Let X be a manifold, π X : T ∗ X → X its cotangent bundle. Set ˙ T ∗ X = T ∗ X \ T X∗ X and denote by ˙π X the projection ˙ T ∗ X → X.

Andrea D’Agnolo Pierre Schapira

Abstract. In this paper we show how the techniques of sheaf theory allow us to treat the Cauchy problem (in the language of D-modules) in various sheaves of ramified holomorphic functions.

1. Review on sheaves. In this section all manifolds and morphisms of manifolds will be real analytic. For what follows refer to [12].

Let X be a manifold, π X : T ^∗ X → X its cotangent bundle. Set ˙ T ^∗ X = T ^∗ X \ T _X ^∗ X and denote by ˙π X the projection ˙ T ^∗ X → X.

If A, B are subsets of X, one denotes by C(A, B) the normal cone of A along B, a closed conic subset of T X.

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.

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

One denotes by D ^b (X) the derived category of the category of bounded complexes of sheaves of k-vector spaces on X, where k is a commutative field. (In sections 2 and 3 we shall take k = C.)

If A ⊂ X is a locally closed subset, one denotes by k A the constant sheaf on A with fiber k.

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 [11]).

Let f : Y → X be a morphism of manifolds 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 .

On a complex manifold X, we shall denote by O _X the sheaf of holomorphic functions and by D _X the sheaf of rings of finite order differential operators, and

Appeared in: D-modules and microlocal geometry (Lisbon, 1990), de Gruyter, Berlin, 1993, pp. 37–44.

1991 Mathematics Subject Classification. 58 G, 32 B, 18 F 20.

1

one shall use the usual terminology of D-modules theory (cf. [18] for an introduction to this theory). In particular one denotes by char(M) the characteristic variety of a coherent D X -module M, and if f : Y → X is non-characteristic for M, one denotes by M _Y the inverse image (as a D-module) of M.

2. Ramified holomorphic functions. Here we shall deal with various sheaves of ramified holomorphic functions that we shall now describe. In this section we take k = C.

Let X be a complex manifold and let Z be a smooth hypersurface defined by an equation g = 0, where g : X → C. Recall some classical constructions (cf [5] and [4]).

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). Set f X ^∗ = f C ^∗ × C X and consider the Cartesian diagram:

(2.1)

X f ^∗ −→ C f ^∗

p

↓ ^p ↓

X −→ ^g C .

Notice that the functor p X! is exact and that p ^! _X = p ⁻¹ _X . We set

L ^ram _Z/X = g ⁻¹ p ! k _{C f}

.

Notice that, by adjunction, there is a natural morphism (that we shall use later):

(2.2) τ : L ^ram _Z/X −→ k _X .

If G is a sheaf on X, or more generally, an object of D ^b (X), the complex of ramified sections of G along Z is naturally described by Rp _X∗ p ⁻¹ _X G. By the Poincar´e-Verdier duality one gets:

Rp _X∗ p ⁻¹ _X G = RHom(p _X! k _{X f}

, G)

= RHom(L ^ram _Z/X , G).

In particular we set:

O _Z/X ^ram = RHom(L ^ram _Z/X , O X ).

One easily sees that O _Z/X ^ram is concentrated in degree zero.

We shall also deal with ramified sections of “logarithmic type”.

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

L ¹ _{0}/C := RHom _D

(N , O C ).

Remark that the complex O ¹ _{0}/C := RHom(L ¹ _{0}/C , O ^C ) is concentrated in de- gree zero and represents a sheaf of holomorphic functions on C with logarithmic ramification at 0.

We set:

L ¹ _Z/X := g ⁻¹ L ¹ _{0}/C ,

and notice that there is a natural morphism:

(2.3) τ : L ¹ _Z/X −→ k _X .

We shall consider the complex RHom(L ¹ _Z/X , G). In particular we set O _Z/X ¹ = RHom(L ¹ _Z/X , O _X ).

Again this complex is concentrated in degree zero.

We shall also have to consider sheaves whose sections are sums of ramified sec- tions of the preceding type. More precisely consider the following geometrical frame:

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

Let K _i , i = 1, . . . , r, be objects of D ^b (X) endowed with morphisms:

τ _i : K _i → k _X .

We define the complex K as being the third term of a distinguished triangle:

(2.4) K −→ ⊕ ^r _i=1 K i h

−→ ⊕ ^r−1 _i=1 k X

−→, +1

where h is the composite of the map ⊕ ^r _i=1 τ _j and the map ⊕ ^r _i=1 k _X → ⊕ ^r−1 _i=1 k _X , given by (a 1 , . . . , a _r ) 7→ (a 2 − a 1 , . . . , a _r − a _r−1 ).

We apply this construction and set

Σ _i O _Z ^ram

_/X = RHom(K, O _X ), where K is defined by (2.4) with K _i = L ^ram _Z

_/X .

Similarly we set:

Σ i O _Z ¹

_/X = RHom(K, O X ), where K is defined by (2.4) with K _i = L ¹ _Z

_/X .

Remark that both complexes Σ i O _Z ^ram

_/X and Σ i O _Z ¹

_/X are concentrated in degree zero.

3. The Cauchy problem for ramified holomorphic functions. We shall consider the following geometrical situation:

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 _i ⁻¹ (0).

Let M be a left coherent D X -module such that, for a neighborhood V of ˙ T _Z ^∗ Y : (3.1)

( (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)-(ii) implies that f is non-characteristic for M. One denotes by M _Y the restriction (as a coherent D-module) of M to Y .

The Cauchy-Kowalevski theorem for ramified (resp. ramified of logarithmic type)

holomorphic functions in the framework of D-modules may be expressed by the two

following propositions:

Theorem 3.1. The natural morphism:

(3.2) RHom _D

(M, Σ _i O _Z ^ram

_/X )    _Z −→ RHom _D

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

Theorem 3.2. The natural morphism:

(3.3) RHom _D

(M, Σ _i O _Z ¹

_/X )   

Z −→ RHom _D

(M _Y , O _Z/Y ¹ )   

Z

is an isomorphism.

Remark 3.3. This last theorem was proved in [9] but the proof we shall present in this section is totally different.

Let us now explain how Theorem 3.1 gives an extension to D-modules of [7]’s results.

Let X be an open subset of C ⁿ with 0 ∈ X, let z = (z 1 , z ⁰ ) = (z 1 , . . . , z n ) be the coordinates on X and let (z; ζ) be the associated coordinates in T ^∗ X. Set D = ∂/∂z. Consider the Cauchy problem:

(3.4)

 P (z, D) u(z) = 0, D 0 ^h u(z)  

Y = w _h (z ⁰ ), 0 ≤ h < m.

Here P = P (z, D) is a linear partial differential operator of order m on X with holomorphic coefficients, the hyperplane Y = {z ∈ X; z 1 = 0} is non-characteristic for P and the w _h (z ⁰ )’s are holomorphic functions on Y ramified along the hypersur- face Z = {z ∈ X; z 1 = z 2 = 0} of Y . Let f : Y → X be the embedding. Suppose that P has characteristics with constant multiplicities transversal to Y × _X T ^∗ X at

t f ⁰⁻¹ (T _Z ^∗ Y ) ∩ char(M).

Let Z 1 , . . . , Z r be the smooth hypersurfaces of X whose conormal bundles are the union of the bicharacteristics of P issued from ^t f ⁰⁻¹ ( ˙ T _Z ^∗ Y ).

In [7], Hamada, Leray and Wagschal proved that the holomorphic solution of (3.4), defined in a neighborhood of Y \ Z, extends holomorphically as a sum of ramified functions along the Z _j ’s.

We apply Theorem 3.1 for the choice M = D X /D X P . Then (3.1) is satisfied and the complex RHom D

(M, Σ i O _Z ^ram

i

/X )    _Z is concentrated in degree zero. Its sections may be expressed as sums P

ϕ _i where the ϕ _i ’s are holomorphic functions ramified along the Z _i ’s satisfying P ϕ _i = 0. Hence we get existence and uniqueness for the solution of the Cauchy problem (3.4) when the data on X are sections of Σ _i O _Z ^ram

_/X and the data on Y are sections of O _Z/Y ^ram .

In other words, Theorem 3.1 contains the theorem of [7], and extends it to arbitrary coherent D-modules.

4. An inverse image theorem for sheaves. In this section we will give a

general theorem of inverse image for sheaves on real manifolds and show how to

deduce Theorems 3.1 and 3.2. We emphasize the fact that this theorem is stated

in a purely real geometrical frame and does not make use of any complex structure

nor of any results of partial differential equations.

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

Z = {y} for y ∈ Y ). Let F be an object of D ^b (X) such that (i) 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:

(ii) ^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:

(iii) K _i is weakly R-constructible, (iv) f is non-characteristic for K i ,

(v) a morphism τ _i : K _i → k _X is given.

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

(vi) ^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:

(vii) an isomorphism ψ i : L → f ⁻¹ (K i ) is given,

(viii) f ⁻¹ (τ _i ) ◦ ψ _i induces an isomorphism RΓ _Z L → RΓ _Z k _Y .

Assume moreover that for an open neighborhood V of ˙π _Y ⁻¹ (Z):

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

Adapting an idea of [9] and using [12, Theorem 6.7.1], one can then prove the following statement:

Theorem 4.1. Let be given f , F , K _i , ψ _i and L satisfying the hypotheses (i)–(ix) and let K be constructed as in (2.4). Then the natural morphism induced by the ψ _i ’s:

f ⁻¹ RHom(K, F )  

Z −→ RHom(L, f ⁻¹ F )  

Z

is an isomorphism.

Proof of Theorems 3.1 and 3.2. In the situation of Theorems 3.1 and 3.2 set:

F = RHom D

(M, O X ).

By [12, Theorem 11.3.3], one knows that SS(F ) ⊂char(M) (in fact there is equality but we need only this inclusion which is easily deduced from the Cauchy-Kowalevski theorem in the precised version due to [15]). All the hypotheses of Theorem 4.1 are then obviously checked except for (vii), (viii).

In the case of Theorem 3.1 they follow from the isomorphism:

RΓ _{0} p ! k _{C f}

→ RΓ ∼ _{0} k ^C ,

while in the case of Theorem 3.2 they follow from the isomorphism:

RΓ _{0} L ¹ _{0}/C → RΓ ^∼ _{0} k C . We have:

RHom _D

(M, Σ _i O _Z ^∗

_/X ) = RHom _D

(M, RHom(K, O _X ))

= RHom(K, F )

for ∗ =ram or 1.

On the other hand by the Cauchy-Kowalevski-Kashiwara theorem of [8] one has:

f ⁻¹ RHom _D

(M, O _X ) ∼ = RHom D

(M _Y , O _Y ) and hence

RHom D

(M Y , O _Z/Y ^∗ ) = RHom D

(M Y , RHom(L, O Y ))

= RHom(L, f ⁻¹ F ) for ∗ =ram or 1, which completes the proof. 

Conclusions. Let us put the emphasis on the fact that all our proofs rely only on the following tools: the Cauchy-Kowalevski theorem in the precised version by Leray [15] from which the inclusion

SS(RHom _D

(M, O _X )) ⊂ char(M) is deduced, the isomorphism

f ⁻¹ RHom _D

(M, O _X ) −→ RHom ^∼ _D

(M _Y , O _Y ),

which is deduced by Kashiwara [8] from the classical Cauchy-Kowalevski theorem by purely algebraic tools, and the techniques of [11], [12]. We never use, neither pseudo-differential operators and quantized contact transformations, nor any esti- mate.

Similar methods should be applied to regain the results of [19] and further developments of Theorem 4.1 allow to recover a result of [10] on the hyperbolic Cauchy problem in the frame of hyperfunctions (cf. [1]).

Of course our purely sheaf theoretical methods do not allow us to treat holomor- phic functions with meromorphic singularities for systems satisfying Levi conditions (for these questions we refer to [6], [13], [16]).

Let us also mention the work [14] of Leichtnam who treats general ramified solutions, and the works [17], [20] for other developments.

The results of this paper were announced in [2] and will be developed in [3].

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