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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
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