L’INSTITUTFOURIER DE ANNALES

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A L

E S

E D L ’IN IT ST T U

F O U R

ANNALES

DE

L’INSTITUT FOURIER

Giovanni MORANDO

Tempered solutions ofD-modules on complex curves and formal invariants Tome 59, n^o4 (2009), p. 1611-1639.

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TEMPERED SOLUTIONS OF D-MODULES ON COMPLEX CURVES AND FORMAL INVARIANTS

by Giovanni MORANDO (*)

Abstract. — LetX be a complex analytic curve. In this paper we prove that the subanalytic sheaf of tempered holomorphic solutions of D-modules onX induces a fully faithful functor on a subcategory of germs of formal holonomicD- modules. Further, given a germMof holonomicD-module, we obtain some results linking the subanalytic sheaf of tempered solutions ofMand the classical formal and analytic invariants ofM.

Résumé. — Soit X une courbe analytique complexe. Dans cet article nous démontrons que le faisceau sous-analytique des solutions holomorphes tempérées desD-modules surX induit un foncteur pleinement fidèle sur une sous-catégorie des germes desD-modules holonomes formels. De plus, étant donné un germeM de D-module holonome, nous obtenons des résultats qui lient le faisceau sous- analytique des solutions tempérées deMavec les invariants formels et analytiques classiques deM.

Introduction

The search for algebraic or topological invariants of complex linear partial differential equations is classical and widely developed.

At the very first step of the study of linear differential equations, two main types of equations are distinguished: regular and irregular. To give an idea of the difference between the two kinds of equations, let us recall that, in dimension1, the solutions of the former equations have moderate growth while the solutions of the latter have exponential-type growth.

Keywords: D-modules, irregular singularities, tempered holomorphic functions, subanalytic.

Math. classification:34M35, 32B20, 34Mxx.

(*) Research supported in part by grant CPDA061823 of Padova University and in part by fellowship SFRH/BPD/29934/2006 of Fundação para Ciência e Tecnologia at Centro de Álgebra da Universidade de Lisboa.

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The more general algebraic approach to the study of linear differential equations consists in considering differential equations as sheaves of modules over the ringD_X of linear differential operators on a manifoldX. In this framework, in [7] and [8], M. Kashiwara gives a proof of the Riemann- Hilbert correspondence which is a generalization of the 21st Hilbert’s problem. ForX a complex analytic manifold, M. Kashiwara defines the functor THom and he gives an explicit inverse to the functor of holomorphic solutions from the bounded derived category of complexes of DX-modules with regular holonomic cohomology to the bounded derived category of complexes of sheaves with constructible cohomology. This implies the clas- sic result that the functor of holomorphic solutionsS(·)is an equivalence between the category of regular meromorphic connections onX with poles along a closed submanifoldZ and the category of linear representations of finite dimension of the fundamental group ofXrZ.

The irregular case is more complicated. In complex dimension 1, the classification of meromorphic connections obtained through the formal classification and the Stokes coefficients is nowadays well understood. Let us simply recall that, roughly speaking, the difference between a regular connection and an irregular one is based on the presence of functions of the typeexpϕ(ϕ∈z^−1/lC[z^−1/l],l ∈Z>0) in the solutions of the latter. The polynomials ϕ, appearing at exponent in the solutions of a meromorphic connectionM, are called determinant polynomials ofM. Their presence is made explicit in the Levelt-Turrittin’s Formal Theorem (Theorem 1.13) and in the Hukuhara-Turrittin’s Asymptotic Theorem (Theorem 1.15) which are of analytic nature. It is interesting to look for a topological description of the determinant polynomials.

In higher dimension, the study of irregularD-modules is much more complicated. In [20] (see also [19]), C. Sabbah defines the notion of good model in dimension 2 and he conjectures the analogue of the Levelt-Turrittin’s Formal Theorem, further he proves it for meromorphic connections of rank 65. Recently, T. Mochizuki proved Sabbah’s conjecture in any dimension in the algebraic case, see [15] and [16].

Given a complex analytic manifold X, in [10], M. Kashiwara and P. Schapira defined the complex of sheaves of tempered holomorphic func- tionsO_X^t

sa. The entries of the complex O_X^t

sa are not sheaves on a topological space but on the subanalytic site, X_sa. The open sets of X_sa are the subanalytic open subsets of X, the coverings are locally finite coverings. If X has dimension 1, then O^t_X

sa is a sheaf on X_sa and, for U a relatively compact subanalytic open subset ofX, the sections ofO_X^t_sa(U)

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are the holomorphic functions onU which extend as distributions onX or, equivalently, which have moderate growth at the boundary ofU.

Further in an example in [11], M. Kashiwara and P. Schapira explicited the sheaf of tempered holomorphic solutions of D_Cexp(1/z). Such example suggests that tempered holomorphic functions and the subanalytic site could be useful tools in the study of ordinary differential equations. Roughly speaking, one of the ideas underlying the irregular Riemann-Hilbert correspondence in dimension 1 is to enrich, by ad hoc tools taking in account determinant polynomials and Stokes coefficients, the structure of the category of sheaves ofC-vector spaces where the functor of holomorphic solutions takes values (see the notion ofΩ-filtered local systems in [1] or [14]).

The approach through subanalytic sheaves allows to enrich the topology of the space where the sheaf of solutions lives.

In this paper we go into the study of the subanalytic sheaf of tempered holomorphic solutions of germs of D-modules. Denote by S^t M the subanalytic sheaf of tempered holomorphic solutions of a holonomic D_X-module M. Let X ⊂ C be an open neighborhood of 0, Mod(CXsa) the category of sheaves ofC-vector spaces onXsa. We denote byGMk be the category of modules over the ring of linear differential operators with formal Laurent power series “without ramification” (see Section 1.3 for a precise definition) and with Katz invariant< k. Roughly speaking, up to ramification, every meromorphic connection is formally equivalent to an element ofGMk, forkbig enough. We prove that

S^t · ⊗Dexp(1/z^k)

:GMk −→Mod(CXsa)

is a fully faithful functor (Theorem 3.5). Further we prove that, given a germ of holonomicDX-module Mwith Katz invariant< k, the datum ofS^t M ⊗ D_Xexp(1/z^k+1)

is equivalent to the data of the holomorphic solutions, the determinant polynomials and their “rank” (Theorem 3.7).

Let us also recall that many sheaves of function spaces have been used in the study of irregular ordinary differential equations. For example, one can find in [14] the definitions of the sheavesA⁶^r (r∈R) defined on the real blow-up of the complex plane at the origin. In [6], P. Deligne defined the sheaves Fe^k, successively studied in detail in [12]. Roughly speaking, the solutions ofD_Cexp(ϕ)with values in A⁶^r (resp.Fe^k) depend only on the degree and the argument of the leading coefficient ofϕ (resp. the degree and the leading coefficient ofϕ).

In conclusion we can say that tempered solutions on the subanalytic site give a good topological description of the determinant polynomials of a

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given meromorphic connection. As further development, it would be interesting to describe precisely the image category of the functor of tempered solutions in order to give a full topological description of the space of determinant polynomials. It would also be interesting to give a good notion of Fourier transform for tempered holomorphic solutions of algebraic D- modules in the same spirit of [14].

The present paper is subdivided in three sections organized as follows.

Section 1 is devoted to the definitions, the notations and the presen- tation of the main results that will be needed in the rest of the paper. In particular we recall classical results on the subanalytic sets and site, on the tempered holomorphic functions and on the germs of D-modules on complex curves. We recall the Levelt-Turrittin’s Formal Theorem and the Hukuhara-Turrittin’s Asymptotic Theorem. The latter theorem allows to endow holomorphic solutions of meromorphic connections on sufficiently small sectors with a graduation with respect to the space of determinant polynomials.

The functions of the formexp(ϕ),ϕ∈z⁻¹C[z⁻¹], are the responsible for the non-tempered-growth of the solutions of an irregularD-module. This motivates the study of exp(ϕ) that we develop in Section 2. In partic- ular, given ϕ ∈ z⁻¹C[z⁻¹] and U a relatively compact subanalytic open subset ofC, we give a necessary and sufficient topological condition onU so that exp(ϕ) ∈ O^t_C_sa(U). Further, given ϕ₁, ϕ₂ ∈ z⁻¹C[z⁻¹], we prove that the condition “for any U ⊂ C relatively compact subanalytic open set,exp(ϕ₁)∈ O^t

Csa(U)if and only ifexp(ϕ₂)∈ O^t

Csa(U)” is equivalent to

“ϕ1and ϕ2 are proportional by a real positive constant”.

InSection 3we apply the results of Section 2 to the study of the functor of tempered holomorphic solutions of germs ofDX-modules on a complex curveX. We prove thatS^t · ⊗DXexp(1/z^k)

:GMk →Mod(CX_sa) is a fully faithful functor and that, given a germ ofDX-moduleM with Katz invariant< k, the datum ofS^t M ⊗ Dexp(1/z^k+1)

is equivalent to the data of the holomorphic solutions ofM, the determinant polynomials of Mand their rank.

Acknowledgements. — We thank P. Schapira for proposing this problem to our attention, C. Sabbah and N. Honda for many fruitful discussions and A. D’Agnolo for many useful remarks.

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1. Notations and review

In this section we recall the definitions and the classical results concern- ing:

(i) subanalytic sets, the subanalytic site and sheaves on it, (ii) the subanalytic sheaf of tempered holomorphic functions, (iii) germs ofD-modules on a complex curve.

1.1. The subanalytic site

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

Definition 1.1.

(i) A set X ⊂ M is said semi-analytic at x ∈ M if the following condition is satisfied. There exists an open neighborhood W of x such thatX∩W =∪_i∈I∩_j∈JXij whereIandJ are finite sets and either X_ij ={y ∈W; f_ij(y)>0} or X_ij ={y ∈W; f_ij(y) = 0}

for somefij ∈ A(W). Further,X is saidsemi-analyticifX is semi- analytic at anyx∈M.

(ii) A setX ⊂M is said subanalytic if the following condition is sat- isfied. For anyx∈M, there exist an open neighborhood W of x, a real analytic manifoldN and a relatively compact semi-analytic setA⊂M ×N such thatπ(A) =X∩W, whereπ:M ×N→M is the projection.

GivenX⊂M, denote by

◦

X (resp.X,∂X) the interior (resp. the closure, the boundary) ofX.

Proposition 1.2 (See [2]). — Let X and Y be subanalytic subsets ofM. Then X ∪Y, X ∩Y, X, X^◦ and X rY are subanalytic. Moreover the connected components ofX are subanalytic, the family of connected components ofX is locally finite andX is locally connected.

Proposition 1.3 below is based on Łojasiewicz’s inequality, see [2, Corol- lary 6.7].

Proposition 1.3. — LetU ⊂Rⁿ be an open set,X, Y closed subana- lytic subsets ofU. For any x₀∈X∩Y, there exist an open neighborhood W ofx0,c, r∈R>0such that, for any x∈W,

dist(x, X) + dist(x, Y)>cdist(x, X∩Y)^r.

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Definition 1.4. — Let∈R>0,γ:]−, [→M an analytic map. The setγ(]0, [)is said asemi-analytic arc with an endpoint atγ(0).

Theorem 1.5 (Curve Selection Lemma.). — Let Z 6= ∅ be a suban- alytic subset of M and let z0 ∈ Z. Then there exists an analytic map γ:]−1,1[−→M, such thatγ(0) =z0andγ(t)∈Z fort6= 0.

For the rest of the subsection we refer to [10].

Let X be a complex analytic curve, we denote by Op(X)the family of open subsets ofX. Forka commutative unital ring, we denote byMod(kX) the category of sheaves ofk-modules onX.

Let us recall the definition of the subanalytic siteXsaassociated toX. An elementU ∈Op(X)is an open set forX_saif it is open, relatively compact and subanalytic inX. The family of open sets ofXsa is denotedOp^c(Xsa).

ForU ∈Op^c(X_sa), a subsetS of the family of open subsets ofU is said an open covering ofU inXsa ifS⊂Op^c(Xsa)and, for any compactK ofX, there exists a finite subsetS0⊂S such thatK∩(∪V∈S0V) =K∩U.

We denote by Mod(kXsa) the category of sheaves of k-modules on the subanalytic site. With the aim of defining the category Mod(kX_sa), the adjective “relatively compact” can be omitted in the definition above. In- deed, in [10, Remark 6.3.6], it is proved thatMod(kX_sa)is equivalent to the category of sheaves on the site whose open sets are the open subanalytic subsets ofX and whose coverings are the same asXsa.

Given Y ∈Op^c(Xsa), we denote byYX_sa the site induced byXsa onY, defined as follows. The open sets ofYXsaare open subanalytic subsets ofY. A covering ofU ∈Op(Ysa)for the topologyYX_sa is a covering ofU inXsa. We denote by %:X −→X_sa, the natural morphism of sites associated toOp^c(Xsa)−→Op(X). We refer to [10] for the definitions of the functors

%_∗ : Mod(k_X)−→Mod(k_X_sa)and%⁻¹ : Mod(k_X_sa)−→Mod(k_X)and for Proposition 1.6 below.

Proposition 1.6.

(i) %⁻¹ is left adjoint to%_∗.

(ii) %⁻¹ has a left adjoint denoted by%! : Mod(kX)−→Mod(kX_sa).

(iii) %⁻¹ and%_! are exact and %_∗is exact on R-constructible sheaves.

(iv) %_∗ and%! are fully faithful.

Through%_∗, we will considerMod(kX)as a subcategory ofMod(kX_sa).

The functor %_! is described as follows. Let F ∈Mod(k_X), then%_!(F) is the sheaf onXsa associated to the presheafU 7→F U

.

Remark 1.7. — It is worth to mention that, given an analytic mani- foldX, there exists a topological spaceX⁰such that the category of sheaves

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onX_sa with values in sets is equivalent to the category of sheaves on X⁰ with values in sets. A detailed description of the semi-algebraic case and the o-minimal case are presented respectively in [3] and [5].

1.2. Definition and main properties of O^t_X

sa

For this subsection we refer to [10].

LetX be a complex analytic curve, denote byX the complex conjugate curve and byX_Rthe underlying real analytic manifold. We denote byXsa

the subanalytic site relative toX_R.

Denote byOX (resp.DX) the sheaf of holomorphic functions (resp. linear differential operators with holomorphic coefficients) onX. Denote by DbX_R the sheaf of distributions onX_Rand, for a closed subsetZ ofX, by ΓZ(DbX_R)the subsheaf of sections supported byZ. One denotes byDb^t_X

sa

the presheaf of tempered distributions onX_sadefined as follows, Op^c(Xsa)3U 7−→ Db^t_X_sa(U) := Γ(X;DbX_R)

ΓXrU(X;DbX_R). In [10], using some results of [13], it is proved thatDb^t_X

sais a sheaf onXsa. This sheaf is well defined in the categoryMod(%_!D_X). Moreover, for any U ∈Op^c(Xsa),Db^t_X

sa isΓ(U,·)-acyclic.

Denote by D^b %!DX

the bounded derived category of %!DX-modules.

The sheafO_X^t

sa∈D^b %!DX

of tempered holomorphic functions is defined as

O^t_X_sa:=RHom%_!D_X %!O_X,Db^t_X

R

.

In [10], it is proved that, since dimX = 1, R%_∗O_X and O_X^t

sa are concentrated in degree0. Hence we can write the following exact sequence of sheaves onX_sa

0−→ O^t_X_sa −→ Db^t_X_sa −→ Db^∂^¯ ^t_X_sa −→0.

Let us recall thatDb^t_X_sa andO_X^t_sa can be defined without any change on a complex analytic manifoldX (see [10]).

Now we recall the definition of polynomial growth forC^∞functions onX_R and in (1.2) we give an alternative expression forO_X^t

sa(U),U ∈Op^c(Xsa).

Definition 1.8. — Let U be an open subset of X, f ∈ C_X^∞

R(U). One says thatf has polynomial growth at p ∈ X if it satisfies the following condition. For a local coordinate systemx= (x1, x2)aroundp, there exist a compact neighborhoodKofpandM ∈Z>0such that

(1.1) sup

x∈K∩U

dist(x, KrU)^M f(x)

<+∞.

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We say thatf ∈ C_X^∞

R(U)haspolynomial growth onU if it has polynomial growth at anyp∈X. We say thatf is tempered atpif all its derivatives have polynomial growth atp∈X. We say thatf istempered onU if it is tempered at anyp∈X. Denote by C_X^∞,t

sa the presheaf on X_R of tempered C^∞-functions.

It is obvious thatf has polynomial growth at any point ofU. In [10] it is proved that C_X^∞,t

sa is a sheaf on Xsa. ForU ⊂R² a relatively compact open set, there is a simple characterization of functions with polynomial growth onU.

Proposition 1.9. — LetU ⊂R² be a relatively compact open set and letf ∈ C_R^∞2(U). Then f has polynomial growth if and only if there exist C, M∈R>0such that, for any x∈U,

f(x)

6 C dist(x, ∂U)^M. For Proposition 1.10 below, see [10].

Proposition 1.10. — One has the following isomorphism O^t_X_sa 'RHom%!D_X %_!O_X,C_X^∞,t_sa

.

Hence, for U ∈Op^c(Xsa), we deduce the short exact sequence (1.2) 0−→ O^t_X_sa(U)−→ C_X^∞,t

sa(U)−→ C^∂^¯ ^∞,t_X

sa(U)−→0.

Now, we recall two results on the pull back of tempered holomorphic functions. We refer to [9] for the definition of D_X→Y, for f : X → Y a morphism of complex manifolds. For Lemma 1.11, see [10, Lemma 7.4.7].

Lemma 1.11. — Let f : X → Y be a closed embedding of complex manifolds. There is a natural isomorphism inD^b(%_!D_X)

%!DX→Y

⊗L

%_!f⁻¹DY

f⁻¹O_Y^t ' O_X^t .

For Proposition 1.12, see [17, Theorem 2.1].

Proposition 1.12. — Let f ∈ O_C(X), U ∈ Op^c(Xsa) such that f|_U is an injective map,h∈ OX f(U)

. Then h∈ O_C^t

sa f(U)

if and only if h◦f ∈ O_X^t_sa(U).

We conclude this subsection by recalling the definition of the sheaf of holomorphic functions with moderate growth at the origin. We follow [14].

LetS¹ be the unit circle,S¹×R>0 the real blow-up at the origin ofC^×.

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Forτ ∈R,r∈R>0,0< < π, the set Sτ±,r:=n

%e^iϑ∈C^×; %∈]0, r[, ϑ∈]τ−, τ +[o

is called an open sector centered at τ of amplitude 2 and radius r or simply an open sector. Identifying S¹ with [0,2π[⊂ R, we will consider sectors centered atτ ∈S¹. Further, with an abuse of language, we will say that an open sectorS_τ±,r containsϑ∈Ror e^iϑ ∈S¹ if ϑ∈]τ−, τ+[ mod 2π).

The sheaf on S¹×R>0 of holomorphic functions with moderate growth at the origin, denoted A⁶⁰, is defined as follows. For U an open set of S¹×R>0, set

(1.3) A⁶⁰(U) =n

f ∈ O_C(Ur(S¹× {0}))satisfying the following condition: for any (e^iϑ,0) ∈ U there exist C, M ∈ R>0

and an open sector S ⊂ U containing e^iϑ such that f(z)

< C|z|^−M for any z∈So . Clearly,A⁶⁰is a sheaf onS¹×R>0.

In [18], the author defines the functor ν^sa₀ of specialization at 0 for the sheaves on the subanalytic site. One has that,%⁻¹ν₀^sa(O_C^t_sa)' A⁶⁰.

1.3. D-modules on complex curves and good models

In this subsection we recall some classical results on germs ofDX-modules on a complex analytic curveX. For a detailed and comprehensive presen- tation we refer to [14], [9] and [1].

Given a complex analytic curve X andx0∈X, we denote by OX(∗x0) (resp. DX(∗x0)) the sheaf on X of holomorphic functions on X r{x0} meromorphic at x₀ (resp. the sheaf of rings of differential operators of finite order with coefficients in OX(∗x0)). Further, we denote by O(∗x\0) (resp.D(∗x\₀)) the field of formal Laurent power series (resp. the ring of differential operators with coefficients inO(∗x\0)). The ringO(∗x0)comes equipped with a natural valuationv:O(∗x0)→Z∪ {+∞}.

By the choice of a local coordinateznearx0, we can suppose thatX ⊂C is an open neighborhood ofx₀= 0∈C.

The category of holonomicDX(∗0)-modules, denotedModh(DX(∗0)), is equivalent to the category of local meromorphic connections.

Forϕ∈z⁻¹C[z⁻¹], setL^ϕ:=DX(∗0) exp(ϕ).

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Forl∈Z>0, letµ_l:C→C,z7→z^l. We denote byµ^∗_l the inverse image functor forDX(∗0)-modules.

Theorems 1.13 and 1.15 below are cornerstones in the theory of ordinary differential equations. We refer to [14], [21] and [22].

Theorem 1.13 (Levelt-Turrittin’s Formal Theorem). — Let M ∈ Modh(DX(∗0)). There exist l ∈ Z>0, a finite set Σ⊂ z⁻¹C[z⁻¹], a fam- ily,{Rϕ}ϕ∈Σ, of regular holonomicDX(∗0)-modules indexed byΣand an isomorphism inMod(D\X(∗0))

(1.4) µ^∗_lM ⊗Od_∗0' ⊕

ϕ∈ΣL^ϕ⊗ Rϕ⊗Od_∗0.

In the literature (for example [14]) the definition of the Katz invariant of M ∈Modh(DX(∗0))is given starting from the Newton polygon ofM. For sake of simplicity we give an equivalent definition based on the isomorphism (1.4). Clearly, the valuationv induces a map, still denotedv,z⁻¹C[z⁻¹]→ Z∪ {+∞}.

Definition 1.14.

(i) LetM ∈Modh(DX(∗0)). Suppose that (1.4)is satisfied,l is min- imal and Σ 6= {0}. The Katz invariant of M is max

ϕ∈Σ{^−v(ϕ)_l }. If Σ ={0} then the Katz invariant ofMis0.

For k ∈ Z>0, we denote by Modh(DX(∗0))k the full abelian subcategory of Modh(DX(∗0)) whose objects have Katz invariant strictly smaller thank.

(ii) Let Σ⊂ z⁻¹C[z⁻¹] be a finite set and {Rϕ}_ϕ∈Σ a family of reg- ular holonomicD_X(∗0)-modules. A D_X(∗0)-module isomorphic to

⊕

ϕ∈ΣL^ϕ⊗ Rϕ, is said agood model.

We denote byGMkthe full subcategory ofModh(DX(∗0))kwhose objects are good models.

Roughly speaking, Theorem 1.15 below says that the formal isomorphism (1.4) is analytic on sufficiently small open sectors.

Let

P:=

m

X

j=0

a_j(z) d

dz ^j

,

where m ∈ Z>0, a_j ∈ O_C(X) and a_m 6= 0. Denote by C_C⁰ the sheaf of continuous functions onC. For l ∈Z>0,S an open sector, h∈ {1, . . . , l}, letζ_h:S→Cbe thel different inverse functions toz7→z^l defined onS.

Theorem 1.15(Hukuhara-Turrittin’s Asymptotic Theorem). — There exist a finite setΣ⊂z⁻¹C[z⁻¹],l, rϕ∈Z>0 (ϕ∈Σ) such that for anyτ ∈

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R, there exist an open sector S containingτ,f_ϕkh∈ O_C(S)∩ C_C⁰ Sr{0}

(ϕ∈Σ, k= 1, . . . , rϕ, h= 1, . . . , l), satisfying (i)

f_ϕkh(z) exp(ϕ◦ζ_h(z)); ϕ∈Σ, k= 1, . . . , r_ϕ, h= 1, . . . , l is a basis of the C-vector space of holomorphic solutions of P u = 0 onS,

(ii) there existC, M ∈R>0 such that, for anyz∈S, C|z|^M 6|fϕkh(z)|6 C|z|^M⁻¹ (ϕ∈Σ, k= 1, . . . , rϕ, h= 1, . . . , l).

It is well known that, given M ∈ Modh(DX(∗0)), there exists P ∈ DX(∗0)such that M ' _D^D^X^(∗0)

X(∗0)·P. With a harmless abuse of language, we will speak without distinctions about the solutions ofMand the solutions ofP u= 0.

Definition 1.16. — We use the notations of Theorem 1.15. LetM ∈ Mod_h(D_X(∗0)).

(i) Iffϕkh(z) exp(ϕ◦ζh(z))is a holomorphic solution ofMon an open sector, then we say thatϕ◦ζh is adeterminant polynomialofM.

We denote byΩ(M)the set of all determinant polynomials ofM.

Ifϕ◦ζh∈Ω(M), we say thatrϕ∈Z>0is the rankofϕ◦ζh. (ii) Givenϑ∈S¹, we set

Ω_ϑ:=

ⁿ X

j=1

a_jz⁻^j^l; a_j∈C, l, n∈Z>0

.

We denote byModΩ_ϑ(C)the category of finite dimensional vector spaces graded with respect toΩϑ.

Theorem 1.15 implies that, given ϑ ∈S¹, the holomorphic solutions of Mon a sufficiently small sector containingϑ, can be endowed with aΩϑ- graduation. Hence, there exists a functor

S^Ω(·)ϑ: Modh(DX(∗0))−→ModΩ_ϑ(C).

The Ω_ϑ-graduation on the holomorphic solutions of meromorphic connections described above is the first step to have a local irregular Riemann- Hilbert correspondence in dimension 1. We refer to [1] and [14] for a com- plete description ofΩ-filtered local systems.

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2. Tempered growth of exponential functions

This section is subdivided as follows. In the first part we study the family of sets where a function of the formexp(ϕ), ϕ∈z⁻¹C[z⁻¹], is tempered.

In the second part we use the results of the first part in order to prove that such family determines ϕ up to a multiplicative positive constant.

Throughout this sectionX=C.

2.1. Sets where exponential functions have tempered growth

Forϕ∈z⁻¹C[z⁻¹]r{0}andA∈R>0, set

(2.1) Uϕ,A:=n

z∈C^×; Re ϕ(z)

< Ao , further setU0,A:=C.

First we state and prove the analogue of a result of [11].

Proposition 2.1. — Let ϕ∈z⁻¹C[z⁻¹] and U∈Op^c(Xsa) with U6=∅. The conditions below are equivalent.

(i) exp ϕ

∈ O^t_X_sa(U).

(ii) There existsA∈R>0 such thatU ⊂Uϕ,A. Before proving Proposition 2.1, we need the following

Lemma 2.2 ([11]). — Let W 6= ∅ be an open subanalytic subset of P¹(C),∞∈/W. The following conditions are equivalent.

(i) There existsA∈R^>0 such thatRez < A, for any z∈W.

(ii) The function exp(z) has polynomial growth on any semi-analytic arcΓ⊂W with an endpoint at∞. That is, for any semi-analytic arc Γ⊂W with an endpoint at ∞, there exist M, C ∈R>0 such that, for anyz∈Γ,

(2.2)

exp(z)

6C 1 +|z|²^M . Proof.

Clearly, (i)⇒(ii).

Let us prove (ii)⇒(i). Setz:=x+iy and suppose thatxis not bounded onW. There exist, L∈R>0 and a real analytic map

γ: [0, [−→P¹(C) t7−→(x(t), y(t)),

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such that γ(0) = ∞, γ ]0, [

⊂ W and x ]0, [

=]L,+∞[. Since γ is analytic, there existq∈Q,c∈Randµ∈R>0 such that, for anyt∈]0, [,

γ(t) =

x(t), c x(t)^q+O x(t)^q−µ .

Now, if (2.2) is satisfied, thenexp(x)has polynomial growth in a neighbor-

hood of+∞, which gives a contradiction.

Proof of Proposition 2.1.

Clearly, (ii)⇒(i).

(i)⇒(ii). The result is obvious if ϕ= 0. Otherwise, we distinguish two cases.

Case 1: Supposeϕ(z) =¹_z.

Suppose that for anyA∈R>0, there existszA∈U such thatRe _z¹

A

>

A. Then, by Lemma 2.2, there exists a semi-analytic arc with an endpoint at0,Γ⊂U, such thatexp Re¹_z

has not polynomial growth onΓ. That is, for anyM, C∈R>0, there existszM,C ∈Γsatisfying

exp Re 1 zM,C

> C

|zM,C|^M.

Apply Proposition 1.3 with X = Γ and Y =∂U. There exist an open neighborhoodV of0,c, r∈R>0such that, for any z∈Γ∩V,

|z|6c dist(z, ∂U)^r. Hence,

exp Re 1 zM,C

> c⁻¹C dist(zM,C, ∂U)^rM. It follows thatexp ¹_z

is not tempered onU. Case 2: Supposeϕ(z) =

n

P

j=1 aj

z^j, withn∈Z>0 anda_n6= 0.

Let

η(z) :=

ⁿ X

j=1

aj

z^j ⁻¹

= zⁿ

n

P

j=1

ajz^n−j .

There exists a neighborhoodW ⊂Cof0such thatη∈ O_C(W). It is well known that a non-constant holomorphic function is locally the composition of a holomorphic isomorphism and a positive integer power ofz. Since it is sufficient to prove the result in a neighborhood of 0 and up to finite coverings, we can suppose thatU ⊂W andη|_U is injective.

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Consider the following commutative triangle, U ^η //

exp(ϕ)CCCCCCCCη(U)!!

exp(¹_ζ)

C.

Using Proposition 1.12 and Case 1, we have that exp(ϕ)∈ O^t(U)⇔exp

1 ζ

∈ O^t(η(U))

⇔η(U)⊂U1

ζ,A for some A∈R>0

⇔U ⊂U_ϕ,A for someA∈R>0.

Corollary 2.3. — Let ϕ ∈ z⁻¹C[z⁻¹]. Let S be an open sector of amplitude2π,U ∈Op^c(X_sa),∅6=U ⊂S,ζ:S→C, an inverse ofz7→z^l. Thenexp ϕ◦ζ

∈ O_X^t_sa(U)if and only if there existsA∈R>0such that ζ(U)⊂Uϕ,A.

In particular, setting U_ϕ◦ζ,A:=n

z∈S; Re ϕ◦ζ(z)

< Ao ,

one has thatexp(ϕ◦ζ)∈ O^t_X

sa(U)if and only if there existsA∈R>0such thatU ⊂U_ϕ◦ζ,A.

Proof. — Letµl(z) :=z^l. Consider the following commutative diagram, ζ(U) ^µ^l //

exp(ϕ)DDDDDDDD!! U

exp(ϕ◦ζ)

C.

By Proposition 1.12, we have that exp(ϕ◦ζ)∈ O_X^t_sa(U)if and only if exp(ϕ)∈ O^t_X

sa(ζ(U)). Then the conclusion follows by Proposition 2.1.

Now we are going to introduce a class of subanalytic sets which plays an important role in what follows.

Definition 2.4. — For τ ∈ R we say that U ∈ Op^c(X_sa) is concentrated along τ if U 6= ∅ is connected, 0 ∈ ∂U and, for any open sector S containingτ, there exists an open neighborhood W ⊂Cof0 such that U∩W ⊂S.

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Lemma 2.5 below follows easily from the well known fact that a non- constant holomorphic function is locally the composition of a holomorphic isomorphism and a positive integer power ofz.

Lemma 2.5. — LetW ⊂Cbe an open neighborhood of0,f ∈ O(W).

Suppose that f has a zero of order l ∈ Z>0 at 0. There exists τf ∈ R, depending only on the argument off^(l)(0), satisfying the following condi- tions.

(i) For anyτ ∈R,U ∈Op^c(Xsa)concentrated alongτ, there exists an open neighborhoodW⁰of0,W⁰ ⊂W, such thatf|_U∩W0 is injective andf(U∩W⁰)is concentrated along l(τ+τf).

(ii) For any τ ∈ R, V ∈ Op^c(Xsa) concentrated along τ, there exist an open neighborhoodW⁰ of0,V₀, . . . , V_l−1∈Op^c(X_sa), such that Vj is concentrated along ^τ_l −τf +j^2π_l , andf(Vj) =V ∩W⁰ (j = 0, . . . , l−1).

Proposition 2.6 below will play a fundamental role in the next subsection.

Proposition 2.6. — Let n ∈ Z>0, τ0 ∈ R. There exists τ ∈ R such that, for anyϕ=^%e_z^iτn⁰ +ϕe∈z⁻¹C[z⁻¹](%∈R>0, −v(ϕ)e < n), there exist U0, . . . , U_2n−1∈Op^c(Xsa)satisfying

(i) Uj is concentrated along τ+j^π_n, (ii) exp ϕ

,exp −ϕ

∈ O_X^t_sa(U_j), (j = 0, . . . ,2n−1).

Proof. — The result is obvious ifϕ= 0. Otherwise we distinguish three cases.

Case 1: Supposeϕ(z) =¹_z.

Recall (2.1). For A ∈ R>0, one checks easily that the set U1

z,A (resp.

U₋1

z,A) is the complementary of the closed disc of center _2A¹ ,0 (resp.

−_2A¹ ,0

) and radius _2A¹ . Set

U1:=

(x, y)∈R²; |x|<1, p

|x| −x²< y <1 , U₂:=

(x, y)∈R²; |x|<1, −1< y <−p

|x| −x² .

It is easy to see that U1 (resp. U2) is concentrated along ^π₂ (resp. ^3π₂ ) andU₁∪U₂⊂U1

z,1∩U₋1

z,1. Hence, by Proposition 2.1, (2.3) exp(1/z), exp(−1/z)∈ O^t_X_sa(Uj) (j= 1,2).

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Case 2: Suppose that ϕ(z) = _z¹_m, for m ∈ Z>0. Let µ_m : C → C, µm(z) =z^m. Consider the commutative triangle

C^×

exp (1/zDD^mD)DDDDD!!

µ_m //

C^×

exp(1/z)

C.

Consider U₁, U₂ as in Case 1. Applying Lemma 2.5 (ii) with f = µ_m, we obtain that there exist V1,0, . . . , V_1,m−1 ∈ Op^c(Xsa) (resp. V2,0, . . . , V_2,m−1∈Op^c(X_sa)) such that

(i) V_1,j (resp.V_2,j) is concentrated along _2m^π +j^2π_m (resp. _2m^3π +j^2π_m), (ii) µm(Vk,j) =Uk ,

(j = 0, . . . , m−1, k= 1,2).

Clearly,µ_m|_V

k,j is injective. By Proposition 1.12, we have that exp 1/z^m

∈ O^t_X

sa(V_k,j)

resp. exp −1/z^m

∈ O^t_X

sa(V_k,j) if and only if

exp 1/z

∈ O_X^t_sa µm(Vk,j)

=O_X^t_sa(Uk) resp. exp −1/z

∈ O^t_X_sa µ_m(V_k,j)

=O^t_X_sa(U_k) (j = 0, . . . , m−1, k= 1,2). The conclusion follows.

Case 3: Suppose that ϕ(z) =

n

X

j=1

aj

z^j ∈z⁻¹C[z⁻¹], forn∈Z>0,a_j∈C(j= 1, . . . , n) anda_n6= 0.

First, we recall the implicit function theorem for convergent power series.

We denote byC{x} (resp.C{x, y}) the ring of convergent power series in x(resp.x, y). We refer to [4, Theorem 8.6.1, p. 166] for the proof.

Theorem 2.7. — Let F ∈ C{x, y} be such that F(0,0) = 0. There existsη(x)∈ ∪_l∈Z_>0x^1/lC{x^1/l} such thatF(x, η(x)) = 0.

Consider

F(z, η) :=−ηⁿ+zⁿ

n

X

j=1

ajη^n−j

=−ηⁿ+zⁿa1ηⁿ⁻¹+· · ·+zⁿa_n−1η+zⁿan. (2.4)

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By Theorem 2.7, there exist l ∈ Z>0, η(z) ∈ z^1/lC{z^1/l} such that F z, η(z)

= 0. Since an 6= 0, we have that η(z) 6= 0, for z 6= 0. It follows thatη(z)∈z^1/lC{z^1/l} satisfies

(2.5) ϕ η(z)

=

n

X

j=1

aj

η(z)^j = 1 zⁿ.

Further, substituting η(z) in (2.4), one checks that l = 1 and η(z) = zσ(z), for σ an invertible element of C{z} such thatarg σ(0)

= ^arg(a_nⁿ⁾. In particular, there exists an open neighborhoodW ⊂Cof the origin such thatη∈ O_C(W).

Now, by Case 2, there exist V_k,j ⊂W (j = 0, . . . , n−1, k = 1,2) such thatV1,j (resp.V2,j) is concentrated along _2n^π +j^2π_n (resp. ^3π_2n +j^2π_n ) and

(2.6) exp 1

zⁿ

,exp

− 1 zⁿ

∈ O_X^t

sa(V_k,j) (j = 0, . . . , n−1, k= 1,2).

As η has a zero of order1 at 0, by Lemma 2.5 (i), there exists τη ∈R, depending only onarg η(0)

= ^arg(a_nⁿ⁾, such that, up to shrinkingW, (i) η|_V

k,j is injective and (ii) η V1,j

(resp.η V2,j

) is concentrated along τη +_2n^π +j^2π_n (resp.

τ_η+^3π_2n+j^2π_n), (j = 0, . . . , n−1).

Consider the commutative triangle W r{0} ^η //

exp _zn¹

$$I

II II II

II C^×

exp(ϕ(z))

C. By Proposition 1.12, we have that

exp ϕ(z)

∈ O_X^t_sa η(V_k,j)

resp. exp −ϕ(z)

∈ O^t_X_sa η(Vk,j) if and only if

exp ϕ◦η(z)

= exp 1/zⁿ

∈ O_X^t_sa V_k,j

resp. exp −ϕ◦η(z)

= exp −1/zⁿ

∈ O^t_X_sa Vk,j

(j = 0, . . . , n−1, k= 1,2). The conclusion follows from (2.6).

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Remark 2.8. — Recall the definition given in the end of Subsection 1.2 of the sheafA⁶⁰defined onS¹×R>0, considered as the real blow-up at0of C^×. Letτ∈R,U ∈Op^c(X_sa)concentrated alongτ, the set{(τ,0)} ∪U ⊂ S¹×R>0is not open. Further ifexp(ϕ)∈ A⁶_(τ,0)⁰ thenexp(−ϕ)∈ A/ ⁶_(τ,0)⁰ .

We conclude this subsection with an easy lemma which will be useful in the next subsection. First, let us introduce some notation.

Given ϕ∈ z⁻¹C[z⁻¹], ϕ= ^ηe_z_n^iτ + ˜ϕ, for η ∈R>0, n∈ Z>0, τ ∈ R and

˜

ϕ∈z⁻¹C[z⁻¹],−v( ˜ϕ)< n, set Iϕ:=n

ϑ∈[0,2π]; cos(τ−nϑ)<0o .

In other words,Iϕis the support of exp(ϕ)as a section ofA⁶⁰ _S₁_×{0}. Recall the definition of the sets Uϕ,A given in (2.1).

Lemma 2.9.

(i) Let ϕ1, ϕ2 ∈ z⁻¹C[z⁻¹], ϕj := ^η^j_z^enj^iτj + ˜ϕj, for ηj ∈ R>0, nj ∈ Z>0, τ_j∈Randϕ˜_j ∈z⁻¹C[z⁻¹],−v( ˜ϕ_j)< n_j.

Ifn16=n2 orτ16=τ2, thenIϕ₁rIϕ₂6=∅andIϕ₂rIϕ₁6=∅. (ii) Letϑ0 ∈[0,2π[ and ϕ∈ z⁻¹C[z⁻¹]r{0}. If ϑ0 ∈Iϕ, then there

exists an open sector S containing ϑsuch that, for any A ∈R>0, S⊂Uϕ,A. In particular, for anyU ∈Op^c(Xsa)concentrated along ϑ₀,exp(ϕ)∈ O^t_X_sa(U).

(iii) Letϑ0 ∈ [0,2π[ andϕ ∈z⁻¹C[z⁻¹]r{0}. If ϑ0 ∈/ Iϕ, then there exists an open sector S containing ϑsuch that, for any A ∈R^>0, S ⊂X rUϕ,A. In particular, for any U ∈Op^c(Xsa)concentrated alongϑ0,exp(ϕ)∈ O/ ^t_X

sa(U).

Proof. — The result follows from some easy computations.

2.2. Comparison between growth of exponential functions In this subsection we are going to use the results of the previous subsection in order to prove that ifϕ1, ϕ2∈z⁻¹C[z⁻¹]and, for anyλ∈R>0, ϕ₁6=λϕ₂, then the families{Uϕ1,A}A∈R>0 and{Uϕ2,A}A∈R>0 are not cofi- nal.

The main result of this subsection is Proposition 2.10 below.

Proposition 2.10. — Letϕ1, ϕ2∈z⁻¹C[z⁻¹]r{0}.

(i) Suppose that there existsλ∈R>0 such thatϕ₁ =λϕ₂. Then, for anyA∈R>0,Uϕ₁,A=U_ϕ₂_,A

λ. In particular, for anyU ∈Op^c(Xsa), exp(ϕ1)∈ O^t_X_sa(U)if and only if exp(ϕ2)∈ O^t_X_sa(U).

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(ii) Suppose that, for any λ ∈ R>0, ϕ₁ 6= λϕ₂. Then for any open sectorS of amplitude>_max{−v(ϕ^2π

1),−v(ϕ2),2} at least one of the two conditions below is satisfied (resp. for any open neighborhoodS of 0, both conditions below are satisfied).

(a) There exists U ∈ Op^c(Xsa), U ⊂ S such that exp(ϕ1) ∈ O_X^t_sa(U)andexp(ϕ₂)∈ O/ ^t_X_sa(U).

(b) There exists V ∈ Op^c(Xsa), V ⊂ S such that exp(ϕ1) ∈/ O_X^t

sa(V)andexp(ϕ₂)∈ O_X^t

sa(V).

Proof.

(i) Obvious.

(ii) ForS an open sector, setS˜:=

ϑ∈[0,2π[;∃ r >0re^iϑ ∈S . Let

ϕ₁(z) :=η₁e^iτ¹

zⁿ¹ + ˜ϕ₁(z) and ϕ₂(z) := η₂e^iτ²

zⁿ² + ˜ϕ₂(z), forη_j∈R>0,τ_j∈[0,2π[andϕ˜_j(z)∈z⁻¹C[z⁻¹], −v( ˜ϕ_j)< n_j (j= 1,2).

Suppose thatn16=n2 orτ16=τ2.

By Lemma 2.9 (i),I_ϕ₁rI_ϕ₂ 6=∅andI_ϕ₂rI_ϕ₁ 6=∅.

LetS be an open neighborhood of0,ϑ1∈Iϕ₁rIϕ₂ andϑ2∈Iϕ₂rIϕ₁. There existU_k ∈Op^c(X_sa)concentrated alongϑ_k such that U_k ⊂S (k= 1,2). The result follows by Lemma 2.9 (ii),(iii).

Suppose that S is an open sector of amplitude > _max{n^2π

1,n₂,2}. Then, there existsϑ∈S˜ such that either ϑ∈Iϕ₁ rIϕ₂ or ϑ∈Iϕ₂ rIϕ₁. Since ϑ∈S, there exists˜ U ∈Op^c(Xsa)concentrated alongϑsuch that U ⊂S.

The conclusion follows by Lemma 2.9 (ii),(iii).

Now suppose thatn₁=n₂=nandτ₁=τ₂. That is, ϕ₁(z) =η₁e^iτ¹

zⁿ + ˜ϕ₁(z) and ϕ₂(z) = η₂e^iτ¹

zⁿ + ˜ϕ₂(z).

Since, for any λ∈R>0,ϕ16=λϕ2, we have thatn>2.

Set ψ21:=ϕ2− ^η_η²

1ϕ1 and ψ12 :=ϕ1−^η_η¹

2ϕ2. Since ψ21 6= 0 and ψ21 =

−^η_η²

1 ϕ₁−^η_η¹

2ϕ₂

=−^η_η²

1ψ₁₂, thenI_ψ₂₁ =I_−ψ₁₂.

By Proposition 2.6, there existτ∈RandU0, . . . , U_2n−1, V0, . . . , V_2n−1∈ Op^c(X_sa)satisfying the conditions

(i) U_j, V_j are concentrated alongτ+j^π_n, (ii) exp(ϕ1),exp(−ϕ1)∈ O^t_X

sa(Uj)andexp(ϕ2),exp(−ϕ2)∈ O_X^t

sa(Vj), (j = 0, . . . ,2n−1).

Since −v(ψ₁₂) = −v(ψ₂₁) < n, if S is an open sector of amplitude

> ^2π_n , there exists j⁰ ∈ {0, . . . ,2n−1} such that τ+j⁰^π_n ∈ S˜ and either τ+j⁰^π_n ∈/Iψ₁₂ orτ+j⁰^π_n ∈/Iψ₂₁.

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More generically, {τ+j^π_n; j ∈0, . . . ,2n−1} 6⊂I_ψ₁₂ and{τ+j^π_n; j ∈ 0, . . . ,2n−1} 6⊂I_ψ₂₁.

Let us consider the case τ+j⁰^π_n ∈/ Iψ₁₂. Since Vj⁰ is concentrated along τ+j⁰^π_n, Lemma 2.9 (iii) implies

(2.7) exp(ψ12)∈ O/ ^t_X_sa(Vj⁰).

Suppose now thatexp(ϕ1)∈O^t_X_sa(Vj⁰). Sinceexp(−ϕ2),exp(ϕ2)∈O^t_X_sa(Vj⁰) and the product of tempered functions is tempered, we have thatexp(ϕ1−

η₁

η2ϕ₂)∈ O^t_X_sa(V_j0), which contradicts (2.7). Henceexp(ϕ₁)∈ O/ ^t_X_sa(V_j0).

Let us consider the case τ+j⁰^π_n ∈/ I_ψ₂₁. Since U_j⁰ is concentrated along τ+j⁰^π_n, Lemma 2.9 (iii) implies

(2.8) exp(ψ21)∈ O/ _X^t_sa(Uj⁰).

Suppose now thatexp(ϕ₂)∈O_X^t

sa(U_j⁰). Sinceexp(ϕ₁),exp(−ϕ₁)∈O^t_X

sa(U_j⁰) and the product of tempered functions is tempered, we have thatexp(ϕ2−

η2

η₁ϕ₁)∈ O_X^t

sa(U_j⁰), which contradicts (2.8). Hence exp(ϕ₂)∈ O/ _X^t

sa(U_j⁰).

Corollary 2.11. — Let l ∈ Z>0 ω, ϕ1, ϕ2 ∈ z⁻¹C[z⁻¹], such that

−v(ω) > max

j=1,2{^−v(ϕ_l ^j⁾+ 1}. Let S an open sector of amplitude 2π, ζ an inverse ofz7→z^ldefined onS. The following conditions are equivalent.

(i) ϕ1◦ζ6=ϕ2◦ζ.

(ii) At least one of the following two conditions is verified:

(a) there existsU ∈Op^c(X_sa),U ⊂S such thatexp(ϕ₁◦ζ+ω)∈ O_X^t

sa(U)andexp(ϕ2◦ζ+ω)∈ O/ ^t_X

sa(U);

(b) there existsV ∈Op^c(X_sa),V ⊂Ssuch thatexp(ϕ₁◦ζ+ω)∈/ O_X^t

sa(V)andexp(ϕ2◦ζ+ω)∈ O^t_X

sa(V).

Proof.

(ii)⇒(i). Obvious.

(i)⇒(ii). Setµl(z) :=z^l.

Suppose now that ϕ₁ ◦ζ 6= ϕ₂ ◦ζ. It follows that, for any λ ∈ R>0, λ(ϕ1+ω◦µl) 6=ϕ2+ω◦µl. Consider the sector ζ(S) of amplitude ^2π_l . Since −v(ω) > 2, then ^2π_l > −_lv(ω)^2π . Hence by Proposition 2.10 there exists ζ(U) ⊂ ζ(S)such that either exp(ϕ₁+ω◦µ_l) ∈ O^t_X_sa(ζ(U)) and exp(ϕ2+ω◦µl)∈ O/ ^t_X

sa(ζ(U))or viceversa. By Proposition 1.12, it follows that eitherexp(ϕ₁◦ζ+ω)∈ O_X^t

sa(U)andexp(ϕ₂◦ζ+ω)∈ O/ ^t_X

sa(U)or

viceversa.

Remark 2.12. — There is another way to prove Proposition 2.10. Let us briefly summarize it.