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008 Microsupport of tempered solutions of D-modules

associated to smooth morphisms

Teresa Monteiro Fernandes

Abstract

Let f : X → Y be a smooth morphism of complex analytic mani-

folds, and let F be an R-constructible sheaf on Y . Let M be a coherent

DX -module.

We prove that the microsupport of RHomDX(M, tHom(f−1F,OX))

is contained in the 1-microcharacteristic variety of M along V =

X ×Y

T ∗Y .

As a consequence, we obtain a similar estimate for the microsup-

port of the complex of solutions of M in the sheaf of distributions

holomorphic in the fibers of an arbitrary smooth morphism.

1 Introduction

Let X be a complex analytic manifold. Let OX denote the sheaf of holomor-

phic functions on X and let DX denote the sheaf of holomorphic differential

operators.

Let DbR−c(CX) denote the full subcategory of the derived category Db(CX)

of the complexes with R-constructible cohomologies (we shall call such a

complex an R-constructible sheaf for short) .

The research of the author was supported by Fundacao para a Ci e ncia e Tecnologia

and Programa Ciencia, Tecnologia e Inovacao do Quadro Comunitario de Apoio.

Mathematics Subject Classification. Primary: 35A27; Secondary: 32C38.

1

Let tHom(. ,OX ) denote the functor of moderate cohomology introduced

by M. Kashiwara in ([12] ) and studied by M. Kashiwara and P. Schapira in

([15]). This functor is defined on DbR−c(CX) and, in particular, when M is

a real manifold and X is a complexification of M , by taking as F the dual

D′(CM ) of the constant sheaf on M , we get tHom(F,OX ) = DbM , the sheaf

of Schwartz distributions on M .

The notion of microsupport of an object G in Db(CX), SS(G), was

introduced by M.Kashiwara and P.Schapira ([13] , [14]), as a subset of the

cotangent bundle π : T ∗X → X, and it describes the directions of non

propagation for G.

Let M be a coherent DX -module. In other words, locally M is a system

of partial differential equations on X. The estimation of the microsupport

of the solutions of M in tHom(F,OX ) is a challenging problem in Algebraic

Analysis, since the growth conditions require far beyond algebro-geometrical

tools, and it has been settled only in particular situations. One of this

tools is the so called Levi-condition introduced in [7] for one operator and

generalized by the notion of 1-microcharacteristic variety of a D-Module

along an involutive submanifold of T ∗X ([24] and [25], see also [21]). On

the other hand, another difficulty is that F needs not, in general, to be

concentrated in degree zero. However, this property is satisfied by F = CU ,

the constant sheaf supported by a Stein subanalytic relatively compact open

subset U (see [9] and [20]).

The purpose of this paper, the second around the subject (see also [23]),

is to treat a general problem posed by P. Schapira on the propagation (i.e.

microsupport) for the solutions in the sheaves tHom(F,OX) of a system of

differential equations on a complex manifold. Here we treat the case where

F is of the form f−1G, for a smooth morphism f of complex manifolds.

Let us recall that in [23] we proved that if G is C-constructible on Y and

f : X → Y is a smooth morphism, then

SS(RHomDX(M, tHom(f−1G,OX ))) ⊂ T ∗X ∩ ρ−1

V (C1V (M)) (1)

2

where V denotes the involutive submanifold X ×Y

T ∗Y of T ∗X, C1V (M)

denotes the 1-microcharacteristic variety of M along V and ρV : T ∗V →

TV (T ∗X) the canonical projection. Here we identify TV (T ∗X) to the relatif

cotangent bundle T ∗(V/T ∗Y ) and consider the embedding of T ∗X → T ∗V

obtained as a composition of T ∗X → T ∗X ×X

V by the zero section of V (as

a bundle over X) and of T ∗X ×X

V → T ∗V.

In that case, by [12] or [15], we used the crucial fact that tHom(f−1G,OX)

is an object of the bounded derived category of regular holonomic D-modules

which has no counterpart in the general case.

In the sequel we shall note for short C1V (M)|X := T ∗X ∩ ρ−1

V (C1V (M)).

For an overview on classical results (distribution solutions), we refer

[10]. In the case of a microdifferential operator and the sheaf of tempered

microfunctions (also introduced by [1]) we refer [8].

Still in the case of distributions, but with the tools of D-Modules, when

M has real simple characteristics, we refer [6] and [2].

More recently, for larger classes of F ∈ DbR−c(CX) and M, the general

problem was studied in [23] and [16].

On the other hand, in [16], assuming that M is regular along a sub-

bundle V of T ∗X in the sense of [18], for any F ∈ DbR−c(CX), the following

estimate was obtained:

SS(RHomDX(M, tHom(F,OX ))) ⊂ V +SS(F )a

where a denotes the antipodal map on T ∗X and the operation + was defined

in [13].

In this paper we prove that the estimate (1) still holds when G ∈

DbR−c(CY ) (cf Theorem 2.5) .

Our main inspiration is the analysis developped in [7], which we partially

adapted in order to obtain a Cauchy-Kowalevskaia theorem of precise type

with data satisfying tempered growth conditions. The other essential tool

is a Grauert type theorem for subanalytic open subsets of a real analytic

manifold (cf Theorem 3.5) together with a realification procedure based in

3

Proposition 3.1 which allows us to reduce to the case of Stein open subana-

lytic sets.

We shall follow the classical method to prove propagation theorems for

D-Modules.

More precisely, we prove a Cauchy-Kowalevskaia-Kashiwara theorem (cf

Theorem 4.2) with data in tHom(f−1G,OX)|H and tHom((f−1G)H ,OH)),

where H is a closed submanifold non 1-microcharacteristic for M along V .

For this, the main step is, classically, the reduction to M defined by one

operator and f : X = Z × Y → Y being the projection. For U a suban-

alytic open subset of Y , H a submanifold of X non 1-microcharacteristic

for P , we then obtain a precised Cauchy-Kowalevskaia theorem with data

respectively in H0(tHom(f−1CU ,OX )) and in H0(tHom((f−1

CU )H ,OH))

(cf Proposition 4.1).

Together with a Zerner type theorem (cf Lemma 4.3), this entails esti-

mate (1) in the case G = CU , U being a finite union of subanalytic Stein

open subsets of Y .

Then we can deduce that estimate (1) is satisfied for an open subanalytic

relatively compact subset U in a real manifold N complexified by Y and by

realification the proof of the general case follows.

As an example of application we prove (cf Theorem 2.6):

Let X be a complex manifold, let Y be a d-dimensional manifold com-

plexifying a real analytic submanifold N and let f : X → Y be a smooth

morphism. Then tHom(D′(f−1CN ),OX) is concentrated in degree d and

Hd(tHom(D′(f−1CN ),OX) can be understood as a sheaf of distributions

with holomorphic parameters which we shall denote, for the sake of simplic-

ity, DbX|N . We then obtain:

SS(RHomDX(M,DbX|N ) ⊂ C1

V (M)|X .

It is a pleasure to express here my gratitude to M. Kashiwara, P. Schapira,

G. Morando, L. Prelli, A.R. Martins, S. Guillermou, for their helpeful atten-

tion and, in particular, to D. Barlet whom I owe the essential idea to prove

4

Theorem 3.5.

2 Notations and main results

We will mainly follow the notations in [13].

Let X be a complex analytic manifold. We denote by τ : TX → X

the tangent bundle to X and by π : T ∗X → X the cotangent bundle. We

identify X with the zero section of T ∗X. Given a smooth submanifold Y of

X, TY X denotes the normal bundle to Y and T ∗Y X the conormal bundle.

For a submanifold Y of X and a subset S of X, we denote by CY (S) the

normal cone to S along Y , a closed R+-conic subset of TY X.

For a morphism f : X → Y of complex manifolds, we denote by

fπ : X ×Y T ∗Y → T ∗Y and fd : X ×Y T ∗Y → T ∗X

the associated morphisms. Recall that when f is smooth, fd is an embed-

ding, and that when f is an embbeding, fπ is smooth. In which follows,

unless expressly mentioned, we shall be concerned by the case when f is

smooth. Set V = X×Y T ∗Y . Then fπ is smooth and by fd, V is a sub-bundle

of T ∗X. By (5.5.10) of [13] we get an embedding T ∗X → V ×XT ∗X → T ∗V

where, for the left arrow, we consider the embedding of X in the zero section

of V . Moreover, if A is a biconic closed subset of T ∗V , noting τ = fd, we

have, by the preceding identification,

τπτ−1d (A) = A ∩ T ∗X. (2)

We shall identify T ∗(T ∗X) to T (T ∗X) by the Hamiltonean isomorphism. We

shall denote by ρ′V the canonical morphism ρ′V : T (T ∗X) ×T ∗X

V → TV (T ∗X).

Then ρ′V factors as the composition of the canonical morphisms

ρ′V : T (T ∗X) ×T ∗X

V → T ∗V →ρV

TV (T ∗X).

Moreover we have a natural isomorphism of bundles on V : TV (T ∗X) ≃

T ∗(V/T ∗Y ) and, given a bicharacteristic leaf Σ of V , TV (T ∗X)×V Σ ≃ T ∗Σ.

5

For a subset A of T ∗X, we denote by Aa the image of A by the antipodal

map

a : (x, ξ) 7→ (x;−ξ).

The closure of A is denoted by A. Let γ ⊂ TX be a cone; the polar cone

γ to γ is the convex cone in T ∗X defined by

γ = (x; ξ) ∈ TX;x ∈ π(γ) and Re〈v, ξ〉 ≥ 0 for any (x; v) ∈ γ.

For A and B two closed R+-conic subsets of T ∗X one associates a closed

R+-conic subset A+B of T ∗X (cf. Definition 6.2.3 and Proposition 6.2.4 of

[13]). We denote by Db(CX) the triangulated of complexes of C-vector spaces

with bounded cohomologies and by DbR−c(CX) (respectively Db

C−c(CX)) the

triangulated category of complexes with bounded and R-constructible coho-

mologies (respectively with C-constructible cohomologies).

If F is an object of Db(CX), SS(F ) denotes its microsupport, a closed

R+-conic involutive subset of T ∗X. We shall use the following characteriza-

tion of the microsupport ([13]).

Given (x0, ξ0) ∈ Rn × (Rn)∗ and ε ∈ R we set:

Hε(x0, ξ0) = x ∈ Rn; 〈x − x0, ξ0〉 > −ε,

and if there is no risk of confusion we will write Hε instead of Hε(x0, ξ0).

Proposition 2.1 ([13]). Let X be a real analytic manifold and let p ∈

T ∗X. Let F ∈ Db(CX), α ∈ N ∪ ∞, ω. Then the following conditions are

equivalent:

(i) There exists an open conic neighbourhood U of p such that for any

x ∈ π(U) and any R-valued Cα-function ϕ defined on a neighbourhood of x

such that ϕ(x) = 0, dϕ(x) ∈ U , one has

Hj

ϕ≥0(F )x = 0 for any j.

(ii) Let Ω be a local chart in a neighbourhood of p such that Ω is identified

to an open subset of Rn and p = (x0, ξ0). Then there exist a proper closed

6

convex cone γ ⊂ Rn, ε > 0 and an open neighbourhood W of x0 with ξ0 ∈

Int(γ) such that (W + γa) ∩ Hε ⊂ X and

Hj(X; C(x+γa)∩Hε⊗ F ) = 0, for any j, x ∈ W.

Definition 2.2 ( [13]). Let F ∈ Db(CX). We define the closed conic subset

SS(F ) of T ∗X by: p /∈ SS(F ) if and only if F satisfies the equivalent

conditions in the preceding proposition.

Remark 2.3. We shall use the following properties of the microsupport.

Given a distinguished triangle F ′ → F → F ′′ −−→+1

, one has

SS(F ) ⊂ SS(F ′) ∪ SS(F ′′),

(SS(F ′)\SS(F ′′)) ∪ (SS(F ′′)\SS(F ′)) ⊂ SS(F ).

Proposition 2.4 ( [13]). Let Y and X be real analytic manifolds and let

f : Y → X be a smooth morphism. Let F ∈ Db(CX). Then

SS(f−1F ) = fd(f−1π (SS(F )). (3)

On a complex manifold X, we consider the sheaf OX of holomorphic

functions and the sheaf DX of linear holomorphic differential operators of

finite order as well as its subsheaves DX(m) of operators of order at most

m. Let EX denote the sheaf of microdifferential operators on X and EX(m)

denote the sheaf of microdifferential operators of order at most m (see [19],

[21] or [11] for a detailed study). We denote by Mod(DX) (respectively by

Mod(EX )) the abelian category of DX-modules (respectively of EX-modules)

and denote by Modcoh(DX) (respectively Modcoh(EX)) the abelian category

of coherent DX -modules (respectively of coherent EX-modules).

Given a morphism f : Y → X of complex manifolds we shall consider

the associated (derived) functors f−1 from Db(Mod(DX )) to Db(Mod(DY ))

and Rf∗, from Db(Mod(DY )) to Db(Mod(DX )).

7

Recall that the functor of moderate cohomology associated to constructible

sheaves, tHom(. ,OX ), was introduced in [12]. For a detailed study we refer

to [15]. We shall use the following properties:

(i) For an open subanalytic subset U of Cn, one has H0tHom(CU ,OX) =

Ot−UX where the sheaf Ot−U

X is defined by:

Γ(Ω;Ot−U ) = f ∈ Γ(Ω ∩ U ;OX); for any compact K ⊂ Ω there exist

an integer k ≥ 0 and a numberCK ≥ 0 such thatsupK∩U

|f(z)d(z, δU)k | ≤ CK.

(ii)([20], [9]) Assume that U is Stein relatively compact. Then tHom(CU ,OX)

is concentrated in degree 0 and, for any open Stein subset V , for any j ≥ 1,

Hj(V ; tHom(CU ,OX )) = 0. (4)

Let now M ∈ Modcoh(EX), let Char(M) denote its characteristic variety

and let V denote a smooth homogeneous involutive submanifold of T ∗X =

T ∗X \ X. The 1-microcharacteristic variety C1V (M) is a biconic closed an-

alytic subset of TV (T ∗X) and it contains the normal cone CV (Char(M)).

To construct it we consider the subring EV of EX generated by the operators

of order at most 1 with a principal symbol vanishing on V . Recall that

for P ∈ EV (m) = EV ∩ EX(m), the symbol σ1V (P ) is a homogeneous func-

tion on TV (T ∗X) and when M is defined by an operator P ∈ EV (m), P /∈

EV (m + 1)EX(−1) then C1V (M) is the variety of zeros of σ1

V (P ). For a sub-

manifold H ⊂ X, we shall say that H is non 1-microcharacteristic for M

along V if T ∗HX ∩ ρ−1

V (C1V (M)) ⊂ T ∗

HH.

For a coherent DX -module M and V a smooth involutive homogeneous

submanifold of T ∗X, we define

C1V (M) = C1

V(EX ⊗π−1DX

π−1M),

where V = V ∩ T ∗X. For further details we refer to [24], [25] and [21]. We

shall note for short C1V (M)|X the set T ∗X ∩ ρ−1

V (C1V (M)).

Theorem 2.5. Let X and Y be complex manifolds, let f : X → Y be a

smooth morphism. Let V = T ∗Y ×Y

X. Then, for any F ∈ DbR−c(CY ) and

8

any M ∈ Modcoh(DX), we have

SS(RHomDX(M, tHom(f−1F,OX))) ⊂ C1

V (M)|X . (5)

Remark that by Lemma 2.5 of [16] we can reformulate the preceding

estimate as follows: Let ρV denote the composite of ν : T ∗X → T ∗(X/Y )

and of µ : T ∗(X/Y ) → T ∗(V/T ∗Y ). Let y ∈ Y and set Σ = f−1(y). Identify

T ∗Σ to Σ ×X

T ∗(X/Y ). Then

SS(RHomDX(M, tHom(f−1F,OX)))|Σ ⊂ Σ ×

Xρ−1

V (C1V (M)). (6)

Assume that Y is a complexified of a real analytic manifold N of real

dimension d. Let f : X → Y be a smooth morphism. Then it is well

known that tHom(D′(Cf−1(N)),OX) is concentrated in degree d. When f

is a projection, we can describe Hd(tHom(D′(Cf−1(N)),OX)) as a sheaf of

distributions with holomorphic parameters which we denote by DbX|N for

short. We have the following estimate, which is to relate with the results in

[7].

Corollary 2.6. Let M ∈ Modcoh(DX). Then,

SS(RHomDX(M,DbX|N ) ⊂ C1

V (M)|X .

3 Realification

3.1 Microsupport and 1-Microcharacteristic Variety

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

denote by Y the underlying topological space Y endowed with the complex

conjugate structure and, identifying Y to the diagonal ∆ ⊂ Y × Y by the

canonical embedding l, let us regard Y × Y as a complexification of Y . Let

us denote by j : X → X × Y the associated morphism, q : X × Y → Y and

p : X × Y → X the associated projections.

9

Proposition 3.1. For any F ∈ DbR−c(CY ) and any M ∈ Modcoh(DX) we

have a canonical isomorphism:

Rp∗RHomDX×Y

(p−1M, tHom((f × idY )−1Rl∗F,OX×Y )[dimY ])

≃ RHomDX(M, tHom(f−1F,OX)). (7)

Proof. We have pj = idX and j is proper so we may use the adjunction

formula (7.4) of Theorem 7.2 of [13] for j, f−1F and p−1M to obtain a

natural isomorphism

RHomDX×Y

(p−1M, tHom(Rj∗f−1F,OX×Y )[dimY ]) (8)

≃ Rj∗RHomDX(M, tHom(f−1F,OX)).

On the other hand, the equality lf = (f × idY )j and Proposition 2.5.11

of [13] entail the existence of a natural isomorphism such that, for any

F ∈ DbR−c(CY ), Rj∗f

−1F ≃ (f × idY )−1Rl∗F which, composed with (8),

gives (3.1).

q.e.d.

Corollary 3.2. In the situation of Proposition 3.1, we have

SS(RHomDX(M, tHom(f−1F,OX)))

⊂ pπpd(SS(RHomDX×Y

(p−1M, tHom(f × idY )−1Rl∗F,OX×Y ))).

Proof. It is a consequence of the properness of p on the support of

RHomDX×Y

(p−1M, tHom(f × idY )−1Rl∗F,OX×Y )).

q.e.d.

Proposition 3.3. Let V be the submanifold

V = T ∗(Y × Y ) ×Y ×Y

X × Y ⊂ T ∗(X × Y ).

10

Then, for any M ∈ Modcoh(DX), we have

pπp−1d (C1

V(p−1M)) = C1

V (M)|X .

Proof. By (2), we have pπp−1d ρ−1

V(C1

V(p−1M)) = ρ−1

V(C1

V(p−1M)) ∩ T ∗X

and by ([23], Lemma 2.3) we have ρ−1V

(C1V

(p−1M))∩T ∗X = ρ−1V (C1

V (M))∩

T ∗X. q.e.d.

3.2 Subanaytic open sets in real anaytic manifolds

A well known result of H. Grauert (Proposition 7, [4]) implies that any open

subset of Rn admits a fundamental system of Stein open neighbourhoods in

Cn. By open neighbourhood of an open subset Ω of R

n we mean an open

subset Ω′ of Cn such that R

n ∩ Ω′ = Ω.

We shall prove a similar however weaker result (Theorem 3.5 below),

under the assumption that Ω is subanalytic relatively compact. For that

purpose, we need the following essential tool due to H. Hironaka (cf [5], [3]).

Recall that a subset Q of Rn is a quadrant if there is a partition of 1, ..., n

into disjoint subsets I0, I+, I− such that Q = x = (x1, ..., xn) ∈ Rn : xi =

0 if i ∈ I0, xi > 0, if i ∈ I+, and xi < 0, if i ∈ I−. By definition, a

quadrant is a semianalytic subset and is open when I0 is empty.

Theorem 3.4. Rectilinearization Theorem ([5])

Let S be a real analytic manifold of dimension n. Let K be a compact

subset of S. Let X be a subanalytic subset of S. Then there exist finitely

many mappings φi : Rn → S such that:

(1) For each i,there is a compact subset Li of Rn such that

⋃i φi(Li) is

a neighbourhood of K.

(2) For each i, φ−1i (X) is a union of quadrants in R

n.

Theorem 3.5. Let Ω be a relatively compact subanalytic open set of a real

analytic (paracompact) manifold S. Then, given a complexification S′ of S,

there exists a finite family of subanalytic Stein open subsets of S′, Ui, such

that Ω = (∪Ui) ∩ S.

11

Proof. The proof of the case where Ω is semianalytic is due to D. Barlet and

it inspired us the proof of the general case. In particular the result holds for

n ≤ 2.

Let Ω be a relatively compact subanalytic open subset of S. Taking

X = Ω in Theorem 3.4, we may cover Ω by a finite number of open subsets

Ωi such that Ωi ∩ Ω is a finite union⋃

j φi(Qij), for a finite family of real

analytic proper surjective morphisms φi, Qij denoting the intersection of an

open quadrant of Rn with a suitable polydisc BRij

(0). It is sufficient to prove

that each Ωi ∩Ω satisfies our assertion, hence we may assume that Ω is the

image of a single Q by a suitable morphism φ. By the proof of Theorem 3.4, φ

is a proper modification with center in the boundary of Ω, hence φ|Q : Q → Ω

is an analytic isomorphism. Since the family of Stein subanalytic open

subsets is closed under finite intersection, we may assume without loss of

generality that Q = x ∈ Rn, x1 > 0∩BR(0). Let φC be a complexification

of φ defined in a complex polydisc BR′(0) ⊂ Cn and let Ω′′ denote the Stein

open subset of Cn given by BR′(0) ∩ z = (z1, ..., zn),ℜz1 > 0. We get

Ω′′ ∩ Rn = Q. Moreover, up to a shrinking of R′, φC|Ω′′ is an analytic

isomorphism. Hence Ω′ := φC(Ω′′) is a Stein open subanalytic set of S′

satisfying Ω′ ∩ S = Ω. q.e.d.

4 Cauchy-Kowalevskaia Theorem and Propagation

Let X ⊂ Cn × C

d be an open neighbourhood of 0 of the form X = Z × Y ,

with Z ⊂ Cn and Y ⊂ C

d.

We shall consider a differential operator P of the form

P (z, y,Dz ,Dy) = Dmz1

+∑

0≤j<m

aα(z, y)Dαz′D

jz1

, |α| ≤ m, (9)

where α = (α2, ..., αn) ∈ Nn−1, Dα

z′ = Dα2z2

...Dαnzn

and where the coefficients

aα(z, y) are holomorphic in a neighbourhood Ω0 of 0 in X.

Let Ω be an open convex subset in Z and let h ∈ C. We note Hh and

H, respectively, the hyperplane z1 = h and z1 = 0. Let δ be a real positive

12

number. Following J.M. Bony and P. Schapira in [8], Ω is δ − Hh − flat if,

whenever z ∈ Ω and z ∈ Hh satisfy

|z1 − h| ≥ δ|zj − zj |j=2,...,n

then z ∈ Ω.

Let now Ω be an open subset of X. We shall say that Ω is Y −δ−Hh−flat

if, for any y ∈ Y , f−1(y) ∩ Ω is δ − Hh − flat in Z.

In this situation we have:

Proposition 4.1. (Precised Cauchy Theorem) Let U be an open sub-

analytic set of Y . Let P be an operator of the form (9). Then there exist

an open neighbourhood Ω0 of 0 ∈ X and δ > 0 such that, for any h, for any

open subset Ω ⊂ Ω0 which is Y − δ − Hh − flat, for any

f ∈ Γ(Ω;H0(tHom(CZ×U ,OX)))

and for any

(g) ∈ Γ(Ω ∩ Hh × Y ;H0(tHom(CHh×U ,OHh×Y )))m,

the Cauchy Problem

Pu = f, γ(u) = (g), (10)

where γ(u) = (u|Hh×Y , ...,Dmz1

u|Hh×Y ), admits a unique solution

u ∈ Γ(Ω;H0(tHom(CZ×U ,OX ))).

Proof. We shall adapt and follow step by step the proof of Theorem 2.4.3

of [7].

For an open set Ω and f ∈ Γ(Ω;H0(tHom(CZ×U ,OX))), for any compact

K ⊂ Ω and any k ∈ N, we set

(i) |f |K,k = supK∩Z×U

|f(w)d(w, δ(Z × U))k|.

Therefore, if f ∈ Γ(Ω;H0(tHom(CZ×U ,OX))) and (g) ∈ Γ(Ω ∩ Hh ×

Y ;H0(tHom(CHh×U ,OHh×Y )))m, for any compact K ⊂ Ω, there exists an

13

integer k ≥ 0 such that |f |K,k < +∞ and (g) satisfies similar estimates on

Ω ∩ Hh × Y .

Let K be a compact subset of Z × Y . For (z, y) = (z1, z′, y) ∈ K we set

(ii) dz′,K(z, y) = infsup ||z′ − z′||; (z1, z′, y) ∈ δK.

The operator P may obviously be written in the form

P (z, y,Dz ,Dy) = Dmz1

+∑

0≤j<m−1

Dαz′D

jz1

a′αj(z, y), |α| ≤ m, (11)

where the coefficients a′α(z, y) are holomorphic in a neighbourhood Ω0 of 0.

It is clear that, for any open subset Ω ⊂ Ω0 and f ∈ Γ(Ω;H0(tHom(CZ×U ,OX ))),

there exists Cα ≥ 0 such that

|a′αf |K,k < Cα|f |K,k. (12)

We may assume that h = 0. We shall construct the solution u of (10) by

successive approximation, defining recursively a sequence

uν ∈ Γ(Ω;H0(tHom(CZ×U ,OX)))

by

u0 = 0,Dmz1

uν+1 = f+∑

0≤j≤m−1

Dαz′D

jz1

a′αuν ,Djz1

uν+1|H×Y = gj , j = 0, ...,m−1.

(13)

Let us note vν = uν+1 − uν . We get uν =∑

i=0,...,ν−1vi,

Dmz1

v0 = f,Djz1

v0|H×Y = gj , j = 0, ...,m − 1 (14)

and

Dmz1

vν+1 =∑

0≤j≤m−1

Dαz′D

jz1

a′αjvν , Djz1

vν+1|H×Y = 0, |α| ≤ m (15)

Let us fix a given ν, let us assume that δ ≥ 0 is given such that Ω is

Y − δ−H − flat and that vν ∈ Γ(Ω;H0(tHom(CZ×U ,OX))). We will prove

14

that there exists a constant C, depending only on P and Ω0, such that, for

any compact K ⊂ Ω, of the form K = K1 ×K2, with K1 ⊂ Z, K1 being the

closure of a δ − H − flat open subset of Z, and K2 ⊂ Y , we have

|vν+1|K,k ≤ Cδ|vν |K,k. (16)

Having (12) in mind, it is enough to prove that, if

w ∈ Γ(Ω;H0(tHom(CZ×U ,OX)))

and v is the solution of

Dmz1

v = Dαz′D

jz1

w,Djz1

v|H×Y = 0, |α| ≤ m, j < m (17)

then there exists a constant C, only depending on m and of Ω0, such that

|v|K,k ≤ Cδ|w|K,k. (18)

Remark that v is then the solution of the Cauchy Problem

Dm−jz1

v = Dαz′w, |α| ≤ m − j,Dj

z1v|H×Y = 0, |α| ≤ (m − j), j < m. (19)

We have, for such K and k, |w|K,k = sup(z,y)∈K1×(K2∩U)

|w(z, y)|d(y, δU)k . We

may assume |w|K,k < +∞, therefore for any (z, y) ∈ K ∩ (Z × U) we have

|w(z, y)| ≤ |w|K,kd(y, δU)−k .

Let z ∈ Int(K1). By Cauchy integral formula, we get

|Dαz′w(z, y)| ≤ |w|K,k e|α|(|α| + 1)!d(y, δU)−kdz′,K(z, y)−|α|. (20)

Furthermore, we may assume δ < 1 and diameter of Ω0 < 1. Hence

|Dαz′w(z, y)| ≤ |w|K,k e|α|(|α| + 1)!d(y, δU)kdz′,K(z, y)−m+j . (21)

Since Int(K1) is δ − H − flat, one easily checks that, for t ∈ [0, 1], pt :=

(tz1, z′, y) belongs to K ∩ Z × U and that

dz′,K(pt) ≥ dz′,K(z, y) + |(1 − t)z1

δ|

15

hence,

|

∫ 1

0Dα

z′w(pt)z1dt| ≤ |w|K,k e|α|(|α| + 1)!d(y, δU)−kδdz′,K(z, y)−m+j+1.

(22)

Iterating m − j times this integration, we get

|v|K,k ≤ |w|K,ke|α|(|α| + 1)!δm−j ,

and we can choose C = e|α|(|α|+ 1)!. Therefore (18) is proved which entails

(16).

Remark that v0 = G +∑

j=0,...,m−1gj(z

′, y)zj1/j! where G denotes the mth-

primitive of f , vanishing up to the order m on H×Y . If |f |K,k, |gj |K,k < +∞

it is then clear that |v0|K,k < +∞. Denoting C0 = |v0|K,k, arguing by

induction thanks to (16) as in [7], we get

|vν+1|K,k ≤ C0(Cδ)ν+1. (23)

We may also assume that δ < 1/C. Therefore the series∑

vν defines a

holomorphic function u in Ω ∩ Z × U satisfying the estimate |u|K,k ≤

C0∑

(Cδ)ν < +∞. Clearly, u satisfies Pu = f . Furthermore, by (18) and

(17), if a holomorphic function u′ defined in Ω∩Z×U satisfies |u′|K,k < +∞

and Pu′ = 0, γ(u′) = 0, then |u′|K,k ≤ Cδ|u′|K,k, hence u′|K = 0. Therefore

u is the unique solution of (10). q.e.d.

From Proposition 4.1 one obtains the following tempered version of

Cauchy-Kowalevskaia-Kashiwara Theorem (cf[11]):

Theorem 4.2. Let X and Y be complex manifolds, let X → Y be a smooth

morphism and let M ∈ Modcoh(DX). Let F ∈ DbR−c(CY ). Let H be a

smooth manifold of X non 1-microcharacteristic for M along V and let f ′

denote f |H . Then the natural morphism in Db(CH)

RHomDX(M, tHom(f−1F,OX))|H → RHomDH

(MH , tHom(f ′−1F,OH))

(24)

is an isomorphism.

16

Proof. First case:

Assume F = CU where U is a finite union of Stein open subanalytic rel-

atively compact subsets of Y . Since the family of Stein open subanalytic

sets is stable under finite intersection, a simple argument by induction on

the number of Stein open sets shows that it is sufficient to assume that U

is Stein. We may assume the following situation in a neighbohood of x:

X = Z × Y , where Z is an open neigborhood of 0 ∈ Cn, Y is an open

neighbourhood of 0 ∈ Cd and f is the projection.

Considering the associated symplectic coordinates in T ∗X, (z, y; ζ, τ)

with z = (z1, ..., zn), we have V = (z, y; ζ, τ); ζ = (ζ1, ..., ζn) = 0 in a

neighbourhood of (0, 0; 0, dτ) ∈.

V , and we may assume p = (0, 0; dz1, 0)

and H = (z, y), z1 = 0. Moreover, by Remark 2.3, we may assume M =

DX/DXP for a differential operator P of the form (9). The result then

follows immediatly from Proposition 4.1.

Second case:

We assume that Y is the complexification of a real analytic manifold N and

that F ∈ DbR−c(CN ). The assertion is locally checked so we may assume

that F has compact support. Then there exists a finite filtration N =

N0 ⊃ N1 ⊃ ... ⊃ Nl = ∅ such that FNj\Nj+1is a constant sheaf. Since

Nj \ Nj+1 is a locally closed subanalytic set, which is always the difference

of two open subanalytic sets, it is enough to prove the result for F = CU

for an open subanalytic relatively compact subset of N . Moreover we may

assume that N is Rd and Y = C

d. By Theorem 3.5, we can write U as a

finite union U =⋃

j Ωj ∩ Rd, where each Ωj is Stein open subanalytic in

Cd. The assertion holds for

⋃j Ωj by the first case. On the other hand,

Cd \ R

d is a finite union of Stein open subsets, therefore the assertion holds

for (⋃

j Ωj) \ Rd and hence for U .

General case:

It easily follows by Proposition 3.1 and the second case. q.e.d.

The following Lemma is an adaptation of Zerner’s Lemma:

17

Lemma 4.3. Let U be an open subanalytic set of Y . Let φ be a C∞ func-

tion in a neighbourhood of 0 ∈ X such that φ(0) = 0 and dφ(0) = dz1.

Let Ω = (z, y) ∈ X : φ(z, y) < 0. Let u ∈ Γ(Ω;H0(tHom(CZ×U ,OX)))

and assume that Pu extends as a section of H0(tHom(CZ×U ,OX))) in a

neighbourhood of 0.Then u extends to a neighbourhood of 0 as a section of

H0(tHom(CZ×U ,OX ))).

Proof. Having in mind Proposition 4.1 the proof is similar to that of Lemma

2.7 in [23]. We may assume that φ is defined in Ω0 ⊂ Ω, with Ω0 as in

Proposition 4.1, and that Pu extends to Ω0. By classical arguments, there

exists < ǫ << 1, R > 0 such that the open polydisc centered in (−ǫ, 0, 0)

and radius max (R, δR) is contained in Ω0. Then,

Wǫ = (z1, z′, y) : |z1 + ǫ| < δ(R − ||z′||), ||z′|| < R, ||y|| < R ⊂ Ω0

is Y − δ − H−ǫ − flat and is a neighbourhood of 0. Again by Propo-

sition 4.1 the solution of Puǫ = Pu, γ(uǫ) = γ(u) defines a section of

H0(tHom(CZ×U ,OX))) on Wǫ.

q.e.d.

The preceding Lemma has a global version adapting Lemma 3.1.5 of [21]:

Lemma 4.4. Let ω and Ω be two convex subsets in X, with Ω open, ω locally

closed and ω ⊂ Ω. Assume that any real hyperplane whose conormal belongs

to the closure of ξ;∃x ∈ Ω, ρV (x, ξ) ∈ C1V (DX/DXP ) which intersects Ω

also intersects ω. Then, if u ∈ Γ(ω;H0(tHom(CZ×U ,OX))) is such that Pu

extends to Ω, u extends to Ω.

As an easy consequence of the Analytic continuation principle for holo-

morphic functions we obtain:

Lemma 4.5. (Analytic continuation principle) Let f : X → Y be a

smooth morphism. Let Ω be a subanalytic open subset of X such that, for

18

any y ∈ U , Ω ∩ f−1(y) is non empty and connected. Let ω ⊂ Ω be an open

subset such that, for any y ∈ U , ω ∩ f−1(y) is non empty.

If u ∈ Γ(Ω;H0(tHom(Cf−1(U),OX))) vanishes on Ω, then u = 0.

Lemma 4.6. (Propagation) Let X and Y complex manifolds and let f :

X → Y be a smooth morphism. Let U be a finite union of Stein subanalytic

relatively compact open subsets of Y . Then, for any M ∈ Modcoh(DX), we

have

SS(RHomDX(M, tHom(Cf−1U ,OX))) ⊂ C1

V (M)|X . (25)

Proof. We may assume U is Stein. Let p = (x, ξ) /∈ ρ−1V (C1

V (M)). As

above, we may assume that x = 0 ∈ Cn+d,X = Z × Y , where Z is an

open neigborhood of 0 ∈ Cn, Y is an open neighbourhood of 0 ∈ C

d, f is

the projection, V = (z, y; ζ, τ); ζ = (ζ1, ..., ζn) = 0 in a neighbourhood of

(0, 0; 0, dyd) ∈.

V , and p = (0, 0; dz1, 0). Moreover, we may assume M is of

the form DX/DXP for a differential operator of the form

P (z, y,Dz ,Dy) = Dmz1

+∑

0≤j<m

aα(z, y)Dαz′D

jz1

, |α| ≤ m (26)

where α = (α2, ..., αn) ∈ Nn−1, Dα

z′ = Dα2z2

...Dαnzn

and where the coefficients

aα(z, y) are holomorphic in a neighbourhood Ω0 of 0. Let H denote the

complex RHomDX(M, tHom(CZ×U ,OX)). Let Ω0, δ be given by Proposi-

tion 4.1. We shall apply Proposition 2.1 (ii) and prove that, for 0 < ǫ << 1

and 0 < R << 1, denoting by ω the open polydisc of center (−ǫ, 0, ..., 0)

and radius R, BR(−ǫ), and by γ the proper closed convex cone of Cn+d

γ = (z1, z′, y),Rez1 ≤ −δ(||(z′, y)|| + |Imz1|),

we have

Hj(X; C((z,y)+γa)∩Hǫ⊗H) = 0, for any j, (z, y) ∈ ω.

By (4), it is enough to prove

i) H0(X; C((z,y)+γa)∩Hǫ⊗ tHom(CZ×U ,OX)) = 0

19

ii) P defines an isomorphism in H1(X; C((z,y)+γa)∩Hǫ⊗ tHom(CZ×U ,OX )).

Let us prove i):

Let s ∈ Γ(X; C((z,y)+γ)∩Hǫ⊗ tHom(CZ×U ,OX)). Then there exists

(z′, y′) ∈ ω such that s extends as a section s′ of Γ((z′, y′)+int(γ); tHom(CZ×U ,OX))

with support in Hǫ. Let γ′ denote the closed convex cone of Cn defined by

γ′ = (z1, z′, ),Rez1 ≤ −δ(||z′|| + |Imz1|)

Then, for any y′ ∈ U , the holomorphic function in the z variable s′(., y′) is

defined in z + Int(γ′) and has support in z,Rez1 ≥ −ǫ, hence vanishes

which entails s = 0 by Lemma 4.5.

The proof of ii) follows from Lemma 4.4 and the stepwise adaptation of

Proposition 5.1.5 of [13] by a classical argument (see [21], [23]).

q.e.d.

Proposition 4.7. Let X and Y complex manifolds and let f : X → Y be a

smooth morphism. Assume that Y is a complexification of a real manifold

N . Let F belong to DbR−c(CN ). Then (5) holds for f−1F .

Proof. The estimation (5) is locally checked so we may assume that F has

compact support. Then there exists a finite filtration N = N0 ⊃ N1 ⊃ ... ⊃

Nl = ∅ such that FNj\Nj+1is a constant sheaf. Since Nj \ Nj+1 is a locally

closed subanalytic set, which is always the difference of two open subanalytic

sets, by Remark 2.3, it is enough to prove the result for F = CU for an open

subanalytic relatively compact subset of N . Moreover we may assume that

N is Rd and Y = C

d. By Theorem 3.5, we can write U as a finite union

U =⋃

j Ωj ∩Rd, where each Ωj is Stein open subanalytic in C

d. By Lemma

4.6, (5) holds for⋃

j Ωj. On the other hand, Cd \R

d is a finite union of Stein

open subsets, hence (5) holds for (⋃

j Ωj) \ Rd which ends the proof.

q.e.d.

20

We may now achieve the proofs of the main results.

Proof of Theorem 2.5

By Corollary 3.2 we have

SS(RHomDX(M, tHom(f−1F,OX)))

⊂ pπpd(SS(RHomDX×Y

(p−1M, tHom(f × idY )−1Rl∗F,OX×Y ))).

where p : X × Y → X denotes the projection. The result is then an imme-

diate consequence of Proposition 4.7 together with Corollary 3.3.

Proof of Corollary 2.6

It is an immediate consequence of Proposition 4.7.

In the case of a product, f being a projection, there is a particular class

of D-modules for which estimate in Theorem 2.5 is easily improved.

Corollary 4.8. Assume that X = Z × Y and that f : X → Y is the

projection. Denote by p : X → Z the projection. Let M ∈ Modcoh(DZ).

Then

SS(RHomDX(p−1M, tHom(f−1F,OX))) ⊂ V +pdpπ

−1Char(M).

Proof. It is clear that we have

C1V (p−1M) = CV (p−1M).

The result then follows from the equalities T ∗ X ∩ ρ−1V (CV (p−1M)) =

V +Char(p−1M) = V +pdpπ−1Char(M). q.e.d.

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Teresa Monteiro Fernandes

Centro de Algebra da Universidade de Lisboa, Departamento de Matematica,

FCUL,

Edifıcio C6, piso 2, Campo Grande, 1749-16,Lisboa

Portugal

tmf@ptmat.fc.ul.pt

24