WITH APPLICATIONS TO THE CAUCHY PROBLEM

WITH APPLICATIONS TO THE CAUCHY PROBLEM

Andrea D’Agnolo Pierre Schapira

0. Introduction.

1. Review on sheaves.

1.1. Geometry.

1.2. The category D ^b (X).

1.3. The category D ^b (X; p X ).

2. An inverse image theorem for sheaves.

2.1. The main theorem.

2.2. A particular case.

3. Applications to the Cauchy problem.

3.1. Ramified solutions.

3.2. Ramified solutions of logarithmic type.

3.3. Decomposition at the boundary.

Bibliography.

0. Introduction.

There is a wide literature on the so called Cauchy problem. Let us recall in particular the following papers.

(i) In 1976 Hamada, Leray and Wagschal solved the initial value problem for a linear partial differential equation when the data are ramified along the characteristic hypersurfaces. Their proof of this result relies essentially on the precise Cauchy-Kowalevski theorem of Leray (cf [H-L-W]).

(ii) In 1978 Kashiwara and Schapira proposed a new proof and an extension of the previous work to general systems. This time microdifferential operators and complex contact transformations were involved (cf [K-S 1]).

(iii) In 1988 Schiltz showed how the holomorphic solution for the Cauchy prob- lem can be expressed as a sum of functions which are holomorphic in do- mains whose boundary is given by the real characteristic hypersurfaces is- sued from the boundary of a strictly pseudoconvex domain where the data are defined (cf [Sc]).

The aim of this paper is to propose a new approach to the Cauchy problem based on sheaf theory, or more precisely on its microlocal version. By this method we shall in particular recover the above results and even extend (i) and (iii) to general systems of partial differential equations (i.e. to D-modules).

Appeared in: Duke Math. J. 64 (1991), no. 3, 451–472.

1

Let us describe our work with some details.

In all of this paper we shall systematically use the language of derived categories and sheaves (e.g. cf. [K-S 4, Ch. I,II]).

Let f : Y → X be a morphism of complex analytic manifolds. Let M be a left coherent module over the ring D X of holomorphic differential operators and assume f is non-characteristic for M. Our starting point is the Cauchy-Kowalevski theorem as stated by Kashiwara (cf [K 1], [K 2, Th. 2.5.16]):

(0.1)

 



 

the natural morphism:

f ⁻¹ RHom _D

(M, O _X ) −→ RHom _D

(M _Y , O _Y ), is an isomorphism.

Here M Y denotes the induced system on Y and O X is the sheaf of holomorphic functions on X.

Let K be an object of D ^b (X), the derived category of sheaves in X. By an appropriate choice of K to be explained in §3, the complex RHom(K, O _X ) is a complex of D X -modules representing a sheaf of “ramified” holomorphic functions.

Next, let L be an object of D ^b (Y ), ψ : L → f ⁻¹ K be a morphism, and set F = RHom D

(M, O X ). Due to (0.1), theorems as those given in (i), (ii) or (iii), may be stated as follows.

(0.2)

 



 

the morphism induced by ψ:

f ⁻¹ RHom(K, F )  

Z −→ RHom(L, f ⁻¹ F )  

Z , is an isomorphism on some subset Z of Y . Hence we are reduced to the following sheaf theoretical problem.

Problem 0.1. Give conditions on f , F , K, L and ψ as above so that (0.2) holds.

Our answer (Theorem 2.1.1 below) relies on the microlocal theory of sheaves as developed in [K-S 4]. We shall apply it to the Cauchy problem using (0.1) and the inclusion (cf. [K-S 4, Th. 11.3.3]):

(0.3) SS(RHom D

(M, O X )) ⊂ char(M),

where SS(F ) denotes the micro-support of the object F of D ^b (X) (cf. [K-S 4, Def.

5.1.2]).

We now give the plan of this paper.

In §1 we make a short review on the theory of sheaves, mainly to fix the notations.

In particular we recall the notion of micro-support, the functor µhom, the category D ^b (X; p X ) that is the localization of the derived category of sheaves on X at p X ∈ T ^∗ X.

In §2 we state and prove our main theorem, namely Theorem 2.1.1, on the well poseness for the Cauchy problem, in a sheaf theoretical frame.

In §3, as an application of Theorem 2.1.1, we show how to recover the results obtained in the papers cited at the beginning of this introduction.

Let us put the emphasis on the fact that all our proofs rely only on the following

tools: the Cauchy-Kowalevski theorem in the precise version by Leray [L] from

which the inclusion (0.3) is deduced, the isomorphism (0.1) which is deduced by

Kashiwara [K] from the classical Cauchy-Kowalevski theorem by purely algebraic

tools, and of course, the techniques of [K-S 3], [K-S 4]. We never use, neither pseudo-differential operators and quantized contact transformations, nor any esti- mates.

Of course our purely sheaf theoretical methods do not allow us to treat holo- morphic functions with growth conditions (e.g. meromorphic solutions). For these questions we refer to [M-F], [La].

The results of this paper were announced in our note [D’A-S].

Let us mention that further developments of Theorem 2.1.1 allow in particular to recover a result of [K-S 2] on the hyperbolic Cauchy problem in the frame of hyperfunctions (cf. [D’A]).

1. Review on sheaves.

For what follows refer to [K-S 4].

Until §3 all manifolds and morphisms of manifolds will be real and of class C ^∞ . 1.1. Geometry. Let X be a manifold, τ _X : T X → X its tangent bundle and π X : T ^∗ X → X its cotangent bundle. We will identify X with T _X ^∗ X, the zero section of T ^∗ X. Set ˙ T ^∗ X = T ^∗ X \ T _X ^∗ X and denote by ˙π _X the projection ˙ T ^∗ X → X.

Let M be 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 C _M (A) the normal cone of A along M . This is a closed conic subset of T _M X, the normal bundle to M in X.

More generally, if B is another subset of X, one denotes by C(A, B) the normal cone of A along B, a closed conic subset of T X (see [K-S 4, Def 4.1.1]).

If γ is a conic subset of T X one notes γ ^a = −γ. One says that γ is proper if its fibers contain no lines. One denotes by γ ^◦ the polar cone to γ, a convex conic subset of T ^∗ X. Recall that its fibers are given by γ _x ^◦ = {θ ∈ T _x ^∗ X; hv, θi ≥ 0 ∀v ∈ γ _x }.

Let f : Y → X be 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 ).

Let N (resp. M ) be 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.

Let A be 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. Let V be a subset of T ^∗ Y . We refer to

[K-S 4, Def. 6.2.7] for the intrinsic definition of f being non-characteristic for A on

V . Let (x) (resp. (y)) be a system of local coordinates on X (resp. Y ), and let (x; ξ)

(resp. (y; η)) be the associated coordinates on T ^∗ X (resp. T ^∗ Y ). We recall that

f is non-characteristic for A on V iff for every (y; η) ∈ V there are no sequences

{(y _n , (x n ; ξ n ))} in Y × A such that:

 

 

 

 

 

 

 

 

 

 y _n →

n y, x _n →

n f (y),

t f ⁰ (y _n ) · ξ _n →

n η,

|x _n − f (y _n )||ξ _n | →

n 0,

|ξ _n | →

n ∞.

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.

To an object F of D ^b (X) one associates the micro-support SS(F ) of F , a closed conic involutive subset of T ^∗ X (cf [K-S 4]).

Let M be a closed submanifold of X. One associates to F the Sato’s microlocal- ization of F along M , denoted by µ _M (F ) (cf. [K-S 4, Def. 4.3.1]). This is an object of D ^b (T _M ^∗ X). More generally, if G is another object of D ^b (X), the microlocalization of F along G, denoted by µhom(G, F ), is defined in [K-S 4] as:

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. By identifying T _∆ ^∗ (X × X) to T ^∗ X by the first projection T ^∗ X × T ^∗ X → T ^∗ X, one sees that µhom(G, F ) belongs to D ^b (T ^∗ X). Concerning this functor one proves that:

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, 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.

Let f : Y → X be a morphism of manifolds. One defines the relative dualizing complex ω _{Y /X} by ω _{Y /X} = f ^! A X . One sets ω X = a ^! _X A, where a X : X → {pt}.

Recall that one has an isomorphism ω _{Y /X} ⊗ f ⁻¹ ω _X ∼ = ω Y . Recall moreover that ω _X ∼ = or X [dim X] where or _X is the orientation sheaf and dim X is the dimension of X. If there is no risk of confusion, when working on T ^∗ X, we shall write ω _X , or _X , ω _{Y /X} , etc. instead of π _X ⁻¹ ω _X , π ⁻¹ _X or _X , π _Y ⁻¹ ω _{Y /X} , etc.

Let F be an object of D ^b (X) and let V be a subset of T ^∗ Y . One says that f is non-characteristic for F on V if f is non-characteristic for SS(F ) on V .

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 }).

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). 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 may enlarge the category D ^b (X; p X ) and work with ind-objects and pro-objects. This is what has been done in [K-S 4, ch. 6], but there is another way to attack this problem. In this section we will give the definition of microlocal images for complexes with some prescribed conditions on the micro-support and to this end we need the following result on the cutting of the micro-support.

Let X be a vector space and let x ₀ ∈ X. In what follows we will often identify X with T x

X.

Let γ be a (not necessarily proper) closed convex cone of T _x

X. Let ω be an open neighborhood of x 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 1.3.1. 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 ).

(1.3.1)

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 ).

(1.3.2)

After [K-S 4, Prop. 6.1.4] we give the following definition (here x ₀ = 0).

Definition 1.3.2. Take ξ 0 ∈ ˙ T ₀ ^∗ X, ω ⊂ X 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) ∂ω and ∂γ are tangent at their intersection.

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

Remark that, if g(x) < 0 (resp. h(x) < 0) is a local equation for ω (resp. γ) at x ∈ ∂ω ∩ ∂γ, condition (vii) means that −d g(x) ∈ R ⁺ d h(x).

One has the following sharp result.

Proposition 1.3.3. (cf [K-S 4, Prop. 6.1.4, 6.1.8]) 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 .

Sketch of the proof. Keep the same notations as in Definition 1.3.1, 1.3.2. 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 . Then it is possible to show that Φ _X (γ, ω; F ) (resp. Ψ _X (γ, ω; F )) satisfies the requirements. q.e.d.

We can now define microlocal images.

Let F ∈ Ob(D ^b (X; p _X )) and consider the condition:

(1.3.3) ^t f ⁰⁻¹ (p _Y ) ∩ f _π ⁻¹ (SS(F )) ⊂ {p} in a neighborhood of p.

Lemma 1.3.4. (cf [K-S 4, Prop. 6.1.9]) Let F ∈ Ob(D ^b (X; p X )) satisfy (1.3.3), then:

a) There exist F ⁰ ∈ Ob(D ^b (X)) with:

(1.3.4) ^t f ⁰⁻¹ (p _Y ) ∩ f _π ⁻¹ (SS(F )) ⊂ {p},

and a morphism F ⁰ → F (resp. F → F ⁰ ) in D ^b (X) which is an isomorphism at p _X .

b) For F ∈ Ob(D ^b (X; p X )) satisfying (1.3.3), the object f ⁻¹ F ⁰ (resp. f ^! F ⁰ ) of the category D ^b (Y ; p _Y ) does not depend (up to isomorphism) on the choice of F ⁰ .

Definition 1.3.5. Let F ∈ Ob(D ^b (X; p X )) satisfy (1.3.3). We define the microlo- cal inverse image (resp. extraordinary inverse image) of F by f _µ,p ⁻¹ F := f ⁻¹ F ⁰ (resp.

f _µ,p ^! F := f ^! F ⁰ ), where F ⁰ is the complex constructed in Lemma 1.3.4.

f _µ,p ⁻¹ (resp. f _µ,p ^! ) is a functor from the full subcategory of D ^b (X; p _X ) whose objects verify (1.3.3) to D ^b (Y ; p Y ).

Note that the definition of f _µ,p ⁻¹ F , f _µ,p ^! F as in this lemma is coherent with the definition given in [K-S 4,§6].

Proof of Lemma 1.3.4. Since the proofs for f ⁻¹ and f ^! are similar, we will treat

only the statement concerning f ⁻¹ . If p _X ∈ T _X ^∗ X then f is non-characteristic for

F and we may take F ⁰ = F . Assume then that p _X ∈ ˙ T ^∗ X. Following the proof

of Proposition 1.3.3 we may choose a refined cutting pair (γ, ω) on X at p _X so

that F ⁰ = Φ _X (γ, ω; F ) satisfies (1.3.4). This proves a). As for b) we have to show

that if F ⁰ ∼ = F ⁰⁰ at p X , F ⁰⁰ satisfying (1.3.4), then f ⁻¹ F ⁰ ∼ = f ⁻¹ F ⁰⁰ at p Y . It is not restrictive to assume the isomorphism F ⁰ ∼ = F ⁰⁰ being induced by a morphism u : F ⁰ → F ⁰⁰ in D ^b (X). Embedding u in a distinguished triangle F ^{0 u} → F ⁰⁰ → F ₀ ⁺¹ →, we see that f is non-characteristic for F ₀ and that ^t f ⁰⁻¹ (p _Y ) ∩ f _π ⁻¹ (SS(F ₀ )) = ∅.

Then f ⁻¹ F ⁰ ∼ = f ⁻¹ F ⁰⁰ at p Y . q.e.d.

2. An inverse image theorem for sheaves.

2.1 The main theorem. In this paragraph we will give an answer to Problem 0.1.

Let X be a real analytic manifold. We say that K ∈ Ob(D ^b (X)) is weakly coho- mologically 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 and we made use the notion of “lim”

−→ and “lim”

←− (e.g. cf. [K-S 4, §1.11]).

Note that weakly R-constructible complexes on a real analytic manifold are w-c-c (cf [K-S 4, §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 2.1.1. 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 is non-characteristic for K on V and p 1 , . . . , p r are isolated in ^t f ⁰⁻¹ (p Y )∩

f _π ⁻¹ (SS(K)),

(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:

(2.1.1) f ⁻¹ RHom(K, F )   _Z −→ RHom(L, f ⁻¹ F )   _Z , is an isomorphism.

Recall that f _µ,p ⁻¹

K in (v) has been defined in Lemma 1.3.4.

Before going into the proof let us explain how the morphism in (vii) is con-

structed. From ψ and the natural morphism f ⁻¹ K ⊗ ω _{Y /X} → f ^! K, we obtain the

arrow L⊗ω _{Y /X} → f ^! K, and hence the arrow L⊗ω Y → f ^! K ⊗f ⁻¹ ω X ∼ = f ^! (K ⊗ω X ).

Applying the functor RΓ _{y} we obtain:

RΓ _{y} (L ⊗ ω _Y ) → RΓ _{y} f ^! (K ⊗ ω _X )

∼ = RΓ {x} (K ⊗ ω _X ).

Proof. We shall adapt to our situation the proof of [K-S 1,Theorem 3.3]. Recall that one has a distinguished triangle in D ^b R

(T ^∗ X):

Rπ _{Y !} (·) → Rπ _{Y ∗} (·) → R ˙π _{Y ∗} (·) ⁺¹ →

(here D ^b R

(T ^∗ X) denotes the full subcategory of D ^b (X) whose objects have locally constant cohomology along the orbits of the action of R ⁺ on T ^∗ X). If we apply this triangle to the morphism induced by ψ:

R ^t f _! ⁰ f _π ⁻¹ µhom(K, F ) → µhom(L, f ⁻¹ F ), we get the morphism of distinguished triangles:

(2.1.2)

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 ).

Recall that Rπ Y ∗ µhom(L, f ⁻¹ F ) ∼ = RHom(L, f ⁻¹ F ). On the other hand con- sider the diagram:

T ^∗ Y

t

f

←−−−− Y × _X T ^∗ X −−−−→ T ^f

^∗ X

π

 

y ^π

 

y ^π

  y

Y ←−−−− ^∼ Y −−−−→ X. ^f

Since f is non-characteristic for F , ^t f ⁰ is proper on f _π ⁻¹ (SS(F )) and hence it is proper on f _π ⁻¹ (supp(µhom(K, F ))). Then one has the isomorphisms:

Rπ _{Y ∗} R ^t f _! ⁰ f _π ⁻¹ µhom(K, F ) ∼ = Rπ Y ∗ R ^t f _∗ ⁰ f _π ⁻¹ µhom(K, F )

∼ = Rπ ∗ f _π ⁻¹ µhom(K, F )

∼ = f ⁻¹ Rπ X∗ µhom(K, F )

∼ = f ⁻¹ RHom(K, F ),

where the third isomorphism follows from the fact that, µhom(K, F ) being in D ^b R

(T ^∗ X), Rπ _X∗ µhom(K, F ) ∼ = i ⁻¹ µhom(K, F ), i denoting the immersion of the zero-section X in T ^∗ X.

Thus, the second vertical arrow in (2.1.2) is nothing but the morphism (2.1.1),

and it is enough to prove that the first and the third vertical arrows in (2.1.2) are

isomorphisms at every y ∈ Z.

A. First vertical arrow.

Consider the following diagram where δ _Y , δ _X denote the diagonal embeddings and δ the graph embedding:

Y × Y −−−−→ X × Y ^f

−−−−→ X × X ^f

δ

x 

 ^δ

x 

 ^δ

x 

 Y ←−−−− ^∼ Y −−−−→ ^f X.

We shall denote by q ₁ , q ₂ the first and second projections defined on a product of two spaces.

Recall (cf [K-S 4, §4.3]) that:

Rπ _{Y !} µhom(L, f ⁻¹ F ) ∼ = δ _Y ⁻¹ RHom(q ₂ ⁻¹ L, q ₁ ^! f ⁻¹ F ) ⊗ ω _Y ^⊗−1 , On the other hand one has the chain of isomorphisms:

Rπ Y ! R ^t f _! ⁰ f _π ⁻¹ µhom(K, F ) ∼ = Rπ ! f _π ⁻¹ µhom(K, F )

∼ = f ⁻¹ Rπ X! µhom(K, F )

∼ = f ⁻¹ δ _X ⁻¹ RHom(q ₂ ⁻¹ K, q ^! ₁ F ) ⊗ ω ^⊗−1 _X

∼ = δ _Y ⁻¹ f ˜ ⁻¹ RHom(q ₂ ⁻¹ K, q ^! ₁ F ) ⊗ ω ^⊗−1 _X , The map f being non-characteristic for F , we get:

f ˜ ⁻¹ RHom(q ₂ ⁻¹ K, q ^! ₁ F ) ⊗ ω ^⊗−1 _X ∼ =

∼ = f ₂ ⁻¹ f ₁ ⁻¹ RHom(q ₂ ⁻¹ (K ⊗ ω _X ), q ₁ ^! F )

∼ = f ₂ ⁻¹ RHom(q ₂ ⁻¹ (K ⊗ ω _X ), q ^! ₁ f ⁻¹ F ).

It is then enough to prove the isomorphism for every y ∈ Z, x = f (y):

RHom(q ₂ ⁻¹ (K ⊗ ω _X ), q ₁ ^! f ⁻¹ F ) _(y,x) −→ ^∼

−→ RHom(q ∼ ₂ ⁻¹ (L ⊗ ω _Y ), q ^! ₁ f ⁻¹ F ) _(y,y) . (2.1.3)

For an open neighborhood V ⊂ Y of y and for U ⊂ Y ranging over an open neighborhood system of y, we have by (vi):

“lim”

−→

U 3y

RΓ(V × U ; RHom(q ₂ ⁻¹ (L ⊗ ω _Y ), q ₁ ^! f ⁻¹ F )) ∼ =

∼ = “lim”

−→

U 3y

RHom(RΓ _c (U ; L ⊗ ω _Y ), RΓ(V ; f ⁻¹ F ))

∼ = RHom(“lim”

←−

U 3y

RΓ _c (U ; L ⊗ ω _Y ), RΓ(V ; f ⁻¹ F ))

∼ = RHom(RΓ {y} (Y ; L ⊗ ω _Y ), RΓ(V ; f ⁻¹ F )), (2.1.4)

(recall that the first equality follows from the Poincar´e-Verdier duality). Similarly to (2.1.4) we get:

“lim”

−→

W 3x

RΓ(V × W ; RHom(q ⁻¹ ₂ (K ⊗ ω _X ), q ^! ₁ f ⁻¹ F )) ∼ =

∼ = RHom(RΓ _{x} (X; K ⊗ ω _X ), RΓ(V ; f ⁻¹ F )),

where now V ⊂ Y is an open neighborhood of y and W ⊂ X ranges over an open neighborhood system of x. Recalling the hypothesis (vii), (2.1.3) follows.

B. Third vertical arrow.

We have to prove that the natural morphism:

(2.1.5) 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).

This will follow from the following Proposition 2.1.2. q.e.d.

Proposition 2.1.2. 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 p _Y and assume:

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

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

Assume moreover that 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) p ₁ , . . . , p _r are isolated in ^t f ⁰⁻¹ (p _Y ) ∩ f _π ⁻¹ (SS(K)).

(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.

Then the natural morphism (2.1.5) induced by ψ is an isomorphism.

For the proof of this proposition we will need the following lemma which is an immediate consequence of [K-S 4, Prop. 6.7.1].

Lemma 2.1.3. Let F , K be objects of D ^b (X). Let V be an open neighborhood of p Y and assume:

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

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

Assume that there exists p ∈ ^t f ⁰⁻¹ (p Y ) with:

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

Then the natural morphism R ^t f _! ⁰ f _π ⁻¹ µhom(K, F ) _p

→ µhom(f ⁻¹ K, f ⁻¹ F ) _p

is an isomorphism.

Proof of Proposition 2.1.2. Following the proof of Proposition 1.3.3 and due to assumptions (iii) and (iv), we can find refined cutting pairs (γ _i , ω _i ) (i = 1, . . . , r) on X at p _X such that, setting F _i = Φ _X (γ _i , ω _i ; F ), K _i = Φ _X (γ _i , ω _i ; K), and for K ₀ being the third term of a distinguished triangle ⊕ ^r _i=1 K _i → K → K ₀ ⁺¹ →, we have:

— f _π ⁻¹ SS(F _i ) ∩ ^t f ⁰⁻¹ (p _Y ) ⊂ {p _i },

— f _π ⁻¹ SS(K _i ) ∩ ^t f ⁰⁻¹ (p _Y ) ⊂ {p _i } and p _i ∈ SS(K / ₀ ) for i = 1, . . . , r,

— F ∼ = ⊕ ^r _i=1 F _i and K ∼ = ⊕ ^r _i=0 K _i in D ^b (X; f _π ^t f ⁰⁻¹ (p _Y )).

Moreover, since f is non-characteristic for the F i ’s, we have:

— f ⁻¹ F ∼ = ⊕ ^r _i=1 f ⁻¹ F _i in D ^b (Y ; p _Y )

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 _i=1 ⊕ ^r _j=0 µhom(K _j , F _i )) _p

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

∼ = ⊕ ^r _i=1 µhom(f ⁻¹ K i , f ⁻¹ F i ) p

∼ = ⊕ ^r _i=1 µhom(f _µ,p ⁻¹

K, f ⁻¹ F _i ) _p

∼ = ⊕ ^r _i=1 µhom(L, f ⁻¹ F _i ) _p

∼ = µhom(L, f ⁻¹ F ) _p

.

Here the second isomorphism follows from the fact that µhom is a microlocal func- tor, the third one follows from Lemma 2.1.3 and the forth from assumption (iv).

q.e.d.

2.2. A particular case. Let f : Y → X be a morphism of manifolds. Let Z be a closed subset of Y . Let F be an object of D ^b (X) such that:

(2.2.1) f is non-characteristic for F .

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

(2.2.2) ^t f ⁰⁻¹ (p Y ) ∩ f _π ⁻¹ (SS(F )) ⊂ {p 1 , . . . , p r }.

Let K _i (i = 1, . . . , r) be objects of D ^b (X) such that:

(2.2.3) K _i is w-c-c,

(2.2.4) f is non-characteristic for K i , (2.2.5) a morphism τ _i : K _i → A _X is given.

Assume that for every p _Y ∈ ˙π _Y ⁻¹ (Z) and for p ₁ , . . . , p _r as in (2.2.2):

(2.2.6) ^t f ⁰⁻¹ (p _Y ) ∩ f _π ⁻¹ (SS(K _i )) ⊂ {p _i }.

Let L be an object of D ^b (Y ) such that:

(2.2.7) an isomorphism ψ _i : L → f ⁻¹ K _i is given,

(2.2.8) f ⁻¹ (τ _i ) ◦ ψ _i induces an isomorphism RΓ _Z L → RΓ _Z A _Y . Assume moreover that for an open neighborhood V of ˙π _Y ⁻¹ (Z):

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

We shall give here a way to build up a complex K ∈ Ob(D ^b (X)), in order to satisfy the hypotheses of Theorems 2.1.1.

Let h be the composite of the map ⊕ ^r _i=1 τ _j : ⊕ ^r _i=1 K _i → ⊕ ^r _i=1 A _X and the map

⊕ ^r _i=1 A _X → ⊕ ^r−1 _i=1 A _X , given by (a ₁ , . . . , a _r ) 7→ (a ₂ − a ₁ , . . . , a _r − a _r−1 ).

Let K be defined by embedding h in a distinguished triangle:

(2.2.10) K −→ ⊕ ^r _i=1 K _i −→ ⊕ ^{h r−1} _i=1 A _X −→ . ⁺¹ It is now straightforward to verify the following:

Lemma 2.2.1. K is w-c-c in D ^b (X) and the following estimate holds:

SS(K) = ∪ i SS(K i ) ∪ T _X ^∗ X.

By completing the morphism of distinguished triangles (where the vertical arrows are the natural ones):

(2.2.11)

f ⁻¹ K −−−−→ ⊕ ^r _i=1 f ⁻¹ K j −−−−→ ⊕ ^r−1 _i=1 f ⁻¹ A X +1

−−−−→

x 

 x  

L −−−−→ ⊕ ^r _i=1 L −−−−→ ⊕ ^r−1 _i=1 L −−−−→ , ⁺¹

we get a morphism

(2.2.12) ψ : L −→ f ⁻¹ K.

Lemma 2.2.2. The morphism ψ induces an isomorphism:

RΓ _Z L −→ RΓ ^∼ _Z f ⁻¹ K.

Proof. One has the morphism of distinguished triangles:

RΓ _Z f ⁻¹ K → ⊕ ^r _i=1 RΓ _Z f ⁻¹ K _j → RΓ _Z f ⁻¹ ⊕ ^r−1 _i=1 A _X ⁺¹ →

↓ ↓ ↓

RΓ Z f ⁻¹ A X → ⊕ ^r _i=1 RΓ Z f ⁻¹ A X → RΓ _Z f ⁻¹ ⊕ ^r−1 _i=1 A X +1

→ .

The third vertical arrow is the identity and the second one is an isomorphism by (2.2.7) and (2.2.8). Hence RΓ Z f ⁻¹ K ∼ = RΓ Z A Y ∼ = RΓ Z L. q.e.d.

We can then state the following:

Proposition 2.2.3. Let be given f , F , K i and L satisfying the hypotheses (2.2.1)–

(2.2.9). For K and ψ constructed in (2.2.10) and (2.2.12), the hypotheses of The- orem 2.1.1 are satisfied.

The proof is immediate if one just notice that, due to (2.2.4), the hypothesis (vii) of Theorem 2.2.1 reads as

RΓ _{y} L −→ RΓ ^∼ _{y} (f ⁻¹ K).

3. Applications to the Cauchy problem.

In this section, as an application of Theorem 2.1.1, we will show how to recover the results obtained in the papers cited at the beginning of the Introduction.

For our purpose, the base ring A introduced at the beginning of §1.2 is now the field C. But in order to avoid confusion with the complex line, we still denote it by A.

3.1 Ramified solutions. (cf [H-L-W].) Let Y be a complex analytic manifold and let Z be a smooth complex hypersurface of Y . First, we shall construct a sheaf L on Y such that, for G ∈ Ob(D ^b (Y )), RHom(L, G) describes in some sense the

“ramified sections of G along Z” (cf [G] and [D]).

Let p : f C ^∗ → C be the universal covering of C ^∗ = C \ {0}. Recall that one can choose a coordinate t ∈ C ∼ = f C ^∗ so that p(t) = exp(2πit). Choose a complex analytic function g : Y → C such that Z = g ⁻¹ (0) (this is possible locally). Set Y f ^∗ = f C ^∗ × C Y and consider the Cartesian diagram:

(3.1.1)

Y f ^∗ −−−−→ f C ^∗

p

 

y ^p   y

Y −−−−→ C. ^g

We will call the complex Rp Y ∗ p ⁻¹ _Y G the complex of ramified sections of G along Z.

Notice that p Y ! is an exact functor and that p ^! _Y = p ⁻¹ _Y . By the Poincar´e-Verdier duality one gets:

Rp _{Y ∗} p ⁻¹ _Y G = RHom(p _{Y !} A _{Y f}

, G)

= RHom(g ⁻¹ p ! A _{C f}

, G).

Set L = g ⁻¹ p _! A _{C f}

and notice that, by adjunction, there is a natural morphism:

(3.1.2) τ : L −→ A _Y .

Let us now describe what will be our geometrical frame.

(3.1.3) X is a complex analytic manifold, Y is a smooth hypersurface of X, Z is a smooth hypersurface of Y , Z _i (i = 1, . . . , r) are smooth hypersurfaces of X pairwise transversal, transversal to Y and such that Z i ∩ Y = Z for every i. Let f : Y → X be the embedding. Assume to be given complex analytic functions g : Y → C, g i : X → C with dg 6= 0, dg i 6= 0, such that g _i ◦ f = g and Z = g ⁻¹ (0), Z _i = g ⁻¹ _i (0).

Let M be a left coherent D _X -module such that, for a neighborhood V of ˙ T _Z ^∗ Y : (3.1.4)

( (i) f π is non characteristic for C(char(M), ˙ T _Z ^∗

X) on ^t f ⁰⁻¹ (V ), (ii) char(M) ∩ ^t f ⁰⁻¹ (T _Z ^∗ Y ) ⊂ ∪ _i T _Z ^∗

X ∪ T _X ^∗ X.

Note that (3.1.4)-(ii) implies that f is non-characteristic for M (i.e. f is non- characteristic for char(M)).

Set K _i = g _i ⁻¹ p _! A _{C f}

(i = 1, . . . , r) and F = RHom _D

(M, O _X ). Of course SS(K _i ) ⊂ T _Z ^∗

X ∪ T _X ^∗ X. One has that L = f ⁻¹ K _i and the natural morphisms τ _i : K _i → A _X may be constructed as in (3.1.2).

Lemma 3.1.1. The morphism τ induces an isomorphism:

RΓ _Z L −→ RΓ ^∼ _Z A _Y . Proof. Since g is a smooth map, one has that:

RΓ _Z L ' RΓ _Z g ⁻¹ p _! A _{C f}

' g ⁻¹ RΓ _{0} p _! A _{C f}

, and similarly:

RΓ Z A Y ' RΓ _Z g ⁻¹ A ^C ' g ⁻¹ RΓ _{0} A ^C . It is therefore enough to prove the isomorphism:

RΓ _{0} p ! A _{C f}

−→ RΓ ∼ _{0} A ^C . Since p _! A _{C f}

is R ⁺ -conic for the action of R ⁺ on C, one has:

RΓ(C, RΓ _{0} p _! A _{C f}

) ∼ = RΓ c (C, p _! A _{C f}

)

= RΓ c ( f C ^∗ , A _{C f}

)

∼ = A[2],

and hence RΓ _{0} p _! A _{C f}

∼ = A {0} [2] ∼ = RΓ {0} A C follows. q.e.d.

The complex O ^ram _Z|Y := RHom(L, O _Y ) is concentrated in degree 0 and its sections are the holomorphic functions ramified along the hypersurface Z and similarly for the complex O ^ram _Z

i

|X = RHom(K _i , O _X ). For K constructed from the K _i ’s as in (2.2.10), set Σ _i O _Z ^ram

i

|X = RHom(K, O _X ). Applying Theorem 2.1.1 (cf. Proposition

2.2.3) we get:

Theorem 3.1.2. Keeping the same notations as above, the natural morphism:

(3.1.5) RHom _D

(M, Σ _i O ^ram _Z

i

|X )    _Z −→ RHom _D

(M _Y , O ^ram _Z|Y )    _Z , is an isomorphism.

Let us now explain how Theorem 3.1.2 gives an extension to D _X -modules of [H-L-W]’s result.

Let X be an open subset of C ⁿ with 0 ∈ X. Take a local system z = (z ₁ , z ⁰ ) = (z ₁ , . . . , z _n ) of complex coordinates on X. Let (z; ζ) be the associated coordinates in T ^∗ X. Set D = ∂/∂z. Let P = P (z, D) be a linear partial differential operator of order m on X with holomorphic coefficients for which the hyperplane Y = {z ∈ X; z ₁ = 0} is non-characteristic. Denote by f : Y → X the embedding and set Z = {z ∈ X; z 1 = z 2 = 0}. Suppose that P has characteristics with constant multiplicities transversal to Y × _X T ^∗ X at ^t f ⁰⁻¹ (T _Z ^∗ Y ) ∩ char(P ).

Proposition 3.1.3. Under the previous hypotheses there exist smooth hypersur- faces Z 1 , . . . , Z r of X as in (3.1.3) and satisfying (3.1.4) for M = D X /D X P .

This result is classical. For the reader’s convenience we recall the proof following [S, Prop. 2.2.2].

Proof. Let p(z; ζ) be the principal symbol of P (z, D). Decompose it as a product of irreducible polynomials p(z; ζ) = Q r

i=1 p _i (z; ζ) ^d

, and set q(z; ζ) = Q r

i=1 p _i (z; ζ).

One has char(P ) = q ⁻¹ (0). The fact that P has characteristics with constant multiplicities transversal to Y × _X T ^∗ X at ^t f ⁰⁻¹ (T _Z ^∗ Y ) ∩ char(P ) means that the Poisson bracket {z 1 , q} does not vanish at any point of ^t f ⁰⁻¹ (T _Z ^∗ Y ) ∩ char(P ).

Since ∂q(z; ζ)/∂ζ ₁ = −{z ₁ , q} and q(z; 1, 0, . . . , 0) 6= 0 for z ∈ Y , the equation q(z; τ, 1, 0, . . . , 0) = 0 has r distinct roots for z ∈ Z. Hence Λ ⁰ = ^t f ⁰⁻¹ (T _Z ^∗ Y ) ∩ char(P ) is the disjoint union of r isotropic smooth manifolds Λ ⁰ ₁ , . . . , Λ ⁰ _r of T ^∗ X.

The Hamiltonian vector field H q is transversal to Y × X T ^∗ X at Λ ⁰ _i and therefore the union of the integral curves of H _q issuing from Λ ⁰ _i is a Lagrangian manifold Λ i of T ^∗ X, contained in char(P ). Let Z i = π X (Λ i ). Since the projection on X of the vector field (H _q )| _Λ

is a vector field transversal to Y , Z _i is a smooth hyper- surface of X transversal to Y and, Λ i being Lagrangian, Λ i = ˙ T _Z ^∗

X. Finally, f π

is non-characteristic for C(char(P ), ˙ T _Z ^∗

X) on ^t f ⁰⁻¹ (T _Z ^∗ Y ) since ˙ T _Z ^∗

X ⊂ char(P ).

q.e.d.

Consider the Cauchy problem:

(3.1.6)

 P (z, D) u(z) = v(z),

D ₁ ^h u(z)   _Y = w _h (z ⁰ ), 0 ≤ h < m.

In [H-L-T], Hamada, Leray and Wagschal proved that the Cauchy problem (3.1.6) has a unique solution u(z) = P

ϕ i (z) for any v(z) = P

ψ i (z), where w h is a ramified holomorphic function on Y along Z and ϕ _i (resp. ψ _i ) is ramified on X along Z _i .

We can then apply Theorem 3.1.2 to M = D X /D X P .

One has M _Y ∼ = (D Y ) ^m and hence RHom _D

(M _Y , O _Z|Y ^ram ) ∼ = (O _Z|Y ^ram ) ^m represents the sheaf of ramified Cauchy data. The complex RHom _D

(M, O _X )| _Y is concen- trated in degree zero. From (2.2.10) we get a distinguished triangle:

⊕ ^r−1 _i=1 RHom _D

(M, O _X ) −→ ⊕ ^r _i=1 RHom _D

(M, O _Z ^ram

_|X ) −→

−→ RHom _D

(M, Σ i O ^ram _Z

i

|X ) −→ . ⁺¹

(3.1.7)

By (3.1.5), the complex RHom D

(M, Σ i O _Z ^ram

i

|X )| Z is concentrated in degree zero and its sections may be expressed as a sum P

i ϕ _i , where the ϕ _i ’s are holomorphic functions ramified along Z _i , satisfying the equation P ϕ _i = 0. That is to say: by taking the zero-th cohomology group in (3.1.5), Theorem 3.1.2 ensures the existence and the uniqueness for the solution of the Cauchy problem (3.1.6) with v = 0, the solution being a sum P

i ϕ _i , where ϕ _i is a holomorphic function on X ramified along Z _i .

Moreover the vanishing of H ¹ (RHom _D

(M, Σ _i O _Z ^ram

i

|X ))| _Z , which follows from (3.1.5), ensures the solubility of the equation

(3.1.8) P (z, D)u(z) = v(z)

for u and v of the form P

i ϕ _i as above.

Remark 3.1.4. In [Le], Leichtnam solves the Cauchy problem P f = g where f and g are ramified on X \ ∪Z _i (with the above notations). Theorem 2.1.1 does not allow us to treat Leichtnam’s result, but nevertheless we hope to recover it by similar geometrical methods in the future.

3.2. Ramified solutions of logarithmic type. In [K-S 1] a theorem on the existence and uniqueness for the solution of (3.1.1) is given when the ramification involved are of logarithmic type. We will give here a new proof of their theorem.

Let z ∈ C be a coordinate and set D = ∂/∂z. Consider the left coherent D ^C - module N = D C /D C DzD.

For L ¹ _{0}|C := RHom D

(N , O ^C ) set O _{0}|C ¹ := RHom(L ¹ _{0}|C , O ^C ) = N ^∞ . This represents a complex of holomorphic functions with logarithmic ramification.

Take M C a left coherent D C -module.

If one makes the choice F = RHom _D

(M ^C , O ^C ), then:

RHom D

(M ^C , O _{0}|C ¹ ) ∼ = RHom(L ¹ _{0}|C , F ).

Moreover one has the following Lemma 3.2.1.

a) There is a natural map

τ : L ¹ _{0}|C −→ A ^C ; b) the morphism induced by τ :

RΓ _{0} L ¹ _{0}|C −→ RΓ _{0} A ^C , is an isomorphism.

Proof. To prove a) notice that L ¹ _{0}|C = RHom _D

(N , O ^C ) is concentrated in degree 0 and is represented by the complex

0 −→ O ^{C DzD} −→ O C −→ 0.

A section of L ¹ _{0}|C is then represented by a function ϕ ∈ O C with DzDϕ = 0. We

define τ by τ (ϕ) = zDϕ.

As for b), consider the exact sequence of D ^C -modules:

0 −→ D C /D C D −→ D C /D C DzD −→ D C /D C zD −→ 0.

Here, for P ∈ D C , the second arrow is given by [P ] 7→ [P z D] and the third by [P ] 7→ [P ], [P ] denoting the class of P modulo the corresponding ideal.

Applying RΓ _{0} RHom _D

(·, O C ) to this exact sequence, we get the distinguished triangle:

RHom _D

(D ^C /D ^C zD, RΓ _{0} O C ) −→ RΓ _{0} L ¹ _{0}|C −→

−→ RΓ _{0} A ^C −→ . ⁺¹

Recall that RΓ _{0} O C is concentrated in degree 1. One concludes by observing that the first term of the triangle is 0 due to the fact that zD is an automorphism of H ¹ _{0} (O ^C ). q.e.d.

Let us adopt the notations 3.1.0 and take M as in (3.1.4). Set L = g ⁻¹ L ¹ _{0}|C and K _i = g _i ⁻¹ L ¹ _{0}|C . Let K be the complex defined by (2.2.10). Setting O _Z|Y ¹ = RHom(L, O _Y ), Σ _i O ¹ _Z

i

|X = RHom(K, O _X ), we recover the results of [K-S 1, Prop.

4.2] by simply applying Theorem 2.1.1:

Theorem 3.2.2. With the same notations as above, the natural morphism:

RHom _D

(M, Σ _i O _Z ¹

i

|X )    _Z −→ RHom _D

(M _Y , O _Z|Y ¹ )    _Z , is an isomorphism.

3.3. Decomposition at the boundary. In [Sc] Schiltz shows how the solution of the non-characteristic Cauchy problem may be expressed as a sum of functions which are holomorphic in domains determined by the characteristic real hypersur- faces issuing from the boundary of the domain where the data are defined.

We are going here to recover his result and extend it to general systems. Let us describe our geometrical frame.

Let Y be a smooth complex analytic hypersurface of a complex analytic manifold X and let f : Y → X be the embedding. In what follows we will consider T ^∗ X endowed with the underlying real symplectic structure. Recall that if σ is the symplectic two-form on T ^∗ X, the corresponding real two-form is given by σ ^R = 2Re σ. We consider the following situation:

(3.3.1) ω is an open subset of Y with smooth boundary Z and Ω _i (i = 1, . . . , r) are open subsets of X with smooth boundaries Z _i such that Ω _i ∩ Y = ω and the Z _i ’s are pairwise transversal and transversal to Y .

Set T _ω ^∗ Y = SS(A ω ) ⊂ T ^∗ Y . Recall that SS(A ω ) is identified to the subset of T _Z ^∗ Y consisting of the conormals pointing outside ω.

Let M be a left coherent D _X -module such that for a neighborhood V of ˙ T _ω ^∗ Y : (3.3.2)

( (i) f _π is non characteristic for C(char(M), ˙ T _Ω ^∗

X) on ^t f ⁰⁻¹ (V ), (ii) char(M) ∩ ^t f ⁰⁻¹ (T _ω ^∗ Y ) ⊂ ∪ i T _Ω ^∗

X ∪ T _X ^∗ X.

Remark that (3.3.2)-(ii) implies that f is non-characteristic for M.

Let F = RHom _D

(M, O _X ), set K _i = A _Ω

, an object of D ^b (X), and L = A _ω , an

object of D ^b (Y ). Of course f ⁻¹ K _i = L and moreover one has canonical morphisms

τ _i : K _i → A _X . We can then define K by (2.2.10). Set Σ _i RΓ _Ω

O _X = RHom(K, O _X ).

Theorem 3.3.6. With the same notations as above, the canonical morphism:

(3.3.3) RHom D

(M, Σ i RΓ Ω

O _X )| _Y −→ RHom _D

(M Y , RΓ ω O _Y ), is an isomorphism.

Proof. It is enough to prove the isomorphism at each point y ∈ Y . If y / ∈ ω both complexes are zero.

If y ∈ ω one has

Σ _i RΓ _Ω

O _X | _ω ∼ = O X | _ω , RΓ _ω O _Y | _ω ∼ = O Y | _ω ,

and hence the isomorphism (3.3.3) is nothing but the Cauchy-Kowalevski theorem (0.1).

If y ∈ ∂ω we apply Theorem 2.1.1 (cf. Proposition 2.2.3) for Z = {y}. The only non trivial property to check is

RΓ _{y} (A _ω ) ∼ = RΓ _{y} (A _Y ).

Since ω has a smooth boundary, A ω ∼ = RHom(A ω , A Y ), and hence RΓ _{y} (A ω ) ∼ = RHom(A _{y} , RHom(A ω , A Y ))

∼ = RHom(A _{y} ⊗ A _ω , A _Y )

∼ = RΓ {y} (A _Y ).

q.e.d.

Let us explain why Theorem 3.3.6 contains the result of [Sc].

Let X be an open subset of C ⁿ with 0 ∈ X. Let z = (z ₁ , . . . , z _n ) denote the complex coordinates in C ⁿ and let (z; ζ) be the associated coordinates in T ^∗ X. Let ω be an open subset of Y = {z ∈ X; z ₁ = 0} with smooth boundary Z. Denote by f : Y → X the embedding. Consider a linear partial differential operator with holomorphic coefficients P = P (z, D) on X for which Y is non-characteristic.

Assume P has characteristics with constant multiplicities transversal to Y × _X T ^∗ X at ^t f ⁰⁻¹ (T _Z ^∗ Y ) ∩ char(P ). Let M = D _X /D _X P .

Proposition 3.3.7. Under the previous hypotheses there exist open subsets Ω i of X as in (3.3.1) and satisfying (3.3.2).

Proof. Take q(z; ζ) as in Proposition 3.1.3. Set Λ ⁰ = ^t f ⁰⁻¹ (T _Z ^∗ Y ) ∩ char(M), a

disjoint union of r isotropic smooth manifolds Λ ⁰ ₁ , . . . , Λ ⁰ _r of (T ^∗ X) ^R , the cotangent

bundle endowed with the underlying real analytic symplectic structure. From the

fact that ∂q(z; ζ)/∂ζ ₁ = 0 and that {z ₁ , q} 6= 0 on ^t f ⁰⁻¹ (T _Z ^∗ Y ) ∩ char(M), one

easily sees that the integral leaves of H _{Re q} ^R , H _{Im q} ^R , are transversal to Y × _X T ^∗ X

at Λ ⁰ _i . Therefore the union of these integral leaves issuing from Λ ⁰ _i is a Lagrangian

manifold Λ _i of (T ^∗ X) ^R , contained in char(M). One constructs the hypersurfaces

Z _i of X ^R similarly as in Proposition 3.1.3 and then chooses Ω _i as the half spaces

delimited by Z _i and containing ω. q.e.d.

Considering the distinguished triangle:

⊕ ^r−1 _i=1 RHom _D

(M, O _X ) −→ ⊕ ^r _i=1 RHom _D

(M, RΓ _Ω

O _X ) −→

−→ RHom _D

(M, Σ _i RΓ _Ω

O _X ) −→, ⁺¹

and apply the functor RΓ(Y ; ·| _Y ). By the Cauchy-Kowalevski theorem (0.1) we find that if Y is Stein, the sequence:

0 → ⊕ ^r−1 _i=1 Γ(Y ; Hom _D

(M, O _X )| _Y ) → ⊕ ^r _i=1 Γ(Y ; Hom _D

(M, RΓ _Ω

O _X )| _Y ) → Γ(Y ; Hom _D

(M, Σ _i RΓ _Ω

O _X )| _Y ) → 0

is exact. Hence, by (3.3.3), we get that the holomorphic solution of the Cauchy problem:  P (z, D) u(z) = 0,

D ₁ ^h u(z)  

Y ∈ Γ(ω; O _Y ), 0 ≤ h < m.

may be written as a sum u = P r

i=1 u _i where u _i ∈ Γ(Ω _i ∩ V ; O _X ) satisfies the equation P u i = 0, V being an open neighborhood of Y in X.

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Dept. de Math´ ematiques, Universit´ e Paris-Nord, F-93430 Villetaneuse, France;