An Existence Theorem for Tempered Solutions of D-Modules on Complex Curves

An Existence Theorem for Tempered Solutions of D-Modules on Complex Curves

By

GiovanniMorando^∗

Contents

§1. Subanalytic Sets

§1.1. Review on subanalytic sets

§1.2. Subanalytic subsets ofR²

§2. Tempered Holomorphic Functions

§2.1. The subanalytic site

§2.2. Definition and main properties ofO_X^t_sa

§2.3. Pull-back of tempered holomorphic functions

§3. Existence Theorem

§3.1. Some classical results

§3.2. Existence theorem for tempered holomorphic functions

§4. Tempered Holomorphic Solutions

§4.1. Classical results onD-modules

§4.2. Existence theorem for holonomicDX-modules

§4.3. The case of a single operator

§4.4. R-constructibility for tempered holomorphic solutions References

Communicated by M. Kashiwara. Received June 20, 2006. Revised November 8, 2006.

2000 Mathematics Subject Classification(s): Primary 32S40; Secondary 34M35; 32B20;

58J15.

Key words: D-modules, Ordinary differential equations, Tempered holomorphic, Suban- alytic.

Research supported in part by grant CPDA061823 of Padova University.

∗Dipartimento di Matematica Pura e Applicata, Universit`a di Padova, Via Trieste, 63, Padova 35121, Italy.

Institut de Math´ematiques de Jussieu, Universit´e Paris 6, 175 rue du Chevaleret, 75013 Paris, France.

Abstract

Let X be a complex curve, Xsa the subanalytic site associated to X, M a holonomic DX-module. Let O^t_Xsa be the sheaf on Xsa of tempered holomorphic functions andSol(M) (resp.Sol^t(M)) the complex of holomorphic (resp. tempered holomorphic) solutions ofM. We prove that the natural morphism

H¹(Sol^t(M))−→H¹(Sol(M))

is an isomorphism. As a consequence, we prove thatSol^t(M) isR-constructible in the sense of sheaves on Xsa. Such a result is conjectured by M. Kashiwara and P.

Schapira in [15] in any dimension.

Introduction

The problem of existence for ordinary linear differential equations (and even non-linear) is classical and the litterature presents many results on this subject. In particular, existence theorems for solutions with growth conditions have been obtained by many authors such as J.-P. Ramis and Y. Sibuya ([21]), B. Malgrange ([20]) and N. Honda ([7]). In [21] and [20], the authors proved existence for functions with Gevrey-type growth conditions at the origin on sectors of sufficiently small amplitude. Using similar techniques, in [7], the author proved existence for ultra-distributions with support onR_≥0.

The functional spaces considered in [21] and [20] correspond to sheaves on the real blow-up at the origin of C. Essentially they are sheaves on the unit circle. Indeed, growth conditions did not allow a global sheaf theoretical approach.

Nonetheless, tempered distributions were a basic tool in M. Kashiwara’s functorial proof of the Riemann-Hilbert correspondence in [9] and [10]. In order to use tempered distributions functorially, M. Kashiwara introduced the new functor THom of tempered cohomology. Such a functor represented the first step in a different approach to sheaves which, through [13], led to the full use of sheaves on sites in [14]. Indeed in [14], M. Kashiwara and P. Schapira combined classical analytical results of S. Lojasiewicz ([16], see also [19]) with sheaves on sites. They realized tempered distributions, temperedC^∞functions and WhitneyC^∞functions as sheaves on the subanalytic site. They also defined tempered holomorphic functions O_X^t_sa as the complex of the solutions of the Cauchy-Riemann system in the space of tempered distributions.

In a subsequent paper, [15], M. Kashiwara and P. Schapira extended the notion of microsupport of sheaves to the subanalytic site. In this way they established the framework for the study of tempered holomorphic solutions of D-modules. They also gave an example which is the starting point of the study of tempered holomorphic solutions of an irregular ordinary differential equation.

Given a complex analytic manifoldX, we denote by Op^c_X

sathe category of relatively compact subanalytic open subsets ofX and by X_sa the subanalytic site, that is the site whose underlying category is Op^c_Xsaand whose coverings are the finite coverings. We denote by Mod(k_X) (resp. Mod(k_Xsa)) the category of sheaves of k-modules on the siteX (resp. X_sa). Let:X −→X_sa be the natural morphism of sites.

Given aDX-moduleM, it is natural to compare SolM:=R_∗RHom_DX

M,OX

and

Sol^tM:= RHom_!DX

_!M,O^t_Xsa (for the definition of_!, see Section 2).

Along his proof of the Riemann-Hilbert correspondence, M. Kashiwara proved that, ifMis a regularDX-module, thenSol^tM−→^∼ SolM.

In [15], the authors studiedSol^tMcomparing it toSolM, for a particular example on a complex curve X.

In the present paper, we go into the study ofSol^t(M) forMa holonomic D-module on a complex curve X. In particular we prove an existence theo- rem for tempered solutions of ordinary differential equations in the subanalytic topology, thus refining the classical results on small open sectors. Such a result has two consequences.

First, we obtain that the natural morphism (0.0.1) H¹(Sol^tM)−→H¹(SolM) is an isomorphism.

Second, we prove that the complexSol^t(M) isR-constructible in the sense of [15]. In that paper the authors conjectured such a result in any dimension.

Our results being on a complex curve, it is natural to look for exten- sions of them in higher dimensions. In [22], C. Sabbah conjectured and widely developed the higher dimensional version of Hukuhara-Turrittin’s Theorem.

Recently Y. Andr´e announced the proof of Sabbah’s conjecture. Such results would be at the base of a possible extensions of our results.

The contents of the present paper are subdivided as follows.

InSection 1, we briefly review subanalytic sets recalling the classical re- sults that we will need. We study in detail relatively compact subanalytic open subsets ofR². We give a decomposition ofU ∈Op^c_R2

sausing sets biholomorphic to open sectors. Such a result will be essential in Section 3.

In Section 2, we recall definitions and basic results of sheaves on the subanalytic site and tempered holomorphic functions on a complex curve. In Subsection 2.2 we prove a result concerning the composition of a tempered holomorphic function and a biholomorphism.

InSection 3, we consider an open discX ⊂Ccentered at 0 and

(0.0.2) P :=z^N d

dzI_m+A(z),

where m∈ Z>0, N ∈ N, A∈ gl(m,OC(X)) and I_m is the identity matrix of order m. The aim of this section is to study the solvability ofP in the space of tempered holomorphic functions on U ∈ Op^c_Xsa with 0 ∈ ∂U. We prove that there exist an open neighborhood W ⊂ C of 0 and a finite subanalytic covering {U_j}_j∈J of U ∩W such that for any g_j ∈ O^t_Xsa(U_j)^m there exists u_j∈ O_X^t_sa(U_j)^msuch thatP u_j=g_j (j∈J). We start the section by recalling Hukuhara-Turrittin’s Theorem which is a basic tool in the study of ordinary differential equations.

InSection 4, we deal withDX-modules on a complex analytic curveX.

We begin by recalling some classical results onD_X-modules. In Subsection 4.2, we prove a first consequence of the results of Section 3, that is, (0.0.1) is an isomorphism. In Subsection 4.4 we prove a second consequence of the results of Section 3, that is, Sol^t(M) isR-constructible in the sense of sheaves onX_sa. We thank P. Schapira for proposing this problem to our attention and for many fruitful discussions and A. D’Agnolo for many useful remarks.

§1. Subanalytic Sets

In the first subsection, we recall the definition and some classical results on subanalytic sets. In the second subsection we focus on relatively compact subanalytic open subsets of R². We prove some results mixing the complex and the real analytic structure on R². Indeed, we describe the local structure of relatively compact subanalytic open subsets ofR²via biholomorphic images of open sectors (Theorem 1.4).

§1.1. Review on subanalytic sets

LetM be a real analytic manifold,Athe sheaf of real-valued real analytic functions on M.

Definition 1.1. LetX⊂M.

(i) X is said to besemi-analytic atx∈Mif there exists an open neighborhood W of xsuch that X∩W =∪_i∈I ∩_j∈JX_i,j whereI and J are finite sets and either X_i,j ={y ∈W; f_i,j(y)> 0} or X_i,j ={y ∈W; f_i,j(y) = 0} for somef_i,j∈ A(W). X is saidsemi-analytic if it is semi-analytic at each x∈M.

(ii) X is said subanalytic if for any x∈ M there exist an open neighborhood W ofx, a real analytic manifoldN and a relatively compact semi-analytic set A⊂M×N such thatπ(A) =X∩W, whereπ:M×N →M is the projection.

(iii) Let N be a real analytic manifold. A mapf :X →N is said subanalytic if its graph,

Γ_f :=

(x, y)∈X×N;y =f(x) , is subanalytic inM ×N.

GivenX ⊂M, denote byX^◦ (resp. X,∂X) the interior (resp. the closure, the boundary) ofX.

Proposition 1.1 (See [2]). Let X and Y be subanalytic subsets of M. ThenX∪Y,X∩Y,X,X^◦ andX\Y are subanalytic. Moreover the connected components of X are subanalytic, the family of connected components of X is locally finite and X is locally connected.

Definition 1.2, Theorem 1.1 and Proposition 1.2 below are stated and proved in [4] for the more general case of o-minimal structures.

Definition 1.2 (Cylindrical Cell Decomposition). Let n ∈ Z_>0. A cylindrical cell decomposition (ccd for short)

C_k

k∈K of Rⁿ is a finite par- tition ofRⁿinto subanalytic setsC_kobtained inductively onnin the following way. The setsC_k are called cells.

n= 1: The cells defining a ccd of R are open intervals ]a, b[ or points {c}, where a∈R∪ {−∞},b∈R∪ {+∞}, a < b, andc∈R.

n >1: A ccd D_h

h∈H ofRⁿ is given by a ccd C_k

k∈K ofRⁿ⁻¹,l_k∈Nand subanalytic continuous functions

ζ_k,1, . . . , ζ_k,lk:C_k →R

such that, for any x∈C_k, ζ_k,j(x)< ζ_k,j+1(x),j= 1, . . . , l_k−1 (k∈K).

The cells D_h are the graphs ofζ_k,j, Γ_ζk,j :=

x, ζ_k,j(x)

∈C_k×R

(1≤j≤l_k), and the sets

(1.1.1)

(x, y)∈C_k×R;ζ_k,j(x)< y < ζ_k,j+1(x) for 0≤j≤l_k, where ζ_k,0=−∞andζ_k,lk+1= +∞.

Theorem 1.1 (See [4], Theorem 2.10). Let A₁, . . . , A_d be relatively compact subanalytic subsets ofRⁿ. There exists a cylindrical cell decomposition of Rⁿ adapted to eachA_j. That is, eachA_j is a union of cells.

Proposition 1.2 (See [4], Theorem 3.4). LetZ be a subanalytic subset of Rⁿ. The following properties are equivalent.

(i) Z is closed and bounded.

(ii) Every subanalytic continuous map ζ :]0,1[→Z extends by continuity to a map[0,1[→Z.

(iii) For any subanalytic continuous function ζ : Z → R, ζ(Z) is closed and bounded.

For Theorem 1.2 below, see [2, Theorem 6.4].

Theorem 1.2 (Lojasiewicz’s Inequality). Let M be a real analytic ma- nifold,K⊂M. Letf, g:K→Rbe subanalytic functions with compact graphs.

If f⁻¹ {0}

⊂g⁻¹ {0}

, then there existc, r∈R_>0 such that, for anyx∈K,

|f(x)| ≥c|g(x)|^r . For Theorem 1.3 below, see [12, Proposition 8.2.3].

Theorem 1.3 (Curve Selection Lemma). LetZ be a subanalytic subset of M and let z₀∈Z. Then there exists an analytic map

γ:]−1,1[−→M , such that γ(0) =z₀ andγ(t)∈Z fort= 0.

§1.2. Subanalytic subsets of R²

Notation 1.1. Given a real analytic manifoldM, we denote by Op_Msa (resp. Op^c_M

sa) the category of subanalytic open (resp. relatively compact subanalytic open) subsets ofM.

Let

˜

π:R≥0×]−π,3π[−→R² (, ϑ)−→e^iϑ . One has that, givenU ∈Op^c_R2

sa with 0∈/ U, ˜π⁻¹(U) is a subanalytic open subset ofR>0×]−π,3π[, relatively compact inR².

For R ∈ R>0, η, ξ : [0, R] −→]−π,3π[ subanalytic continuous functions such that η()< ξ(), for any∈]0, R[, denote

B_η^ξ:=

(, ϑ)∈]0, R[×]−π,3π[ ;η()< ϑ < ξ()

.

Remark thatB_η^ξ ⊂[0, R]×]−π,3π[.

Proposition 1.3. LetU ∈Op^c_R2

sa,0∈∂U. There exists an open neigh- borhood W ⊂R² of 0, such that U ∩W is a finite union of sets of the form

˜

π(B^ξ_η)∩W.

Proof. The set ˜π⁻¹(U) is a subanalytic open subset of R_>0×]−π,3π[, relatively compact inR². Let ∈R_>0, < π. Take a cylindrical cell decompo- sition ofR² adapted to

˜

π⁻¹(U)∩

R>0×

− ,2π+ . The conclusion follows.

For z ∈ Cand ∈R_>0, denote by B(z, ) the open ball of center z and radius .

Let us introduce semi-analytic arcs and prove an easy result which states the local equivalence between semi-analytic arcs and graphs of subanalytic func- tions.

Definition 1.3. Let γ :]−1,1[−→ R² be an analytic map, δ ∈ R_>0 such that γ|_[0,δ] is injective. We call

Γ :=γ ]0, δ[

a semi-analytic arc with an endpoint atγ(0).

Recall that, given a functionη, we denote by Γ_η the graph ofη.

Lemma 1.1. Let R ∈ R>0, η : [0, R[→ R a subanalytic continuous map. There exist δ ∈ R>0 and an analytic map γ :]−1,1[−→ R² such that γ(0) =

0, η(0) and

(1.2.1) γ

]−1,1[\{0}

= Γ_η∩

]0, δ[×R .

In particular, there exist a semi-analytic arc Γ with an endpoint at 0, η(0) and an open neighborhood W ⊂R² of

0, η(0)

, such that Γ_η∩W = Γ∩W .

Proof. Letp₁:R²→Rbe the projection on the first coordinate.

By Theorem 1.3 there exists an analytic mapγ:]−1,1[−→R² such that γ(0) =

0, η(0) and

(1.2.2) γ

]−1,1[\{0}

⊂ Γ_η\

0, η(0) .

Remark that we can suppose that γ|_[0,1[ and γ|_]−1,0] are injective. Since γ

]−1,1[

is arcwise-connected,p₁

γ

]−1,1[

is arcwise-connected as well.

Hence, since {0} p₁

γ

]−1,1[

⊂ R_≥0, there exists δ ∈ R_>0 such that p₁

γ

]−1,1[

= [0, δ[.

Further, by (1.2.2),

(1.2.3) p₁

γ

]−1,1[\{0}

=]0, δ[ . Let us prove that if 0 < x < δ, then

x, η(x)

∈ γ

]−1,1[\{0} , this will conclude the proof. Let x ∈]0, δ[. By (1.2.3), there exists y ∈ R such that (x, y) ∈ γ

]−1,1[\{0}

. By (1.2.2), it follows that y = η(x). Hence x, η(x)

∈γ

]−1,1[\{0} .

Roughly speaking, from Lemma 1.1 and Proposition 1.3, it follows that (∂U)∩W is a finite collection of semi-analytic arcs with an endpoint at 0.

Let us now introduce biholomorphic images of open sectors. We start with a well known result on the local nature of holomorphic functions on C. For Proposition 1.4 below, see [6, Theorem 2.1].

Proposition 1.4. Let U ⊂C be an open neighborhood of 0, ϕ :U → C a non constant holomorphic map such that 0 is a zero of order n for ϕ.

There exist an open neighborhood U ⊂U of 0, ∈ R>0, and a holomorphic isomorphism ψ:U→B(0, ) such that, for anyz∈U,

ϕ|U(z) = ψ(z)_n

.

Definition 1.4. Letα, β∈R,r∈R>0, α < β. The set S_α,β,r:=

e^iϑ∈C^×; 0< < r, ϑ∈]α, β[

is called an open sector of amplitude β −α and radius r or simply an open sector.

We will need to stress on the amplitude and the direction of a sector so we will also use the following slightly different notation

S_τ±η,r:=S_{τ−η,τ+η,r} forτ ∈Randη, r∈R>0.

Corollary 1.1. Let U ⊂Cbe an open neighborhood of 0,ϕ:U →Ca non constant holomorphic map such that ϕ(0) = 0.

(i) There existr, τ ∈R_>0 such thatB(0, r)⊂U and, for any ϑ∈R,ϕ|_Sϑ±τ,r is an injective map.

(ii) Suppose that, givenα, β∈R, there existµ, δ, R∈R_>0 such that ϕ

]0, δ[×{0}

⊂S_{α+µ,β−µ,R} . Then, there exist η, r∈R>0 such that

ϕ

S_0±η,r

⊂S_α,β,R .

Proof. It is based on Proposition 1.4 and the fact that holomorphic iso- morphisms are conformal maps.

We are now ready to state and prove the main result of this section. Denote byOCthe sheaf of holomorphic functions on C.

Theorem 1.4. Let U ∈ Op^c_R2

sa, 0 ∈ ∂U. There exist an open neigh- borhood W ⊂ C of 0, a finite set J, open sectors S_j,k, ϕ_j,k ∈ O_C

S_j,k (j∈J, k= 1,2)such that

(i) ϕ_j,k(0) = 0 andϕ_j,k|_Sj,k is injective(j∈J, k= 1,2),

(ii)

U∩W =

j∈J

ϕ_j,1

S_j,1

∩ϕ_j,2 S_j,2

.

Proof of Theorem 1.4. By Proposition 1.3, it is sufficient to prove the statement for U = ˜π(B^ξ_η)∩W, forW ⊂Can open neighborhood of 0.

First we need two technical lemmas.

Lemma 1.2. Let S be an open sector, ϕ∈ O_C S

such thatϕ(0) = 0 andϕ|_S is injective. Suppose that there exists ∈R>0 such thatϕ(S)∩B(0, ) is contained in an open sector of amplitude strictly smaller than 2π.

Then there exist r ∈R>0, an open neighborhood V ⊂Cof 0 and ζ₁, ζ₂ : [0, ]→]−π,3π[ subanalytic continuous functions such that, for any∈[0, ], ζ₁()< ζ₂()and

ϕ

S∩B(0, r)

= ˜π B_ζ^ζ2

1

∩V .

Proof. We limit to give a sketch of the proof which is essentially of topo- logical nature.

There existη∈[0,2π],µ∈R>0,µ < π, such thatϕ(S)∩B(0, )⊂S_η±µ,. Remark that [η−µ, η+µ]⊂]η−π, η+π[⊂]−π,3π[. Take a ccd ofR² adapted to

˜ π⁻¹

ϕ

S

∩B(0, ) ∩

R_>0×]η−µ, η+µ[

.

Since, for anyδ∈R_>0,ϕ S

∩B(0, δ) has just one connected component having 0 in its boundary, the conclusion follows.

Lemma 1.3. Let R ∈ R_>0, η : [0, R] −→]−π,3π[ a subanalytic con- tinuous map. There exist an open neighborhood V ⊂ C of 0, τ, r, ∈ R>0, ϕ∈ OC

B(0, r)

andζ₁, ζ₂: [0, ]→]−π,3π[ subanalytic continuous functions satisfying the following conditions.

(i) ϕ|_S−τ,τ,r is injective.

(ii) For any ∈[0, ],−π < ζ₁()< η()< ζ₂()<3πand

(1.2.4) ϕ

S_0,τ,r

= ˜π

B_η^ζ2 ∩V ,

(1.2.5) ϕ

S_−τ,0,r

= ˜π

B_ζ^η

1

∩V .

Proof. Remark that it is sufficient to prove the statement for η|[0,] for some ∈R>0, < R. We set for shortη :=η|_[0,].

Since η(0)∈]−π,3π[, there exist , µ₁∈R>0,µ₁ < π, such that [η(0)− µ₁, η(0) +µ₁]⊂]−π,3π[ and

(1.2.6) Γ_η\

0, η(0)

⊂]0, R[×]η(0)−µ₁, η(0) +µ₁[.

First, let us show that there exist an open neighborhoodW ⊂Cof ]−1,1[, ϕ∈ OC(W) andδ∈R>0 such thatϕ(0) = 0 and

(1.2.7) ϕ

]−1,1[\{0}

× {0}

= ˜π

Γ_η∩

]0, δ[×R .

By Lemma 1.1, there exist δ∈R>0 and an analytic mapγ:]−1,1[→R² such that γ(0) =

0, η(0) and γ

]−1,1[\{0}

= Γ_η∩

]0, δ[×R .

Since ˜π◦γ is an analytic map, there exist a complex neighborhood Wof ]−1,1[ and ϕ ∈ O_C(W) such that ϕ|_{]−1,1[×{0}} = ˜π◦γ|_]−1,1[. In particular, ϕ(0) = 0 and

(1.2.8) ϕ

]−1,1[\{0}

× {0}

= ˜π

Γ_η∩

]0, δ[×R .

Hence, (1.2.7) follows.

Now, remark that (1.2.6) implies that

(1.2.9) π˜

Γ_η\

0, η(0)

⊂S_η(0)±µ1,R. Combining (1.2.8) and (1.2.9), we have that

ϕ

]0,1[×{0}

⊂π˜

Γ_η∩

]0, δ[×R

⊂π˜ Γ_η\

0, η(0)

⊂S_η(0)±µ1,R . (1.2.10)

Since [η(0)−µ₁, η(0) +µ₁] ⊂]−π,3π[ and µ₁ < π, there exists µ₂ ∈ R>0

such that µ₁< µ₂< π and

(1.2.11) [η(0)−µ₂, η(0) +µ₂]⊂]−π,3π[.

Let r ∈R_>0 be such that B(0, r)⊂W. Then, Corollary 1.1 (ii) applies and there existτ∈R_>0such that, up to shrinking r,

(1.2.12) ϕ

S_0,τ,r

⊂S_η(0)±µ2,R.

Further, by Corollary 1.1 (i), up to shrinking τ and r, we have thatϕ|_S0,τ,r is injective.

Then, Lemma 1.2 applies and there existr ∈R>0, an open neighborhood V ⊂Cof 0 andζ₁, ζ₂: [0, ]→]−π,3π[ subanalytic continuous functions such that, for any∈[0, ], ζ₁()< ζ₂() and

ϕ(S_0,τ,r) = ˜π(B_ζ^ζ2

1)∩V .

Then, (1.2.11) and (1.2.12) imply that one can choseζ₁=η. Hence (1.2.4) follows.

Clearly, (1.2.5) can be proved using the same arguments.

End of the Proof of Theorem 1.4. As said above, by Proposition 1.3, it is sufficient to prove the statement for U = ˜π(B_η^ξ)∩W, for W ⊂C an open neighborhood of 0.

ConsiderB^ξ_η, by Lemma 1.3, there existζ₁, ζ₂: [0, ]→]−π,3π[,r, τ ∈R>0, ϕ₁, ϕ₂ ∈ OC

B(0, r)

, V₁, V₂ ⊂Copen neighborhoods of 0 such that, for any ∈ [0, ], η() < ζ₂() < 3π, −π < ζ₁() < ξ(), ϕ₁|_S0,τ,r ϕ₂|_S−τ,0,r are injective and

˜ π

B_η^ζ2

∩V₁=ϕ₁(S_0,τ,r),

˜ π

B_ζ^ξ

1

∩V₂=ϕ₂(S_−τ,0,r).

We distinguish two cases: ξ(0) =η(0) and η(0)< ξ(0).

(i) Suppose ξ(0) =η(0).

We have that

−π < ζ₁(0)< η(0) =ξ(0)< ζ₂(0)<3π . It follows that there exists ∈R>0 such that, for any∈[0, ],

ζ₁()< η()≤ξ()< ζ₂().

Hence, consideringη, ξ, ζ₁, ζ₂ as restricted to [0, ], we have that B_η^ξ =B_η^ζ2∩B^ξ_ζ

1 .

Now, up to take smallerτ, , we can suppose that ˜π(B_ζ^ξ

1) and ˜π(B_η^ζ2) are contained in an open sector of amplitude strictly smaller than 2π. In particular,

˜

πis a bijection on B_η^ζ2∪B^ξ_ζ

1. It follows that

˜ π

B_η^ξ

= ˜π B_η^ζ2

∩˜π B_ζ^ξ

1

.

TakingV :=V₁∩V₂, the conclusion follows.

(ii) Suppose η(0)< ξ(0).

Up to take smaller τ, there exist ∈R>0 andα, β : [0, ]→Rconstant functions such that, for any∈[0, ],

η()< α()< ζ₂()< ζ₁()< β()< ξ(). It follows that, consideringη, ξ, ζ₁, ζ₂ as restricted to [0, ],

B_η^ξ =B^ζ_η²∪B_α^β∪B_ζ^ξ

1 . The conclusion follows.

Detailing the proof of Theorem 1.4, one can give a more precise statement in the following way.

Remark. LetU,W,ϕ_j,kandS_j,k as given in Theorem 1.4. Givenr, η ∈ R_>0, there exist an open neighborhood W ⊂W of the origin, a finite set J and open sectors S_j,k ⊂ S_j,k (j ∈ J) such that the amplitude (resp. the radius) ofS_j,k is smaller thanη (resp. r) and

U∩W=

j∈J

ϕ_j,1

S_j,1

∩ϕ_j,2 S_j,2

.

§2. Tempered Holomorphic Functions

In the first subsection we recall the definition and some classical results on the subanalytic site X_sa underlying a complex curve X and sheaves on X_sa. In the second subsection we recall the definition of the subanalytic sheaf of tempered holomorphic functions on a complex curve. In the third subsection we prove a result on the pull back of tempered holomorphic functions through biholomorphisms. We refer to [15] and [14] for the first and the second subsec- tions.

§2.1. The subanalytic site

LetXbe a complex analytic manifold, denote byX the complex conjugate manifold and byX_Rthe underlying real analytic manifold. Forka commutative ring, we denote by Mod(k_X) the category of sheaves ofk-modules onX.

We endow Op^c_X

sa := Op^c_X

Rsa with a Grothendieck topology, called the subanalytic topology, by deciding that an usual open covering U =∪i∈IU_i in

Op^c_X

sa is a covering for the subanalytic topology if there exists a finite subset J ⊂Isuch thatU =∪j∈JU_j. Denote byX_sathis site and call it thesubanalytic site. Further, denote by Cov_sa(U) the family of coverings ofU ∈Op^c_Xsa for the subanalytic topology and by Mod(k_Xsa) the category of sheaves of k-modules on the subanalytic site.

One can show (see [14, Remark 6.3.6]) that Mod(k_Xsa) is equivalent to the category of sheaves on the siteX_sa,lf, where the class of open sets ofX_sa,lf is Op_Xsa and, forU ∈Op_Xsa, the family of coverings ofU forX_sa,lf consists of subanalytic open coverings{U_σ}σ∈Σof U such that for any compactK of X, there exists a finite subsetJ ⊂Σ such thatK∩

∪_j∈JU_j

=K∩U.

Let PSh(k_Xsa) be the category of presheaves ofk-modules onX_sa. Denote by f or : Mod(k_Xsa)→ PSh(k_Xsa) the forgetful functor which associates to a sheafF onX_saits underlying presheaf. It is well known thatf oradmits a left adjoint·^a : PSh(k_Xsa)→Mod(k_Xsa).

ForF ∈PSh(k_Xsa), let us briefly recall the construction ofF^a. ForU ∈Op^c_Xsa andS={U₁, . . . , U_n} ∈Cov_sa(U), set

(2.1.1) F(S) :=

(s₁, . . . , s_n)∈ n j=1

F(U_j); s_j|Uj∩Uk=s_k|Uj∩Uk, j, k= 1, . . . , n

.

If S is a covering ofU andS is a refinement ofS, then there exists a natural restriction morphism F(S)−→

_SS F(S).

Then, forU ∈Op^c_X

sa, set

(2.1.2) F⁺(U) := lim−→

S∈Covsa(U)

F(S).

It turns out that F^a F⁺⁺.

Now, let s ∈ F^a(U). Since the inductive limit considered in (2.1.2) is filtrant, s can be identified to an n-uple (s₁, . . . , s_n) ∈ F(S), for S = {U_j}j=1,...,n∈Cov_sa(U),s_j ∈F(U_j) (j= 1, . . . , n).

Further, ifs∈F^a(U) can be identified to s₁∈F(S₁) and tos₂ ∈F(S₂), forS₁, S₂∈Cov_sa(U), then there exists a refinement S∈Cov_sa(U) ofS₁and S₂and ¯s∈F(S) such thatscan be identified to ¯s.

For Proposition 2.1 below, see [14, Proposition 2.1.12].

Proposition 2.1. Consider the complex inMod(k_Xsa)

(2.1.3) F−→^ϕ F −→^ψ F .

The following conditions are equivalent.

(i) (2.1.3)is exact.

(ii) For any U ∈ Op^c_Xsa and any t ∈ F(U) such that ψ(t) = 0, there exist {U_j}j∈J ∈Cov_sa(U) ands_j∈F(U_j)such that ϕ(s_j) =t|Uj (j∈J).

We shall denote by

:X −→X_sa ,

the natural morphism of sites associated to Op^c_Xsa −→ Op_X = {U ⊂ X;U open}. We refer to [14] for the definitions of the functors _∗ : Mod(k_X)−→Mod(k_Xsa) and⁻¹: Mod(k_Xsa)−→Mod(k_X) and for Proposi- tion 2.2 below.

Proposition 2.2.

(i) _∗ is right adjoint to ⁻¹.

(ii) ⁻¹ has a left adjoint denoted by _! : Mod(k_X)−→Mod(k_Xsa). (iii) ⁻¹ and_! are exact,_∗ is exact on constructible sheaves.

(iv) _∗ and_! are fully faithful.

Through _∗, we will consider Mod(k_X) as a subcategory of Mod(k_Xsa).

The functor _! is described as follows. If U ∈Op^c_X

sa andF is a sheaf on X, then_!(F) is the sheaf onX_sa associated to the presheafU →F

U .

§2.2. Definition and main properties of O^t_Xsa

Througout this subsection,X will be a complex analytic curve with struc- ture sheaf OX. For higher dimensions we refer to [14].

Denote by D_X the sheaf of differential operators with holomorphic coef- ficients on X. Denote by Db_XR the sheaf of distributions on X_R and, for a closed subset Z of X, by Γ_Z(Db_XR) the subsheaf of sections supported byZ.

One denotes byDb^t_X

sa the presheaf oftempered distributions onX_Rdefined as follows

Op_XsaU −→ Db^t_Xsa(U) := Γ(X;Db_XR)

Γ_X\U(X;Db_XR). In [14] it is proved thatDb^t_X

sais a sheaf onX_sa. This sheaf is well defined in the category Mod(_!DX). Moreover, for anyU ∈Op^c_Xsa,Db^t_X

sa is Γ(U,·)-acyclic.

One defines the sheaf O^t_Xsa ∈ D^b _!D_X

of tempered holomorphic func- tions as

O^t_Xsa:=RHom_!D

X

_!O_X,Db^t_X

R

.

In [14] it is proved that, since dimX = 1, R_∗OX and O_X^t_sa are concen- trated in degree 0 . Hence we can write the following exact sequence of sheaves onX_sa

0−→ O_X^t_sa −→ Db^t_X

sa

∂¯

−→ Db^t_X

sa−→0. Lemma 2.1. Let X=C, X_R=R²,U, V ∈Op^c_R2

sa. (i) H^j(U,O^t_Xsa) = 0, forj >0.

(ii) The following sequence is exact

(2.2.1) 0→ O_X^t_sa(U∪V)→ O^t_Xsa(U)⊕ O_X^t_sa(V)→ O_X^t_sa(U∩V)→0. Proof. (i) By the definition ofDb^t_X

sa, given h∈ Db^t_X

sa(U), there exists h ∈ Db_XR(R²) such that h

U = h. It is well known that there exists g ∈ Db_XR(R²) such that ¯∂g =h. This implies that ¯∂

g|U

=h. So we have the exact sequence

0−→ O_X^t_sa(U)−→ Db^t_Xsa(U)−→ D^∂¯ b^t_Xsa(U)−→0. SinceDb^t_X

sa is acyclic with respect to the functor Γ(U;·), forU ∈Op^c_X

sa, it follows that, for all j∈Z_>0,H^j(U,O^t_Xsa) = 0.

(ii) Obvious from (i).

Now we recall the definition of polynomial growth for C^∞ functions on X_Rand in (2.2.5) we give an alternative expression for tempered holomorphic functions on U ∈Op^c_X

sa.

Definition 2.1. LetU be an open subset ofX_R,f ∈ C_X^∞_R(U). One says that f haspolynomial growth at p ∈X if it satisfies the following condition.

For a local coordinate systemx= (x₁, x₂) aroundp, there exist a sufficiently small compact neighborhoodK ofpand M ∈Z>0 such that

(2.2.2) sup

x∈K∩Udist(x, K\U)^Mf(x)<+∞.

We say that f ∈ C^∞_XR(U) has polynomial growth on U if it has polynomial growth at anyp∈X. We say thatf istempered atpif all its derivatives have polynomial growth at p∈X. We say thatf istempered on U if it is tempered at any p∈X. Denote byC_X^∞,t the presheaf onX_Rof temperedC^∞-functions.

It is obvious that f has polynomial growth at any point of U. If no confusion is possible we will write “f is tempered” instead of “f is tempered onU”.

In [14] it is proved thatC_X^∞,t is a sheaf onX_sa.

For U ⊂R² a relatively compact open set, we can characterize functions with polynomial growth on U by means of a family of norms.

For x ∈ R², f ∈ C_R^∞2(U), g = (g₁, . . . , g_m) ∈

C^∞_R2(U)_m

and M ∈ Z>0, denote

δ_∂U(x) := dist(x, ∂U), (2.2.3)

||f||^M_U := sup

x∈U δ_∂U(x)^M|f(x)| ,

||g||^M_U := max

||g_j||^M_U; j= 1, . . . , m .

Proposition 2.3. Let U ⊂R² be a relatively compact open set and let f ∈ C_R^∞2(U). Thenf has polynomial growth if and only if there existsM ∈R_>0 such that

(2.2.4) ||f||^M_U <+∞ ,

or equivalently: there existC, M ∈R_>0 such that for anyx∈U, f(x)≤Cδ_∂U(x)^−M .

Proof. Suppose thatf satisfies (2.2.4), that is, sup

x∈U δ_∂U(x)^M|f(x)|<+∞.

Let K be a compact neighborhood of U. For any p ∈ U, K is a compact neighborhood ofpsuch that

sup

x∈K∩Udist(x, K\U)^M|f(x)| ≤sup

x∈U δ_∂U(x)^M|f(x)|

<+∞. Hence,f has polynomial growth.

Conversely, suppose that hhas polynomial growth. That is, for p∈∂U, there exists a compact neighborhoodK_p ofpverifying (2.2.2).

Set

V :=

x∈K_p; δ_∂U\Kp(x)> δ_∂U(x)

.

Then for any x∈V, δ_∂U(x) =δ_∂U∩Kp(x).

Sincep∈V, there exists ∈R>0 such thatB(p, )⊂V. Set Z_p :=B(p, )∪(K_p∩∂U).

Then Z_p∩∂U =K_p∩∂U and, for anyx ∈Z_p∩U, δ_∂U(x) = δ_∂U∩Kp(x) = δ_∂U∩Zp(x).

Hence,

sup

x∈Zp∩Uδ_∂U(x)^M|f(x)|= sup

x∈Zp∩Uδ_∂U∩Zp(x)^M|f(x)|

≤ sup

x∈Kp∩Uδ_∂U∩Kp(x)^M|f(x)|

<+∞. Since∂U is compact, the conclusion follows.

Lemma 2.2 below is an easy consequence of Cauchy’s Formula. See [23, Lemma 3].

Lemma 2.2. Let U be a relatively compact open subset of X, f ∈ OX(U) with polynomial growth onU. Then f ∈ O^t_Xsa(U).

For Proposition 2.4 below, see [14].

Proposition 2.4. One has the following isomorphism O^t_Xsa RHom_!D

X

_!O_X,C_X^∞,t_R .

Hence, forU ∈Op^c_Xsa, we deduce the short exact sequence (2.2.5) 0−→ O_X^t_sa(U)−→ C_X^∞,t_R (U)−→ C^∂¯ _X^∞,t_R (U)−→0.

§2.3. Pull-back of tempered holomorphic functions

Recall that, forU a relatively compact open subset ofR² andz∈R², we set δ_∂U(z) := dist(z, ∂U).

Lemma 2.3. Let X be an open subset of R², f : X → R² be a C^∞- subanalytic map. Let U ∈ Op^c_Xsa, V ∈ Op^c_R2

sa satisfying f(U) = V and f

∂U

=∂V. Let h∈ C_R^∞2(V).

Then h has polynomial growth on V if and only if h◦f has polynomial growth on U.

Proof. Consider the subanalytic continuous functionsδ_∂U, δ_∂V◦f|_U :U → R≥0. Sincef

∂U

=∂V andf(U) =V, δ_∂V ◦f|_U−1

{0}

=∂U . In particular,

δ_∂V ◦f|_U−1 {0}

=δ⁻¹_∂U {0}

.

By Theorem 1.2, there exist a, b, α, β∈R>0 such that, for anyx∈U,

(2.3.1) a

δ_∂V ◦f|_U(x) _α

≤δ_∂U(x), and

(2.3.2) b

δ_∂U(x)_β

≤δ_∂V ◦f|_U(x).

(i) Suppose that h◦f has polynomial growth on U, that is, there exist C, M ∈R_>0such that, for any x∈U,

h

f(x)≤C

δ_∂U(x)_−M .

By (2.3.1), we obtain h

f(x)≤Ca^−M

δ_∂V ◦f|_U(x)_{−M α} .

Sincef(U) =V, it follows that, for anyy∈V, h(y)≤Ca^−M

δ_∂V(y)_{−M α} ,

that is,hhas polynomial growth onV.

(ii) Suppose that h has polynomial growth on V, that is, there exist C, M∈R_>0such that, for any y∈V,

|h(y)| ≤C

δ_∂V(y)_−M .

Sincef(U) =V, we have, for anyx∈U, h(f(x))≤C

δ_∂V ◦f(x)_−M .

By (2.3.2), we obtain

h(f(x))≤Cb^−M

δ_∂U(x)_−Mβ ,

that is,h◦f has polynomial growth onU.