagrangian subspace of E then λ ^ρ is an R-Lagrangian subspace of E ^ρ (as it follows from the formula (λ ^ρ ) ^⊥ = (λ ^⊥ ) ^ρ ).

The following definition slightly generalizes that of [S 1], [K-S 2, Lemma 3.4].

Definition 1.1. Let ρ ⊂ E be isotropic, λ ⊂ E be R-Lagrangian, and set µ = λ∩iλ (an isotropic space). The generalized Levi form L _λ/ρ is the hermitian form on ρ ^µ defined by setting for v, w ∈ ρ ^µ

L _λ/ρ (v, w) = σ ^µ (v, w ^c ),

where (·) ^c denotes the conjugate with respect to the isomorphism E ^µ ∼ = C ⊗ ^R λ ^µ . We will denote by s ^± (λ, ρ) the numbers of positive (resp. negative) eigenvalues of L _λ/ρ .

The inertia index τ (λ 1 , λ 2 , λ 3 ) of three R-Lagrangian subspaces λ j ⊂ E, j = 1, 2, 3, is defined as the signature of the quadratic form q on λ ₁ ⊕ λ ₂ ⊕ λ ₃ given by q(x ₁ , x ₂ , x ₃ ) = σ ^R (x ₁ , x ₂ ) + σ ^R (x ₂ , x ₃ ) + σ ^R (x ₃ , x ₁ ).

Let e λ ^ρ denote the R-Lagrangian subspace (λ ∩ ρ ^⊥ ) + ρ ⊂ E.

Proposition 1.2. The following equalities hold:

sgn L _λ/ρ = − 1

2 τ (λ, iλ, e λ ^ρ )

rk L _λ/ρ = dim ^C (ρ ∩ µ ^⊥ ) − dim ^C (ρ ∩ [(ρ ^⊥ ∩ λ) + (ρ ^⊥ ∩ iλ)])

= dim ^C ρ − dim ^C (λ ^ρ ∩ iλ ^ρ ) − dim ^C (λ ∩ iλ)+

+ 2 dim ^C (ρ ^⊥ ∩ µ) − dim ^R (λ ∩ ρ).

For a proof, cf. [K-S 1, Lemma 3.4], [D’A-Z 2, Proposition 3.3].

As we will recall (cf. Lemmas 1.5, 1.7 below), the generalized Levi form of Defi- nition 1.1 agrees with classical concepts.

§ 1.2. Homogeneous symplectic spaces. Let X be a complex analytic manifold of dimension n, and denote by τ _X : T X → X, π _X : T ^∗ X → X its tangent and cotangent bundle respectively. Denote by ˙ T ^∗ X the space T ^∗ X with the zero section removed, and more generally, for V ⊂ T ^∗ X, let ˙ V = V ∩ ˙ T ^∗ X. Let α X be the canonical one-form on T ^∗ X, and σ _X = d α _X the symplectic two-form.

Let H : T ^∗ T ^∗ X → T T ^{∼ ∗} X be the Hamiltonian isomorphism. The vector field

−H(α _X ), called Euler vector field, is the infinitesimal generator of the action of C ^× = C \ {0} on T ^∗ X. In this paper all submanifolds that we consider are locally conic (i.e. tangent to H(−α X )).

A submanifold V ⊂ T ^∗ X is called isotropic (resp. Lagrangian, resp. involutive) if at each p ∈ V , the space T p V has the corresponding property in the symplectic vector space (T _p T ^∗ X, σ _X (p)). The submanifold V is called regular if α _X | _V 6= 0.

Let Λ be a submanifold of the underlying real analytic manifold T ^∗ X ^R to T ^∗ X.

The submanifold Λ is called R-Lagrangian if at each p ∈ Λ, the space T _p Λ is Lagrangian in the real symplectic vector space (T _p T ^∗ X ^R , 2 Re σ _X (p)). For p ∈ Λ, set

(1.1)

 

 

 

 

 

 

 

 

 



(E(p), σ(p)) = (T _p T ^∗ X, σ _X (p)), ν(p) = CH(α _X (p)),

λ 0X (p) = T p (π _X ⁻¹ π X (p)), λ(p) = T p Λ,

c(λ(p), λ 0X (p)) = dim ^R (λ(p) ∩ λ 0X (p)),

γ(λ(p), λ _0X (p)) = dim ^C (λ(p) ∩ iλ(p) ∩ λ _0X (p)).

If no confusion may arise, we will drop the point p in the previous notations.

Definition 1.3. An R-Lagrangian manifold Λ ⊂ ˙ T ^∗ X is called of real type at p ∈ Λ if dim ^R (ν ∩ λ) = 1.

Notice that Λ is of real type if and only if it exists a complex contact trans- formation χ : T ^∗ X → T ^∗ X at p such that χ(Λ) is the conormal bundle to a real hypersurface of X.

Let M be a real analytic submanifold of the underlying real analytic manifold X ^R to X. Via the natural isomorphism T (X ^R ) ∼ = (T X) ^R the complex tangent plane to M at x ∈ M is defined as

T _x ^C M = T _x M ∩ iT _x M.

Moreover, we will identify the conormal bundle T _M ^∗ X ^R to an R-Lagrangian manifold of T ^∗ X, that we will denote by T _M ^∗ X. For Λ = T _M ^∗ X and p ∈ ˙ T _M ^∗ X we simplify the above notations (1.1), as well as those of Definition 1.1, by writing

(1.2)

 

 



 

 

λ _M (p) = T _p T _M ^∗ X,

s ^± (M, p) = s ^± (λ _M , λ _0X ), c M (p) = c(λ(p), λ 0X (p)), γ M (p) = γ(λ(p), λ 0X (p)), eventually dropping the point p.

§ 1.3. Levi forms. Let us now recall the classical definition of Levi form of a real submanifold.

Definition 1.4. Let M ⊂ X be a real analytic submanifold. Let p ∈ ˙ T _M ^∗ X, and set x = π _X (p). The Levi form L _M (p) of M at p is the restriction to T _x ^C M of the Hermitian form ∂∂φ(x), where φ is a real analytic function vanishing on M such that dφ(x) = p.

The independence of Definition 1.4 from the choice of the local equation φ for M may be deduced from the next Lemma which states the relation between the Levi forms of Definition 1.1 and Definition 1.4.

Lemma 1.5. Let p ∈ ˙ T _M ^∗ X, and set x = π _X (p). Then L _M (p) ∼ L _λ

_/λ

,

where ∼ means that the two forms have same signature and same rank.

For a proof refer to [K-S 1, Proposition 11.2.7] in the case cod ^R _X M = 1, and to [D’A-Z 1, Propositions 1.1, 1.3] for the general case.

Assume now that X is a complexification of a real analytic manifold M .

Definition 1.6. ([S-K-K]) Let M ⊂ X be a real analytic manifold, X a com- plexification of M , and let V ⊂ ˙ T ^∗ X be a regular involutive submanifold. Let p ∈ T _M ^∗ X ∩ V . The generalized Levi form L _T

_X (V, p) is the Hermitian form on (T p V ) ^⊥ of matrix

({f _j , f _k ^c }(p)) _1≤j,k≤l ,

where {·, ·} denotes the Poisson bracket, where f ₁ = · · · = f _l = 0 is a system of holomorphic equations for V at p such that df ₁ ∧ · · · ∧ df _l ∧ α _X (p) 6= 0, and where f _j ^c denotes the holomorphic conjugate to f _j (i.e. the holomorphic function at p satisfying f _j ^c | _T

M

X = f _j | _T

M

X ).

Lemma 1.7. ( [K-S 2, Lemma 3.4]) Let p ∈ ˙ T _M ^∗ X ∩ V and set ρ = (T p V ) ^⊥ . Then L _T

M

X (V, p) = −L _λ

_/ρ .

Notice that the more general case where M is a real analytic submanifold of X is treated in [D’A-Z 2].

Notations 1.8. Let Λ ⊂ T ^∗ X be a real type Lagrangian manifold. On the line of Lemma 1.7 we will use the following notations

(1.3)

 



 

L _Λ (V, p) = −L _λ/ρ , s ^± (Λ, V, p) = s ^± (λ, ρ), s ^± (M, V, p) = s ^± (Λ, V, p),

where in the last equality we assumed Λ = T _M ^∗ X for a real submanifold M ⊂ X.

2. Review on sheaves

In this section all manifolds and morphisms of manifolds will be real of class C ^∞ . We collect here some definitions and results from [K-S 3].

§ 2.1. General notations. Let X be a manifold, and denote by D ^b (X) the derived category of the category of bounded complexes of sheaves of A-modules, where A is a fixed ring with finite global dimension (e.g. A = Z). As a general notation, for K ⊂ X locally closed, A _K denotes the sheaf which is zero on X \ K and which is the constant sheaf on K with stalk A.

The micro-support SS(F ) of a complex F ∈ Ob(D ^b (X)) is a closed conic in- volutive subset of T ^∗ X which describes the directions of non-propagation for the cohomology of F . For p ∈ T ^∗ X the category D ^b (X; p) is the localization of D ^b (X) with respect to the null system {F ∈ Ob(D ^b (X)); p / ∈ SS(F )}. Recall that the objects of D ^b (X; p) are the same as those of D ^b (X) and that a morphism F → G of D ^b (X) becomes an isomorphism in D ^b (X; p) if and only if p / ∈ SS(H), H being the third term of a distinguished triangle F → G → H ⁺¹ →.

The bifunctor µhom(·, ·) is well defined in the category D ^b (X; p), i.e. induces a functor (still denoted by µhom) from D ^b (X; p) ^◦ × D ^b (X; p) to the category D ^b (T ^∗ X; p) (where D ^b (X; p) ^◦ denotes the opposite category to D ^b (X; p)). Recall the relations

Rπ _X∗ µhom(F, G) ∼ = RHom(F, G), (2.1)

H ⁰ µhom(F, G) _p ∼ = Hom _D

_(X;p) (F, G).

(2.2)

§ 2.2. Functors in localized categories. The usual functors of sheaf theory are not generally well defined in the microlocal category. Let us consider the two following cases.

(a) Let f : X → Y be a morphism of manifolds and consider the associated correspondence of cotangent bundles

T ^∗ X

t

f

←− X × _Y T ^∗ Y −→ T ^f

^∗ Y.

Let p ∈ X × Y T ˙ ^∗ Y and set p X = ^t f ⁰ (p), p Y = f π (p). The functor Rf ! : D ^b (X) → D ^b (Y ) is not well defined in D ^b (X; p _X ), in general. The microlocal proper direct image functor

f _! ^µ,p : D ^b (X, p _X ) −→ D ^b (Y ; p _Y ) ^∨

takes value in the category D ^b (Y ; p _Y ) ^∨ of contravariant functors from the category D ^b (Y ; p _Y ) to the category of sets. (Notice that D ^b (Y ; p _Y ) is a full subcategory of D ^b (Y ; p Y ) ^∨ .) For F ∈ Ob(D ^b (X; p X )), f _! ^µ,p F is by definition the pro-object

(2.3) f _! ^µ,p F = “lim”

←−

F

→F

Rf _! F ⁰ ,

where F ⁰ → F ranges over the category of isomorphisms in D ^b (X; p _X ).

(b) Let X, Y , Z be real manifolds and let q ij (i, j = 1, 2, 3) be the projections from X × Y × Z to the corresponding factor (e.g. q ₁₃ : X × Y × Z → X × Z).

The composition functor

· ◦ · : D ^b (X × Y ) × D ^b (Y × Z) −→ D ^b (X × Z)

(K ₁ , K ₂ ) 7→ Rq _13! (q ⁻¹ ₁₂ K ₁ ⊗ q ⁻¹ ₂₃ K ₂ )

has no microlocal meaning, in general. For p W ∈ ˙ T ^∗ W (W = X, Y, Z), the microlocal composition functor

· ◦ _µ · : D ^b (X × Y ; (p X , p Y )) × D ^b (Y × Z; (p Y , p Z )) −→ D ^b (X × Z; (p X , p Z )) ^∨ is then introduced as

(2.4) K ₁ ◦ _µ K ₂ = “lim”

←−

K

→K

, K

→K

Rq _13! (q ₁₂ ⁻¹ K ₁ ⁰ ⊗ q ₂₃ ⁻¹ K ₂ ⁰ ),

where K ₁ ⁰ → K ₁ (resp. K ₂ ⁰ → K ₂ ) ranges over the category of isomorphisms in D ^b (X × Y ; (p X , p Y )) (resp. D ^b (Y × Z; (p Y , p Z ))).

§ 2.3. Simple sheaves and contact transformations. Let Λ ⊂ ˙ T ^∗ X be a (real C ^∞ ) Lagrangian submanifold and let p ∈ Λ. Let F be an element of Ob(D ^b (X)) which satisfies SS(F ) ⊂ Λ at p. The concept of such an F being simple along Λ with shift d ∈ ¹ ₂ Z at p is introduced in [K-S 3, §7.5]. Recall for instance that if M ⊂ X is a submanifold, then A _M is simple along T _M ^∗ X with shift ¹ ₂ cod ^R _X M .

Let χ : T ^∗ X → T ^∗ X be a complex contact transformation at p and let p ⁰ = χ(p).

Denote by Λ ^a _χ the antipodal of its graph, Λ ^a _χ = {(q, q ⁰ ) ∈ T ^∗ (X × X); q ⁰ = −χ(q)}, a Lagrangian submanifold of T ^∗ (X × X).

Recall the notations (1.1), (1.2) and (1.3). In the following we will write for short λ ^χ instead of χ ⁰ (λ(p)).

The next proposition, that collects some results of [K-S 3, ch. 7], shows how it

is possible to define an action of χ on microlocal categories.

Proposition 2.1. Let d χ ∈ ¹ ₂ Z be such that d _χ ≡ 1

2 dim ^R (λ _{0 X×X} (p ⁰ , −p) ∩ T _(p

_,−p) Λ ^a _χ

) mod Z.

Then

(i) there exists a complex K ∈ Ob(D ^b (X × X)) simple along Λ ^a _χ

at (p ⁰ , −p) with shift d _χ .

Moreover, this K enjoys the following properties:

(ii) for F ∈ Ob(D ^b (X; p)), the pro-object K ◦ _µ F is representable in D ^b (X; p ⁰ ), (iii) the functor

K ◦ µ · : D ^b (X; p) −→ D ^b (X; p ⁰ ) is an equivalence of categories,

(iv) for F, G ∈ Ob(D ^b (X; p)), the following isomorphism hold χ _∗ µhom(F, G) _p

∼ = µhom(K ◦ µ F, K ◦ _µ G) _p

,

(v) if F is simple along Λ at p with shift d, then K ◦ µ F is simple along χ(Λ) at p ⁰ with shift d + d _χ − ¹ ₂ [dim ^R X + τ (λ _0X , λ ^χ , λ _0X ^χ )].

3. Levi-simple sheaves

From now on all manifolds and morphisms of manifolds will be complex analytic.

Let X be a complex manifold of dimension n. On the line of [K-S 1, §11.2], due to the complex structure of X, it is possible to associate a shift to simple sheaves.

More precisely, let

(3.1) d(λ, λ _0X ) = − 1

2 [n − dim ^C (λ ∩ iλ) − sgn L _λ/λ

], and notice that by Proposition 1.2

d(λ, λ 0X ) = − 1

2 dim ^R (λ ∩ λ 0X ) + γ(λ, λ 0X ) − s ⁻ (λ, λ 0X )

= − 1

2 [2n − dim ^R (λ ∩ λ 0X )] + dim ^C (λ ∩ iλ) − γ(λ, λ 0X ) + s ⁺ (λ, λ 0X ).

Definition 3.1. Let F ∈ Ob(D ^b (X)), and let Λ ⊂ ˙ T ^∗ X be an R-Lagrangian manifold of real type at p ∈ Λ. We call F Levi-simple along Λ at p if F is simple along Λ at p with shift d(λ, λ _0X ).

The following result is essentially due to [K-S 1, § 11.2].

Proposition 3.2. Let Λ ⊂ ˙ T ^∗ X be an R-Lagrangian manifold of real type at p ∈ Λ.

Then

(i) there exists F ∈ Ob(D ^b (X)) Levi-simple along Λ at p.

Let χ : T ^∗ X → T ^∗ X be a complex contact transformation and let K ∈ Ob(D ^b (X × X)) be simple along Λ ^a _χ

at (χ(p), −p) with shift n. Then

(ii) if F ∈ Ob(D ^b (X)) is Levi-simple along Λ at p, then K ◦ _µ F is Levi-simple

along χ(Λ) at χ(p).

Proof. We follow here the same line of the proof of [K-S 1, Proposition 11.2.8].

Let us first recall that if λ _j ⊂ E (j = 1, 2, 3, 4) are R-Lagrangian subspaces, then τ (λ ₁ , λ ₂ , λ ₃ ) − τ (λ ₁ , λ ₂ , λ ₄ ) + τ (λ ₁ , λ ₃ , λ ₄ ) − τ (λ ₂ , λ ₃ , λ ₄ ) = 0

(3.2)

τ (iλ 1 , iλ 2 , iλ 3 ) = −τ (λ 1 , λ 2 , λ 3 ).

(3.3)

By Proposition 2.1, K ◦ _µ F ∈ Ob(D ^b (X; χ(p))) is simple along χ(Λ) at χ(p) with shift d(λ, λ _0X ) − ¹ ₂ τ (λ _0X , λ ^χ , λ _0X ^χ ). Setting λ ₁ = λ _0X , λ ₂ = λ ^χ , λ ₃ = iλ ^χ , λ ₄ = λ 0X χ in (3.2) and using (3.3) we get

τ (λ _0X ^χ , λ ^χ , λ _0X ) = 1

2 τ (λ, iλ, λ _0X ) − 1

2 τ (λ ^χ , iλ ^χ , λ _0X )

= sgn L _λ

_/λ

− sgn L _λ/λ

. Noticing that dim ^C (λ ∩ iλ) = dim ^C (λ ^χ ∩ iλ ^χ ) we get

(3.4) d(λ, λ 0X ) − 1

2 τ (λ 0X , λ ^χ , λ 0X χ ) = d(λ ^χ , λ 0X ), which proves (ii).

As for (i), since Λ is of real type, we may find a complex contact transformation ψ : T ^∗ X → T ^∗ X such that ψ(Λ) = T _M ^∗ X with cod ^R _X M = 1 and s ⁻ (M, ψ(p)) = 0.

Let L ∈ Ob(D ^b (X × X)) be simple along Λ ^a _ψ at (p, −ψ(p)) with shift n. Since A M [−1] is Levi-simple along T _M ^∗ X at ψ(p), it follows from (ii) that L ◦ µ A M [−1] is Levi-simple along Λ at p. q.e.d.

Corollary 3.3. Let M ⊂ X be a real analytic submanifold. Let p ∈ ˙ T _M ^∗ X and assume ip / ∈ T _M ^∗ X (i.e. T _M ^∗ X is of real type at p). Then F is Levi-simple along T _M ^∗ X at p if and only if

F ∼ = A M [− cod ^R _X M + γ(M, p) − s ⁻ (M, p)].

In particular, if cod ^R _X M = 1 and s ⁻ (M ) = 0, then A M [−1] is Levi-simple at p.

We shall need the following technical result.

Lemma 3.4. Let ρ ⊂ E be a complex l-dimensional isotropic subspace. Then d(λ, λ _0X ) = d(λ ^ρ , λ _0X ^ρ ) − d(λ, λ _0X , ρ) + δ(λ, ρ) + s ⁺ (λ, ρ),

where we set (3.5)

( δ(λ, ρ) = cod ^C _ρ (ρ ∩ (λ ^⊥ + iλ ^⊥ )),

d(λ, λ _0X , ρ) = ¹ ₂ [τ (λ _0X , e λ ^ρ , λ) + l − dim ^C (ρ ∩ λ)].

For a proof cf. [K-S 2, Lemma 3.3], [D’A-Z 2, Proposition 3.5].

As for the notations, if ρ = (T _p V ) ^⊥ for V ⊂ T ^∗ X an involutive subset, we will write

(3.6)

 δ(Λ, V ) = δ(λ, ρ), s ^± (Λ, V ) = s ^± (λ, ρ).

Moreover, to agree with classical notations, if M ⊂ X is a real analytic submanifold and Λ = T _M ^∗ X, we will write

(3.7)

 δ(M, V ) = δ(Λ, V ),

s ^± (M, V ) = s ^± (Λ, V ).

4. Positivity

Let Λ ⊂ ˙ T ^∗ X be an R-Lagrangian manifold at p ∈ Λ and assume Λ is I- symplectic (i.e. λ ∩ iλ = {0}). Let Γ ⊂ T ^∗ X be a C ^× -conic subset. The concept of positive pair (Λ, Γ) is introduced in [S 1] after the work of [M-Sj 1, 2]. This is a notion invariant by complex contact transformations which ensures that if Λ is the set of exterior conormals to a strictly pseudoconvex open subset Ω ⊂ X, then the projection π X (Γ) is contained in X \ Ω.

In order to treat the case of real type Lagrangian manifolds (not necessarily I-symplectic) we adopt the following point of view which is motivated by the next Lemma 4.3 that Pierre Schapira pointed out to us.

Definition 4.1. Let Λ _j (j = 1, 2) be R-Lagrangian manifolds of real type at p ∈ Λ 1 ∩ Λ ₂ , and let F j be Levi-simple along Λ j at p. The pair (Λ 1 , Λ 2 ) is called positive (and we write Λ ₁  Λ ₂ ) if and only if there exists a non zero morphism F 1 → F ₂ in D ^b (X; p).

By (2.2) this is equivalent to

(4.1) H ⁰ µhom(F ₁ , F ₂ ) _p 6= 0.

Let us notice that, by (iv) of Proposition 2.1, this concept of positivity is stable by complex contact transformations. In other words, if χ : T ^∗ X → T ^∗ X is a complex contact transformation, then Λ ₁  Λ ₂ if and only if χ(Λ ₁ )  χ(Λ ₂ ).

Notation 4.2. Let M ⊂ X be a real analytic hypersurface. Let p ∈ ˙ T _M ^∗ X and set x = π X (p). As a general notation, the point p being implicit in our discussion, we denote by M ⁺ the germ of closed half space at x with boundary M and inner conormal p (i.e. such that p ∈ SS(A _M

)).

Lemma 4.3. Let M 1 , M 2 ⊂ X be real hypersurfaces. Let p ∈ ˙ T _M ^∗

X ∩ ˙ T _M ^∗

X and set x = π X (p). The following assertions are equivalent:

(i) H ⁰ (µhom(A _M

1

, A _M

2

)) _p 6= 0, (ii) M ₁ ⁺ ⊃ M ₂ ⁺ at x.

Proof. Set for short C = µhom(A _M

1

, A _M

2

), and consider the distinguished triangle (Rπ _X! C) _x → (Rπ _X∗ C) _x → (R ˙π _X∗ C) _x ⁺¹ → .

One has

(Rπ _X! C) _x ∼ = A _Int(M

1

) ⊗ A _M

2

, (Rπ X∗ C) x ∼ = RΓ _M

1

(A _M

2

) x , (R ˙π X∗ C) x ∼ = µhom(A _M

1

, A _M

2

) p ,

where the first isomorphism follows from [K-S 3, Proposition 4.4.2], the second from (2.1), and the third from ˙π _X ⁻¹ (x) ∩ supp C ⊂ R ⁺ p. Since x / ∈ Int(M ₁ ⁺ ), (Rπ X! C) x = 0 and hence

µhom(A _M

1

, A _M

2

) p ∼ = RΓ _M

1

(A _M

2

) x . One easily concludes. q.e.d.

Recall the following proposition of [D’A-Z 4] (where the notion of positivity was

not explicit).

Proposition 4.4. Let Λ j ⊂ ˙ T ^∗ X (j = 1, 2) be R-Lagrangian manifolds of real type at p ∈ Λ ₁ ∩ Λ ₂ . Then Λ ₁  Λ ₂ if and only if there exists a complex contact transformation χ : T ^∗ X → T ^∗ X such that

( χ(Λ _j ) = T _M ^∗

X, with cod ^R _X M _j = 1, s ⁻ (M _j , p ⁰ ) = 0, (j = 1, 2), M ₁ ⁺ ⊃ M ₂ ⁺ at x ⁰ ,

where p ⁰ = χ(p), x ⁰ = π X (p ⁰ ).

Proof. It is possible to choose a (complex) Lagrangian plane λ ₀ ⊂ E ^ν satisfying:

(i) λ ₀ is transversal to λ _j (j = 1, 2), (ii) s ⁻ (λ ^ν ₁ , λ ₀ ) = 0.

In fact, the set of λ 0 ’s satisfying (i) is open and dense in the Lagrangian Grassman- nian of E ^ν , and the set satisfying (ii) is open and non empty. Let χ : T ^∗ X → T ^∗ X be a complex contact transformation such that (λ 0X χ ) ^ν = λ 0 . Then

χ(Λ _j ) = T _M ^∗

X, cod ^R _X M _j = 1, (j = 1, 2) and s ⁻ (M ₁ , p ⁰ ) = 0. Since T _M ^∗

X  T _M ^∗

X at p ⁰ , one has

H ⁰ µhom(A _M

1

, A _M

2

) _p [−s ⁻ (M ₂ , p ⁰ )] 6= 0, and, by the proof of Lemma 4.3, this is equivalent to

H ^−s

^(M

^,p

⁾ RΓ _M

1

(A _M

2

) x 6= 0.

The functor RΓ _M

1

(·) being left exact, it follows that s ⁻ (M ₂ , p ⁰ ) = 0. Moreover, the relation Γ _M

1

(A _M

2

) _x 6= 0 implies that M ₁ ⁺ ⊃ M ₂ ⁺ at x ⁰ . q.e.d.

Remark 4.5.

(i) Using Proposition 4.4 it is easy to prove that  is a preorder relation. In other words, if Λ _j (j = 1, 2, 3) are R-Lagrangian manifolds of real type at p ∈ Λ ₁ ∩ Λ ₂ ∩ Λ ₃ satisfying Λ ₁  Λ ₂ , Λ ₂  Λ ₃ , then Λ ₁  Λ ₃ .

(ii) Again by Proposition 4.4 it follows that if Λ ₁  Λ ₂ , then λ ₁ ∩iλ ₁ ⊂ λ ₂ ∩iλ ₂ . Proposition 4.6. Let Λ _j (j = 1, 2) be R-Lagrangian manifolds of real type at p ∈ Λ ₁ ∩ Λ ₂ , and let F _j be Levi-simple along Λ _j at p. Then Λ ₁  Λ ₂ if and only if µhom(F ₁ , F ₂ ) _p = A. In particular, taking the zero-th cohomology we get

(4.2) Hom _D

_(X;p) (F ₁ , F ₂ ) ∼ = A.

Proof. Notice that the statement is invariant by complex contact transformations.

We may then assume by Proposition 4.4 that Λ j = T _M ^∗

X for cod ^R _X M j = 1, s ⁻ (M _j ) = 0 (j = 1, 2), M ₁ ⁺ ⊃ M ₂ ⁺ . In this case RΓ _M

1

(A _M

2

) _x = A, and one concludes by noticing that µhom(A _M

, A _M

) _p ∼ = RΓ _M

1

(A _M

2

) _x (cf the proof of Lemma 4.3). q.e.d.

Let us call “restriction morphism” the morphism which corresponds to 1 ∈ A

via (4.2). Notice that if Λ _i (i = 1, 2) are the exterior conormals to some strictly

pseudoconvex open subsets Ω _i ⊂ X, the restriction morphism is nothing but the

morphism A _X\Ω

[−1] → A _X\Ω

[−1] (or, equivalently, A _Ω

→ A _Ω

) induced by the

inclusion Ω ₁ ⊂ Ω ₂ .

Remark 4.7.

(i) Let Λ _j (j = 1, 2) be R-Lagrangian manifolds of real type at p ∈ Λ ₁ ∩ Λ ₂ , and assume Λ 1 is I-symplectic (i.e. λ 1 ∩ iλ ₁ = {0}). If the pair (Λ 1 , Λ 2 ) is positive in the sense of [S] and d(λ ₁ , λ _0X ) = d(λ ₂ , λ _0X ), then Λ ₁  Λ ₂ . The converse implication is not clear to us.

(ii) Let S ⊂ M ⊂ X be real analytic submanifolds. Then T _M ^∗ X  T _S ^∗ X at p ∈ S × _M T ˙ _M ^∗ X if and only if d(λ _M , λ _0X ) = d(λ _S , λ _0X ). In fact one easily checks that in this case µhom(A _M , A _S ) ∼ = A S×

T

X .

5. Generalized microfunctions

Let X be a complex analytic manifold of dimension n, and let O _X be the sheaf of germs of holomorphic functions on X.

Definition 5.1. Let Λ ⊂ T ^∗ X be an R-Lagrangian manifold of real type at p ∈ T ˙ ^∗ X, and let F be Levi-simple along Λ at p. The complex of microfunctions along Λ is the object of D ^b (T ^∗ X; p) defined by:

(5.1) C _Λ = µhom(F, O _X ).

To be rigorous, we should incorporate F and p in the notation of C _Λ . We will not be that precise since, in any case, we will restrict our study to a neighborhood of p.

Remark 5.2. Let M ⊂ X be a submanifold, p ∈ ˙ T _M ^∗ X, ip / ∈ T _M ^∗ X. By Corollary 3.3 one has

(C _T

_X ) _p ∼ = µhom(A M , O _X ) _p [cod ^R _X M − γ(M, p) + s ⁻ (M, p)].

If M is a real analytic manifold and X is a complexification of M , we recover the sheaf of Sato microfunctions up to the orientation sheaf or _M/X (which is locally isomorphic to the constant sheaf A _M ).

Proposition 5.3. ( [K-S 3, Chapter 11]) Let χ : T ^∗ X → T ^∗ X be a complex contact transformation. Then, by choosing a quantization of χ, the following isomorphism holds

χ _∗ (C _Λ ) _χ(p) ∼ = (C χ(Λ) ) _p .

Let D _X be the sheaf of differential operators on X. The complex C _Λ is well defined in the derived category of π _X ⁻¹ D _X -modules. (To see this, just use the left D _X -module structure of the sheaf O _X .)

Let E _X denote the sheaf of finite order microdifferential operators on T ^∗ X. Let

Ω ⊂ X be a strictly pseudoconvex subset with real analytic boundary M , and

denote by Λ the set of exterior conormals to Ω. In this case, it is known (cf [K])

that C _Λ is well defined in the derived category of E _X -modules near p ∈ Λ, but we

do not know if this result still holds for general R-Lagrangian manifolds. However,

we have a partial result.

Proposition 5.4. Let M ⊂ X be a (real analytic) generic submanifold of X. Then the complex C _T

_X is well defined in the category of left E _X -modules

Proof. Consider the diagram

M −→ X

↓ % _φ Z

where Z is a complexification of M , and φ is the induced complex analytic smooth map. The cotangent map ^t φ ⁰ to φ gives an identification between T _M ^∗ X and (Z × _X T ^∗ X) ∩ T _M ^∗ Z. Moreover, one has the isomorphism

(5.2) ^t φ ⁰ _∗ C _T

_X ∼ = RHom E

(E _Z→X , C _M/Z ),

where C _M/Z ∼ = C T

Z are the classical Sato microfunctions on M , and where E _Z→X denotes the transfer module. The action of E X on C T

X is then induced by the natural right φ ⁻¹ _π E _X -module structure of E _Z→X , via the isomorphism (5.2). q.e.d.

Proposition 5.5. The complex C _Λ,p is concentrated in degree ≥ 0, i.e. (H ^j C _Λ ) _p = 0 for every j < 0.

Proof. Let χ : T ^∗ X → T ^∗ X be a complex contact transformation at p as in Proposi- tion 4.4 such that χ(Λ) = T _M ^∗ X, with cod ^R _X M = 1 and s ⁻ (M, p ⁰ ) = 0 for p ⁰ = χ(p).

By quantizing χ, we get

χ _∗ (C _Λ ) _p

∼ = µhom(A M , O _X ) _p

[1]

∼ = RΓ M

(O X ) x

[1],

where x ⁰ = π X (p ⁰ ). The claim follows since the complex RΓ _M

(O X ) x

is in degree

≥ 1 by analytic continuation. q.e.d.

Theorem 5.6. Let Λ _j (j = 1, 2) be R-Lagrangian manifolds of real type at p ∈ Λ 1 ∩ Λ ₂ , and let F j be Levi-simple along Λ j at p. Assume Λ 1  Λ ₂ . Then the natural morphism

(5.3) H ⁰ C _Λ

_,p −→ H ⁰ C _Λ

_,p induced by the restriction morphism is injective.

Proof. By the contact transformation χ at p as in Proposition 4.4, the morphism (5.3) reduces to

H ¹ _M

2

(O _X ) _x

−→ H ¹ _M

1

(O _X ) _x

,

which is injective by analytic continuation (here x ⁰ = π _X (χ(p))). q.e.d.

6. Microlocal direct images of simple sheaves

Let f : X → Y be a smooth map of real analytic manifolds and consider the associated maps

T ^∗ X

t

f

←− X × _Y T ^∗ Y −→ T ^f

^∗ Y.

Set V = X × _Y T ˙ ^∗ Y , and let Λ ⊂ T ^∗ X be a (real) Lagrangian manifold at p ∈ Λ∩V .

If the intersection Λ ∩ V is clean at p (which means that V ∩ Λ is smooth and

T p (V ∩ Λ) = T p V ∩ T p Λ for every p ∈ V ∩ Λ), it is known (cf e.g. [K-S 2]) that, for a sufficiently small open neighborhood U ⊂ T ^∗ X of p, f π t f ⁰⁻¹ (Λ∩U ) = f π (V ∩Λ∩U ) is a Lagrangian manifold in T ^∗ Y .

Via the functor f _! ^µ,p , it is possible to establish a similar correspondence between simple sheaves in X along Λ and simple sheaves in Y along f π t f ⁰⁻¹ (Λ ∩ U ). To explain this correspondence, let us first reduce by contact transformation, to the case where Λ and f _π ^t f ⁰⁻¹ (Λ ∩ U ) are conormal bundles to hypersurfaces. In this frame, one has the following proposition that collects some results of [K-S 3, §7.4].

Proposition 6.1. Let f : X → Y be a smooth morphism of real analytic manifolds.

Let M ⊂ X, N ⊂ Y be real analytic hypersurfaces. Let p ∈ ˙ T _M ^∗ X ∩ V , x = π _X (p), q = f _π (p), y = π _Y (q). Assume

(i) the intersection T _M ^∗ X ∩ V is clean at p,

(ii) for an open neighborhood U 3 p, f _π ^t f ⁰⁻¹ (T _M ^∗ X ∩ U ) = T _N ^∗ Y at q.

Then

(a) there exist local systems of coordinates (x) = (y, t) on X and (y) on Y , at x and y respectively, such that: p = (0; d y ₁ ), f (y, t) = y, N = {y ∈ Y ; y ₁ = 0}, M = {x ∈ X; y 1 = Q(t)}, where Q(t) is a quadratic form,

(b) there exists a morphism in the category of pro-objects of D ^b (Y ):

“lim”

←−

U 3x

Rf _! A _M

_∩U → A _N

[−δ],

where U ranges over the category of open neighborhoods of x and, with the notation (3.5),

δ = #{non positive eigenvalues of Q(t)}

= 1

2 (dim ^R X − dim ^R Y ) − d(λ M , λ 0X , ρ), (6.1)

(c) f _! ^µ,p A _M

∼ = “lim”

←−

U 3x

Rf _! A _M

_∩U exists in D ^b (Y ; y) and the morphism in (b) is an isomorphism in D ^b (Y ; y).

Remark 6.2. The previous proposition is due to [K-S 3, §7.4], in which (c) reads (c)’ f _! ^µ,p A _M

∼ = “lim”

←−

U 3x

Rf _! A _M

_∩U exists in D ^b (Y ; q) and the morphism in (b) is an isomorphism in D ^b (Y ; q).

Since M ⁺ and N ⁺ are closed half-spaces, it is easy to prove that (c)’ is equivalent to (c).

Assume now that f : X → Y is a morphism in the category of complex analytic manifolds. We will need the following result:

Lemma 6.3. Let f : X → Y be a smooth map of complex analytic manifolds.

Under the same hypotheses as in Proposition 6.1, one has:

s ^± (N, q) ≤ s ^± (M, p).

Proof. Let L = π X (T _M ^∗ X ∩ V ) and consider the diagram

T _M ^∗ X ←−−−− T _M ^∗ X ∩ V ^f

^|

∗ MX∩V

−−−−−−−→ T _N ^∗ Y

  y

  y

  y

M ←−−−− L −−−−→ ^{f |}

N

One proves as in [K-S 2] that L ⊂ X is a smooth submanifold, and f | _L is a smooth map. Let φ be a real analytic function on Y , vanishing on N , and such that d φ = q.

It follows that φ ◦ f is a real analytic function on X vanishing on L and such that d(φ ◦ f )(x) ∈ T _L ^∗ X ∩ V = T _M ^∗ X ∩ V . Hence d(φ ◦ f )(x) = p.

Since f is holomorphic, f ^∗ (∂∂φ) = ∂∂(φ ◦ f ). Moreover, f ⁰ (T _x ^C L) = T _y ^C N and hence ∂∂(φ ◦ f )| _T

x

L ∼ ∂∂φ| _T

y

N . By Lemma 1.5 this implies s ^± (L, p) = s ^± (N, q).

Finally, since L ⊂ M and p ∈ L × _M T ˙ _M ^∗ X, one has (cf. [D’A-Z 1]) s ^± (L, p) ≤ s ^± (M, p).

q.e.d.

7. Faithfulness of the microlocal direct image functor

Let f : X → Y be a smooth map of complex analytic manifolds and let n = dim ^C X, n − l = dim ^C Y . The manifold V = X × _Y T ˙ ^∗ Y is then identified to an l-codimensional involutive submanifold of T ^∗ X.

Let Λ ⊂ ˙ T ^∗ X be an R-Lagrangian manifold of real type at p ∈ Λ ∩ V and notice that if the intersection Λ ∩ V is clean at p, ^t f ⁰ f _π ⁻¹ (Λ ∩ U ) is again an R-Lagrangian manifold of real type at q = f _π (p) in T ^∗ Y , for a sufficiently small open neighborhood U ⊂ T ^∗ X of p. We will use notations (1.1), (1.2), (1.3), (3.5), (3.6).

Theorem 7.1. Let Λ j (j = 1, 2) be germs of R-Lagrangian manifolds of real type at p ∈ Λ ₁ ∩ Λ ₂ ∩ V . Assume the following:

(a) the intersection Λ _j ∩ V is clean at p, (b) Λ ₁  Λ ₂ at p,

(c) s ⁻ (Λ ₁ , V ) + δ(Λ ₁ , V ) = s ⁻ (Λ ₂ , V ) + δ(Λ ₂ , V ).

Let F j (j = 1, 2) be Levi-simple along Λ j at p. Then (i) the natural morphism

(7.1) f _! ^µ,p : Hom _D

_(X;p) (F ₁ , F ₂ ) −→ Hom _D

_{(Y ;q)} (f _! ^µ,p F ₁ , f _! ^µ,p F ₂ ) is an isomorphism,

(ii) ^t f ⁰ f _π ⁻¹ (Λ ₁ ∩ U )  ^t f ⁰ f _π ⁻¹ (Λ ₂ ∩ U ).

It is also possible to state a slightly more general (and more symmetrical) version of Theorem 7.1 as follows.

Let X be a complex analytic manifold of dimension n. Let V ⊂ ˙ T ^∗ X be a

germ at p of l-codimensional complex analytic regular involutive submanifold. Let

j : V → T ^∗ X be the embedding and let b : V → [V ] be the (locally defined) projection along the bicharacteristic leaves of V . The quotient space [V ] inherits from T ^∗ X a structure of complex homogeneous symplectic manifold. We may then choose an identification of [V ] near b(p) to an open subset of ˙ T ^∗ Y , where Y is a complex manifold of dimension n − l.

If Λ ⊂ ˙ T ^∗ X is a germ of R-Lagrangian manifold of real type at p ∈ V ∩ Λ such that the intersection V ∩ Λ is clean, it is easy to prove that b(V ∩ Λ) ⊂ T ^∗ Y is a germ of R-Lagrangian manifold of real type at q = b(p).

In order to rephrase the correspondence

(7.2) Λ 7→ b(V ∩ Λ)

in terms of simple sheaves, let us rewrite (7.2) as the correspondence associated to the Lagrangian manifold Γ := T ^∗ Y × T

Y V ^a ⊂ T ^∗ (Y × X) (here a denotes the antipodal map),

(7.3)

Γ ⊂ T ^∗ (Y × X)

p

|

. & ^p

^|

T ^∗ Y T ^∗ X

i.e. b(V ∩ Λ) = p ₁ (Γ ∩ p ^a ₂ ⁻¹ Λ) =: Γ ◦ Λ, where p ₁ , p ₂ are the projections from T ^∗ (Y × X) to the corresponding factor.

Theorem 7.2. Let Λ _j (j = 1, 2) be germs of R-Lagrangian manifolds of real type at p ∈ Λ 1 ∩ Λ ₂ ∩ V . Assume the following:

(a) the intersection Λ _j ∩ V is clean at p, (b) Λ 1  Λ ₂ at p,

(c) s ⁻ (Λ ₁ , V ) + δ(Λ ₁ , V ) = s ⁻ (Λ ₂ , V ) + δ(Λ ₂ , V ).

Let F j (j = 1, 2) be Levi-simple along Λ j at p and let K be simple along Γ at (q, p) with shift n − l. Then

(o) the pro-object K ◦ _µ F _j is representable in D ^b (Y ; q), and K ◦ _µ F _j [s ⁻ (Λ _j , V ) + l − δ(λ _j , ρ)] is Levi-simple along Γ ◦ Λ _j at p,

(i) the natural morphism

(7.4) K◦ _µ : Hom _D

_(X;p) (F ₁ , F ₂ ) −→ Hom _D

_{(Y ;q)} (K ◦ _µ F ₁ , K ◦ _µ F ₂ ) is an isomorphism,

(ii) Γ ◦ Λ ₁  Γ ◦ Λ ₂ .

The proof will be performed by reducing to the case where the maps T ^∗ X ← ⁱ V → T ^{b ∗} Y are induced by a smooth map f : X → Y , thus entering the hypotheses of Theorem 7.1.

8. Proof of the faithfulness

Reduction of Theorem 7.2. In the framework of Theorem 7.2, let χ _X : T ^∗ X → T ^∗ X be a contact transformation such that

(8.1)

 

 

 

 



 

 

 

 

χ _X (Λ ₁ ) = T _M ^∗

X, cod ^R _X M ₁ = 1, s ⁻ (M ₁ , p ⁰ ) = 0, χ X (Λ 2 ) = T _M ^∗

X, cod ^R _X M 2 = 1,

χ _X (V ) = X × _Y T ^∗ Y, for a smooth map f : X → Y, f _π ^t f ⁰⁻¹ (T _M ^∗

X) = T _N ^∗

Y, cod ^R _Y N ₁ = 1,

f _π ^t f ⁰⁻¹ (T _M ^∗

X) = T _N ^∗

Y, cod ^R _Y N ₂ = 1

(where we set p ⁰ = χ X (p)), and let χ Y : T ^∗ Y → T ^∗ Y be the complex contact transformation induced by χ _X via the diagram

T ^∗ X ←−−−− ⁱ V −−−−→ T ^{p ∗} Y

χ

 

y ^χ

 

y ^χ

  y T ^∗ X

t

f

←−−−− X × _Y T ^∗ Y −−−−→ T ^f

^∗ Y,

i.e. χ _Y (e q) = f _π (χ _X (e p)) for e p such that e q = b(e p) (the definition is independent of the choice of e p due to the third equality in (8.1)).

Let q = b(p), q ⁰ = f _π (p ⁰ ) = χ _Y (q). By (b), T _M ^∗

X  T _M ^∗

X at p ⁰ , and hence, by Proposition 4.4, s ⁻ (M ₂ , p ⁰ ) = 0 and M ₁ ⁺ ⊃ M ₂ ⁺ . By Lemma 6.3, s ⁻ (N ₁ , q ⁰ ) = s ⁻ (N ₂ , q ⁰ ) = 0. We are then in the following frame:

(8.1)’

 

 

 

 

 



χ _X (Λ _j ) = T _M ^∗

X, cod ^R _X M _j = 1, s ⁻ (M _j , p ⁰ ) = 0, M ₁ ⁺ ⊃ M ₂ ⁺ ,

χ _X (V ) = X × _Y T ^∗ Y, for a smooth map f : X → Y, f _π ^t f ⁰⁻¹ (T _M ^∗

X) = T _N ^∗

Y, cod ^R _Y N _j = 1, s ⁻ (N _j , q ⁰ ) = 0, Notice that, by construction

(8.2) Γ ◦ Λ j = χ ⁻¹ _Y (f π t f ⁰⁻¹ (χ X (Λ j ))).

Let K _X be simple along Λ ^a _χ

at (p ⁰ , −p) with shift n, and let K _Y ^∗ be simple along Λ ^a _χ

at (q, −q ⁰ ) with shift n − l.

In terms of simple sheaves, the equality (8.2) reads (8.3) K ◦ _µ F _j ∼ = K _Y ^∗ ◦ _µ (f _! ^µ,p (K _X ◦ _µ F _j ))

(recall that K _Y ^∗ ◦ _µ · and K _X ^∗ ◦ _µ · are equivalences of categories) and one has K _Y ^∗ ◦ _µ (f _! ^µ,p (K _X ◦ _µ F _j )) ∼ = K _Y ^∗ ◦ _µ (f _! ^µ,p (A _M

[−1]))

∼ = K _Y ^∗ ◦ _µ A N

[−1 − l − d(λ M

, λ 0X , ρ)],

where the first isomorphism follows from Proposition 3.2 (ii), and the second from Proposition 6.1. By Lemma 3.4 we have

d(λ M

, λ 0X , ρ) = δ(λ M

, ρ) − s ⁻ (M j , V )

and hence K ◦ _µ F _j [s ⁻ (M _j , V ) + l − δ(λ _M

, ρ)] is Levi-simple. Since the numbers appearing in the above shift are stable by contact transformation, (o) follows.

In order to prove (i) and (ii) let us note that we are now in the hypotheses of Theorem 7.1, that may then be applied. q.e.d.

Proof of Theorem 7.1. Using the same techniques as in the previous reduction, by applying a complex contact transformation we may assume from the beginning to be in the following frame

 



 

Λ _j = T _M ^∗

X, cod ^R _X M _j = 1, s ⁻ (M _j , p ⁰ ) = 0, M ₁ ⁺ ⊃ M ₂ ⁺ ,

f _π ^t f ⁰⁻¹ (T _M ^∗

X) = T _N ^∗

Y, cod ^R _Y N _j = 1, s ⁻ (N _j , q ⁰ ) = 0,

and we are then left to prove the following claims:

(i)’ the natural morphism

f _! ^µ,p : Hom _D

_(X;p) (A M

, A M

) −→ Hom _D

_{(Y ;q)} (f _! ^µ,p A M

, f _! ^µ,p A M

) is an isomorphism,

(ii)’ Hom _D

_{(Y ;q)} (A _N

, A _N

) 6= 0.

We recall that, by Proposition 6.1,

(8.4) f _! ^µ,p A _M

∼ = A N

[s ⁻ (M _j , V ) − l + δ(λ _M

, ρ)].

By hypothesis (c), we also know that the shifts appearing in (8.4) for j = 1, 2 are equal, and hence (ii)’ will follow from (i)’.

Before proving (i)’, let us state some preliminary results.

9. An analytic lemma

The authors wish to thank Jean-Pierre Schneiders for useful discussions related to the contents of this section.

Let T ⊂ R ^m be a germ of open neighborhood of 0 and let h be a real analytic function on T . We say that h is C ^ω -quadratic if, for a system of real analytic coordinates (t) = (u, v, w) ∈ T , h(t) = u ² − v ² .

Lemma 9.1. Let h _j (j = 1, 2) be C ^ω -quadratic in T . Let s ⁺ (resp. s ⁻ ) be the number of positive (resp. negative) eigenvalues of Hess h ₁ . Assume

(i) h 1 ≥ h ₂ ,

(ii) Hess h ₂ has s ⁺ positive eigenvalues.

Then, there exists a local system of coordinates (t) = (u, v, w) ∈ T , with (u) = (u 1 , . . . , u _s

), (v) = (v 1 , . . . , v _s

), such that at 0:

h ₁ (t) = Q ₁ (u) − v ² + O(u)O(v) + O(|u| ³ ) + O(|v| ³ ), (9.1)

h ₂ (t) = u ² − [v ² + Q ₂ (v, w)] − R ₂ (v, w) + O(u)O(v, w) + O(|u| ³ ) + O(|v| ³ ), (9.2)

where Q ₁ , Q ₂ are quadratic forms with Q ₁ > 0, Q ₂ ≥ 0 and 0 ≤ R ₂ (v, w) = O(w)O(v, w).

Proof. Since h ₁ is C ^ω -quadratic, we can choose coordinates (e t) = (e u, ev, e w) ∈ T , (e u) = (e u ₁ , . . . , e u _s

), (ev) = (ev 1 , . . . , ev s

), such that h ₁ (e t) = e u ² − ev ² . Write

Hess h 2 = Q 3 (e u, ev) + Q ⁴ (e u, ev, e w) + Q 5 ( e w),

where Q ₃ , Q ₄ and Q ₅ are quadratic forms and Q ₄ (e u, ev, e w) = O(e u, ev)O( e w). The

hypothesis (i) and the independence of h ₁ from e w imply that Q ₄ (e u, ev, e w) = 0 and

Q ₅ ( e w) ≤ 0.

By (ii) and the inequality h 1 | _{e u= e w=0} < 0 it follows that there exists a change of coordinates

(9.3)

 



 

u = e e u(u) e v = ev(u, v) w = e e w(w) such that

h 2 | _(v=w=0) = u ² . On the other hand

h ₁ | _(u=0) = −Q ₆ (v) + O(|v| ³ ),

where Q ₆ > 0 is a quadratic form. It is not restrictive to assume Q ₆ (v) = v ² , and so, by (i),

h ₂ | _(u=0) = −[v ² + Q ₇ (v, w)] − R(v, w) + O(|v| ³ ),

where Q ₇ ≥ 0 is a quadratic form and 0 ≤ R(v, w) = O(w)O(v, w). The claim easily follows q.e.d.

For h _j as above and δ, ε > 0, let

Z _j = {t ∈ T ; h _j (t) ≤ 0}

B δ,ε = {t ∈ T ; |u| < ε |v, w| < δε}.

Proposition 9.2. Let h 2 and h 1 be as in Lemma 9.1, and set d = m − s ⁺ . Then there exists δ > 0 such that for every 0 < ε << 1 one has

(i) RΓ _c (Z _j ∩ B _δ,ε ; A _T ) ∼ = A[d], (ii) the morphism

RΓ c (Z 1 ∩ B _δ,ε ; A T ) −→ RΓ c (Z 2 ∩ B _δ,ε ; A T )

induced by the restriction morphism A Z

→ A _Z

(Z 1 ⊃ Z ₂ ) is an isomor- phism.

Denote by a _T : T → {0} the projection. Recall that one has (9.4) RΓ _c (Z _j ∩ B _δ,ε ; A _T ) ∼ = Ra T ! A _Z

∩B

.

Proof. By Lemma 9.1, we may find a local system of coordinates (t) = (u, v, w) on T such that h 1 (t) and h 2 (t) are given by (9.1) and (9.2). It follows that, for some 0 < δ < 1, the small roots of the equations h ₁ (u, v, w) = 0 and h ₂ (u, v, w) = 0 satisfy

δ|u| ≤ |v, w|.

Consider the projections

T −→ V ^q −→ {0}, ^a

where q is the restriction to T of the projection R ^m _u,v,w → R ^d _v,w , and V = q(T )

(notice that it is not restrictive to assume T = B _a,b for some a, b > 0). If we prove