Temperate holomorphic solutions and regularity of holonomic D-modules on curves

Temperate Holomorphic Solutions and Regularity of Holonomic D-modules on Curves.

ANARITAMARTINS(*)

ABSTRACT- In [8], Kashiwara and Schapira introduced the notion of regularity for ind-sheaves and conjectured that a holonomicD-module on a complex manifold is regular if and only if its complex of temperate holomorphic solutions is regular.

Our aim is to prove this conjecture in the one-dimensional case.

Introduction.

In [8], the authors introduced the notions of microsupport and reg- ularity for ind-sheaves. Let Xbe a complex manifold,Ma coherentDX- module and consider its complex of temperate holomorphic solutions

Sol^t(M):ˆRIhom_bXDX(b_XM;O^t_X):

It is proved in [8] that the microsupport of Sol^t(M) coincides with the characteristic variety of M. Moreover, if M is regular holonomic, then Sol^t(M) is regular. In fact, Kashiwara and Schapira made the following conjecture:

(K-S)-cONJECTURE. Let M be a holonomic DX-module. Then M is regular holonomic if and only if RIhomb_XDX(b_XM;O^t_X)is regular.

In this paper, we prove, in dimension one, that the regularity ofSol^t(M) implies the regularity of the holonomicD_X-moduleM. More precisely, we show thatSol^t(M) is irregular when the holonomicD_X-moduleMhas an irregular singularity. This proof relies in several steps. First we reduce to the caseM ˆ D^m_X=D^m_XP, wherePis a matrix of differential operators of the

(*) Indirizzo dell'A.: Centro de AÂlgebra da Universidade de Lisboa, Complexo 2, Avenida Prof. Gama Pinto,1699 Lisboa Portugal.

E-mail: arita@mat.fc.ul.pt

formz^N@zIm‡A(z);withm;N2N,Imthe identity matrix of ordermand A a mm matrix of holomorphic functions on a neighborhood of the origin. Then we show that it is enough to prove the irregularity of S^t:ˆH⁰(Sol^t(M)) and we give a characterization ofS^t in a sector. From this characterization we easily conclude a contradiction by assuming the regularity ofS^tat (0;0), which completes the desired proof.

The contents of this paper are two sections as follows.

In Section 1, we start with a quick review on sheaves, ind-sheaves, microsupport and regularity for ind-sheaves and we recall the results on the microsupport and regularity ofSol^t(M), proved in [8].

Section 2 is dedicated to the proof of the irregularity of Sol^t(M), whenMis an irregular holonomicDX-module on an open neighborhood Xof 0 in C.

Acknowledgments. We thank P. Schapira who suggested us to solve the (K-S)-conjecture in dimension one, using the tools of [12], and made useful comments during the preparation of this manuscript. Finally, we thank T. Monteiro Fernandes for many useful advises.

1. Notations and review.

We will follow the notations in [8].

SHEAVES. LetXbe a real analyticn-dimensional manifold. We denote byp:TX!Xthe cotangent bundle toX. We identifyX with the zero section ofTXand we denote byT_Xthe setTXnX.

Letkbe a field. We denote by Mod(k_X) the abelian category of sheaves ofk-vector spaces onXand byD^b(kX) its bounded derived category. For a;b2R,a<b(resp.k2Z), we denote byD^[a;b](kX) (resp.D^k(kX)) the full additive subcategory of D^b(k_X) consisting of objects F satisfying H^j(F)ˆ0, for anyj2=[a;b] (resp.j<k).

We denote byR C(k_X) the abelian category ofR-constructible sheaves of k-vector spaces onX and byD^b_{R C}(kX) the full subcategory ofD^b(kX) consisting of objects withR-constructible cohomology.

For an objectF2D^b(k_X), we denote bySS(F) themicrosupport of F, a closedR^‡-conic involutive subset ofTX. We refer to [9] for details.

IND-SHEAVES ON REAL MANIFOLDS. LetXbe a real analytic manifold. We denote by I(kX) the abelian category of ind-sheaves onX, that is, the ca-

tegory of ind-objects of the category Mod^c(kX) of sheaves with compact support onX(see [7]).

Recall the natural faithful exact functor i_X:Mod(k_X)!I(k_X);F7!``lim_!

UXUopen

''F_U:

We usually don't write this functor and identify Mod(k_X) with a full abelian subcategory of I(k_X) and D^b(k_X) with a full triangulated subcategory of D^b(I(kX)).

The category I(k_X) admits an internal hom denoted byIhomand this functor admits a left adjoint, denoted by . If F'``lim_!

i

''F_i and G'``lim_!

j

''G_j, then:

Ihom(G;F)'lim

j

``lim_!

i

''Hom(G_j;F_i);

GF'``lim_!

j

''``lim_!

i

''(GjFi):

The functori_X admits a left adjoint a_X :I(k_X)!Mod(k_X);Fˆ``lim_!

i

''F_i7!lim_!

i

F_i:

This last functor also admits a left adjoint b_X :Mod(k_X)!I(k_X). Both functorsaXandb_X are exact. We refer to [7] for the description ofb_X.

Let X be a real analytic manifold. We denote by R C^c(kX) the full abelian subcategory of R C(k_X) consisting of R-constructible sheaves with compact support. We denote by IR c(k_X) the category Ind(R C^c(k_X)) and by D^b_{IR c}(I(k_X)) the full subcategory of D^b(I(k_X)) consisting of objects with cohomology in IR c(k_X).

THEOREM1.1 ([7]). The natural functor D^b(IR c(kX))!D^b_{IR c}(I(kX)) is an equivalence of categories.

Recall that there is an alternative construction of IR c(k_X), using Gro- thendieck topologies. Denote by Op_Xsa the category of open subanalytic subsets ofX. We may endow this category with a Grothendieck topology by deciding that a familyfU_ig_iin Op_Xsais a covering ofU2Op_Xsaif for any compact subsetKofX, there exists a finite subfamily which coversU\K.

One denotes byX_sathe site defined by this topology and by Mod(k_Xsa) the category of sheaves onXsa(see [1] and [7]). We denote by Op^c_Xsa the sub- category of Op_Xsaconsisting of relatively compact open subanalytic subsets

ofXand forU2Op_Xsa we denote byUXsa the category Op_Xsa\Uwith the topology induced byXsa.

Letr:X!Xsabe the natural morphism of sites. We have functors Mod(k_X)!^r

r ¹Mod(k_Xsa);

and we still denote byrthe restriction ofrtoR C(k_X) and toR C^c(k_X).

We may extend the functor r:R C^c(k_X)!Mod(k_Xsa) to IR c(k_X), by setting:

l: IR c(kX) ! Mod(kXsa)

``lim_!

i

''Fi 7! lim_!

i

rFi:

ForF2IR c(kX), an alternative definition ofl(F) is given by the formula l(F)(U)ˆHomIR c(kX)(kU;F):

THEOREM1.2 ([7]). The functorl is an equivalence of abelian cate- gories.

Most of the time, thanks tol, we identify IR c(kX) with Mod(kXsa).

TEMPERED DISTRIBUTIONS. LetXbe a real analytic manifold. Denote by Db_X the sheaf of distributions on X. For each open subanalytic subset UX, we denote by Db^t_X(U) the space of tempered distributions onU, defined by the exact sequence

0!GXnU(X;DbX)!G(X;DbX)! Db^t_X(U)!0:

It is proved in [7] thatU7! Db^t_X(U) is a sheaf on the subanalytic site Xsa, hence defines an ind-sheaf. We callDb^t_X the ind-sheaf of tempered distributions. This ind-sheaf is well-defined in the category Mod(b_XDX), whereD_X denotes the sheaf of analytic finite-order differential operators.

TEMPERED HOLOMORPHIC FUNCTIONS. Let X be a complex analytic manifold. One defines the ind-sheaf of tempered holomorphic functions as:

O^t_X :ˆRIhombD_X(bO_X;Db^t_XR);

whereXdenotes the complex conjugate manifold,X_Rthe underlying real analytic manifold, identified with the diagonal ofXX, andD_Xthe sheaf of rings of holomorphic differential operators of finite order overX.O^t_X is

actually an object ofD^b(b_XDX) and it is not concentrated in degree 0 if dim X>1. WhenXis a complex analytic curve,O^t_Xis concentrated in degree 0.

Moreover,O_X isr-acyclic andO^t_X is a sub-ind-sheaf ofrO_X.

We end this section by recalling two results of G. Morando, which will be useful in our proof.

THEOREM1.3 ([2]). Let X be an open subset ofCand f 2 OC(X). Let U2Op^c_Xsa such that fj_U is an injective map. Let h2 OC(f(U)). Then hf 2 O^t_X(U)if and only if h2 O^t_C(f(U)).

PROPOSITION1.4 ([2]). Let p2z ¹C[z ¹]andU2Op^c_Csa with02@U.

The conditions below are equivalent.

(i) exp (p(z))2 O^t_C(U).

(ii) There exists A>0such thatRe(p(z))<A, for all z2U.

MICROSUPPORT AND REGULARITY FOR IND-SHEAVES. We refer to [8] for the equivalent definitions for the microsupport SS(F) of an object F2D^b(I(kX)). We shall only recall the following useful properties of this closed conic subset of TX.

PROPOSITION 1.5. (i) For F 2D^b(I(kX)), one has SS(F)\T_XXˆ

ˆsupp(F):

(ii) Let F2D^b(kX). Then SS(iXF)ˆSS(F):

(iii) Let F2D^b(I(k_X)). Then SS(a_X(F))SS(F).

(iii) Let F₁!F₂!F₃ ^‡1!be a distinguished triangle in D^b(I(k_X)).

Then SS(Fi)SS(Fj)[SS(Fk), forfi;j;kg ˆ f1;2;3g.

Let J denotes the functor J:D^b(I(k_X))!(D^b(Mod^c(k_X)))^{^} (where (D^b(Mod^c(k_X)))^{^}denotes the category of functors fromD^b(Mod^c(k_X))^opto Set) defined by:

J(F)(G)ˆHom_D^b_(I(kX))(G;F);

for everyF2D^b(I(k_X)) andG2D^b(Mod^c(k_X)).

DEFINITION1.6 ([8]). LetF2D^b(I(k_X)),LTXbealocallyclosedconic subset andp2TX. We say thatF is regular alongLat pif there exists F⁰isomorphic toFin a neighborhood ofp(p), an open neighborhoodUofp with L\U closed in U, a small and filtrant category I and a functor I!D^[a;b](k_X);i7!F_i such that J(F⁰)'``lim_!

i2I

''J(F_i) and SS(F_i)\UL.

Otherwise, we say thatF is irregular alongLatp.

We say thatFis regular at pifFis regular alongSS(F) atp. If Fis regular at eachp2SS(F), we say thatFis regular.

PROPOSITION1.7 ([8]). (i)LetF 2D^b(I(k_X)). ThenF is regular along any locally closed setLat eachp =2SS(F).

(ii)Let F1!F2!F3 ^‡1!be a distinguished triangle in D^b(I(kX)). If F_j and F_k are regular alongL, so is F_i, for i;j;k2 f1;2;3g, j6ˆk.

(iii)Let F2D^b(k_X). Theni_XF is regular.

TEMPERATE HOLOMORPHIC SOLUTIONS OFD-MODULES. LetXbe a com- plex manifold and letMbe a coherentDX-module. Set

Sol(M)ˆRrRHom_DX(M;O_X);

Sol^t(M)ˆRIhomb_XDX(b_XM;O^t_X):

The equality:

SS(Sol^t(M))ˆChar(M);

(1:1)

was obtained by M. Kashiwara and P. Schapira in [8], where these authors also proved that the natural morphism Sol^t(M)!Sol(M) is an iso- morphism, whenMis a regular holonomicD_X-module. This gives the ``only if'' part of the (K-S)-Conjecture.

2. Proof of the (K-S)-Conjecture in dimension one.

In this section, we considerCendowed with the holomorphic coordinate zandXwill denote an open neighborhood of 0 inC. We shall prove that, for every irregular holonomic DX-module M, Sol^t(M) is irregular, using a similar argument as in the Example of [8].

We shall first reduce the proof to the case where M ˆ D_X=D_XP, for someP2 D_X.

Let M be an irregular holonomic DX-module and let us denote by Char(M) its characteristic variety. SinceMis holonomic it is locally gene- rated by one element and we may assumeMis of the formD_X=I, for some coherent left ideal I of D_X. We may also assume that, locally at 02C, Char(M)T_XX[T_f0g X. Moreover, we may find P2 I such that the kernel of the surjective morphism

D_X=D_XP! M !0;

is isomorphic to a regular holonomic DX-module N (see, for example, Chapter VI of [10]). Therefore, we have an exact sequence

0! N ! D_X=D_XP! M !0;

and we get the distinguished triangle

Sol^t(M)!Sol^t(D_X=D_XP)!Sol^t(N) ^‡1!:

SinceSol^t(N) is regular, by Proposition 1.7,Sol^t(M) will be regular if and only ifSol^t(D_X=D_XP) is. Therefore, we may assume from the beginning that M ˆ DX=DXP, for some P2 DX, having an irregular singularity at the origin.

Let us now recall the following result, due to G. Morando:

THEOREM 2.1 ([2]). LetMbe a holonomic D_X-module. The natural morphism

H¹(Sol^t(M))!H¹(Sol(M));

is an isomorphism.

The Theorem above together with the results in [4] entails that:

H¹(Sol^t(M))'H¹(Sol(M))'C^m_f0g;

for some m2N. Then H¹(Sol^t(M)) is regular and SS(H¹(Sol^t(M)))ˆ

ˆT_f0gX.

As in [8], let us set for short

S^t:ˆH⁰(Sol^t(M))' Ihom_bXDX(b_XM;O^t_X);

S:ˆH⁰(Sol(M))'rHom_DX(M;O_X)'ker(rO_X !^P rO_X):

Remark that, since dimXˆ1, one has a monomorphismS^t! S. Moreover, we have the following distinguished triangle:

S^t!Sol^t(M)!H¹(Sol^t(M))[ 1] ^‡1!: Therefore, one has

SS(S^t)Char(M)[T_f0g XT_XX[T_f0g X;

andS^twill be irregular if and only ifSol^t(M) is.

The problem is then reduced to prove the irregularity ofS^t, for an ir- regular holonomic DX-module of the form M ˆ DX=DXP, with P2 DX.

Moreover, we may assume P is of the form Pˆz^N@_z^m‡^mP¹

kˆ0a_k(z)@_z^k, for someN;m2N.

Let U be an open neighborhood of the origin in C. The problem of finding the solutions of the differential equation Puˆ0 in O_X(U) is equivalent to the one of finding the solutions in OX(U)^m of a system of ordinary differential equations defined by a matrix of differential opera- tors of the form

z^N@zIm‡A(z);

wherem;N2N,Imis the identity matrix of ordermandA2Mm(O_X(U))(¹).

From now on we denote byPthe system Pˆz^N@zIm‡A(z);

(2:1)

and we reduce to the case whereM ˆ D^m_X=D^m_XP, so that S 'rHomDX(D^m_X=D^m_XP;OX);

and

S^t' Ihomb_XDX(b_X(D^m_X=D^m_XP);O^t_X):

Letu₀;u₁;R2R, withu₀<u₁andR>0. We denote the open set fz2C;u₀<argz<u₁;0<jzj<Rg;

byS(u0;u1;R) and call itopen sector of amplitudeu1 u0and radius R.

Let l2N. By choosing a branch, we consider z^1=l as a holomorphic function on subsets of open sectors of amplitude smaller than 2p.

The next goal is to calculate the ind-sheaf S^t. As an essential step we recall the following classical result that gives the characterization of the holomorphic solutions of the matrix of differential operatorsz^N@zIm‡A(z) in some open sectors.

THEOREM2.2 (see [12]). Let us denote by P the matrix of differential operators z^N@_zI_m‡A(z). There exist l2N, a diagonal matrix L(z)2Mm(z ^1=lC[z ^1=l]) and, for any real number u, there exist R>0, u1>u>u0 and Fu 2GLm(OX(S(u0;u1;R))\C⁰(S(u0;u1;R)nf0g)), such that the m-columns of the matrix F_u(z) exp ( L(z)) are C-linearly in- dependent holomorphic solutions of the system Puˆ0. Moreover, for each

(¹) For a commutative ringRwe denote by Mm…R†the ring ofmmmatrices and by GLm…R†the group of invertiblemmmatrices.

uthere exist constants C;M>0so that Fuhas the estimate C ¹jzj^M<jF_u(z)j<Cjzj ^M; for any z2 S(u ₀;u ₁;R):

(2:2)

If there is no risk of confusion we shall writeF(z) instead ofF_u. DEFINITION2.3. We call the matrixF(z)exp( L(z)), given in Theorem 2.2, afundamental solution of P on S(u0;u1;R).

Let us point out that Theorem 2.2 gives a characterization of the ho- lomorphic solutions of the systems of differential operators of the form z^N@zIm‡A(z), not necessarily irregular. However, it follows by Theorem 5.1 of [5] that the matrix Lgiven by Theorem 2.2 will be non-zero if and only ifPis irregular.

Letl2NandL(z) be the diagonal matrix given in Theorem 2.2 for the operator (2.1). For each 1jm, letLj(z)ˆP^nj

kˆ1a^j_kz ^k=lbe the (j;j) entry ofL(z), withn_j 2N,a^j₁; :::;a^j_nj 2C.

COROLLARY 2.4. Let V 2Op^c_Xsa and let us suppose P has a funda- mental solution F(z) exp ( L(z)) on V. Then, G(V;S^t)'C^n(V), where n(V)is the cardinality of the set:

J(V):ˆ fj2 f1; :::; mg;exp ( L_j(z))j_V 2 O^t_X(V)g:

PROOF. By hypothesis, G(V;S) is the m-dimensionalC-vector space generated by them-columns of the matrixF(z) exp ( L(z)). Letkbe the dimension of theC-vector spaceG(V;S^t). Clearlyn(V)k. Let us prove thatkn(V).

Let G₁; :::;G_k be a C-basis of G(V;S^t). Clearly, forhˆ1; :::;k, there existsCh2C^msuch thatGhˆF(z) exp ( L(z))Ch. In particular, thej-th coordinate ofF ¹G_his a complex multiple of exp ( L_j). Further, sinceF ¹ is a matrix of tempered holomorphic functions, F ¹G₁; :::;F ¹G_k are C- linearly independent vectors in O^t_X(V)^m. It follows that there exists fj₁; :::;j_kg f1; :::;mg such that exp ( L_j1(z)); :::;exp ( L_jk(z))2 O^t_X(V).

The conclusion follows. q.e.d.

LEMMA 2.5. Let S be an open sector of amplitude smaller than 2p, p2z ¹C[z ¹], l2N and V 2Op^c_Xsa, with V S and 02@V. Then exp (p(z^1=l))2 O^t_X(V) if and only if there exists A>0 such that Re(p(z^1=l))<A, for all z2V.

PROOF. Let u0;u1;R2R such that 0<u1 u0<2p and SˆS(u₀;u₁;R), and let us denote byUthe open sectorSu₀

l ;u₁ l ;R^1=l

. Let f :X!X be the holomorphic function defined by f(z)ˆz^l. Since u1 u0<2p, we may easily check thatfj_U is an injective map. Moreover, f(U)ˆS and fj_U :U!S is bijective. Set V⁰ˆf ¹(V)\U and let h denotes the holomorphic function defined for each z2S by h(z)ˆexp (p(z^1=l)). By Theorem 1.3, we havehf 2 O^t_X(V⁰) if and only if h2 O^t_X(V). On the other hand, one has exp (p)j_V0ˆhfj_V0and, by Propo- sition 1.4, hf 2 O^t_X(V⁰) if and only if there exists A>0 such that Re(p(z))<A, for allz2V⁰. Combining these two facts, we conclude that exp (p(z^1=l))2 O^t_X(V) if and only if there exists A>0 such that Re(p(z^1=l))<A, for allz2V, as desired. q.e.d.

PROPOSITION2.6. With the notation above, there exist an open sector

S, with amplitude smaller than2pand radius R >0, and a non-empty

subsetIoff1; :::; mgsuch that, for eachj2Iand each open subanalytic subsetV S, the conditions below are equivalent:

(i) there exists A>0such thatRe( L_j(z))<A for all z2V, (ii) there exists0<d<R such that V fz2S;jzj>dg.

Moreover, for each j2 f1; :::;mgnI, there exists A>0 such that, for every z2S,Re( L_j(z))<A.

PROOF. For eachjˆ1; :::;m, ifzˆrexp (iu), one has:

Re( L_j(z))ˆX^nj

kˆ1

a^j_kr ^k=lcos (f^j_k k=lu);

wherea^j_kˆ ja^j_kjandf^j_k ˆarg( a^j_k), for everykˆ1; :::;n_j.

SinceL6ˆ0, we may assume from the beginning thata¹_n1 6ˆ0. For each jˆ1; :::;m and u2R, set c_j(u)ˆcos (f^j_nj n_j=lu). Pick u⁰2R such that c_j(u⁰)6ˆ0, for jˆ1; :::;m, and c₁(u⁰)>0. By continuity, these conditions hold in a neighborhood [u0;u1] ofu⁰. Moreover, we may assume that one has 0<u1 u0<2p.

Let us set:

J:ˆ fj2 f1; :::;mg;c_j(u)<0; 8u2[u₀;u₁]g [ fj2 f1; :::;mg;L_jˆ0g:

Let j2J, withLj(z)6ˆ0, and chooseCj>0 such thatcj(u) Cj, for all u2[u0;u1]. We may assumea^jnj 6ˆ0. Then, for eachu2[u0;u1] andr>0,

one has:

Re( L_j(rexp (iu)))ˆ

ˆr ^{nj=l n}X^{j 1}

kˆ1

a^j_kr^{(nj k)=l}cos (f^j_k k=lu)‡a^j_njcos (f^j_nj n_j=lu)

" #

r ^nj=l X^{nj 1}

kˆ1

a^j_kr^{(nj k)=l} a^j_njC_j

" #

;

and

r!0lim^‡r ^{nj=l n}X^{j 1}

kˆ1

a^j_kr^{(nj k)=l} a^j_njCj

" #

ˆ 1:

Hence, for each j2J, there exists R_j>0 such that Re( L_j(rexp (iu)))0, for every 0<r<R_j andu₀uu₁. Therefore, setting RˆminfRj;j2Jg; one gets that Re( Lj(z))<A, for every A>0,z2S(u0;u1;R) andj2J.

Let us now set

I:ˆ fj2 f1; :::;mg;c_j(u)>0;8u2[u₀;u₁];L_j(z)6ˆ0g:

Letj2I andCj>0 such thatcj(u)>Cj, for allu2[u0;u1]. We may assume a^jnj 6ˆ0. LetV be an open subanalytic subset of the sectorS(u0;u1;R) and suppose that there existsA>0 such that Re( L_j(z))<A, for everyz2V, and that, for each 0<d<R, there existsz_d2V with jz_dj d. For each 0<d<R, let us denote:r_dˆ jz_dj andu_dˆarg(z_d). The sequence fr_dg_d converges to 0 and one has:

d!0lim^‡Re( L_j(r_dexp (iu_d)))ˆ

ˆ lim

d!0^‡r_d^{nj=l n}X^{j 1}

kˆ1

a^j_kr⁽ⁿ_d^{j k)=l}cos (f^j_k k=lu_d)‡a^j_njcos (f^j_nj n_j=lu_d)

" #

lim

d!0^‡r_d^{nj=l n}X^{j 1}

kˆ1

a^j_kr⁽ⁿ_d^{j k)=l}‡a^j_njCj

" #

ˆ ‡ 1;

which is a contradiction. Conversely, ifV is an open subanalytic subset of the setfz2S(u0;u1;R);jzj>dg, for some 0<d<R, thenVis contained on the compact set fz2C;u0argzu1;d jzj Rg, and Re( L_j(z)) is obviously bounded on V. We conclude that I is the desired subset of f1; :::;mg, withf1; :::;mgnIˆJ. q.e.d.

We shall now describe the ind-sheaf of temperate holomorphic solutions of the differential systemPuˆ0 in the open sector given by Proposition 2.6, wherePis the operator (2.1).

THEOREM 2.7. Let M be the D_X-module D^m_X=D^m_XP, where P is the matrix of differential operators (2.1).IfMis irregular, then there exist n>0and an open sector S such that:

``lim_!

R>d>0

''(Cⁿ_SdC^{m n}_S )! Ihom_bXDX(b_XM;O^t_X)_S; (2:3)

where S_dˆ fz2S;jzj>dg, for each0<d<R.

PROOF. LetS(u⁰₀;u⁰₁;R⁰) and I be, respectively, the open sector and the subset off1; :::;mggiven by Proposition 2.6 and let us chooseu0;u1;R2R, withu⁰₀ <u₀<u₁<u⁰₁andR>0, such that the matrix of differential opera- torsPadmits a fundamental solutionF(z) exp ( L(z)) on the open sector S(u0;u1;R). Let us denoteSˆS(u0;u1;R) and letnbe the cardinality of the set I. Remark that I6ˆ ;and so,n>0.

LetVbe a connected subanalytic open subset ofS, relatively compact in X, with 02@V. By Lemma 2.5, for each jˆ1; :::;m, one has exp ( L_j(z))2 O^t_X(V) if and only if there exists A>0 such that Re( L_j(z))<A for each z2V. Thus, by Proposition 2.6, exp ( Lj(z))2 O^t_X(V), for all j2 f1; :::;mgnI, and, for j2I one has exp ( L_j(z))2 O^t_X(V) if and only if VSd, for some 0<d<R.

Since 02@V, it follows that V 6S_d, for all 0<d<R and, hence, exp ( L_j(z))2 O= ^t_X(V), for all j2I.

Therefore, by Corollary 2.4, given a connected subanalytic open subset V ofS, relatively compact inX, eitherV Sd, for some 0<d<R, and in this caseG(V;S^t)'C^m, or elseG(V;S^t)'C^{m n}. By Theorem 1.2, we get

the desired isomorphism (²). q.e.d.

Finally, we may conclude the irregularity of S^t arguing by contra- diction. LetS(resp.n) be the open sector (resp. the positive integer) given by Theorem 2.7. Let us suppose thatS^tis regular atpˆ(0;0). Then, there exist an open neighborhoodU of 0, a small and filtrant categoryLand a functorG:L!D^[a;b](Mod^c(CX)),l7!Gl, such thatJX(S^t)'``lim_!

l

''Gland SS(G_l)\p ¹(U)T_f0gX[T_XX, for all l2L. We may assume from the (²) Notice that, forF2Mod(kXsa), one hasFS:ˆiS!i_S¹F, whereiS:SXsa!Xsa

denotes the natural embedding (see [7]).

beginning thatUis an open ball with center at the origin and thatSU.

Since 02=S, we get:

SS(G_l)\p ¹(S)T_XX; 8l:

Then, for each land k2Z, H^k(G_l)_S is a constant sheaf, since S is con- tractible. In particular, we may find M_l2D^b(Mod(C)) such that (Gl)S'(Ml)S, for eachl. Moreover, one has:

``lim_!

R>d>0

''Cⁿ_SdC^{m n}_S '``lim_!

l

''H⁰(M_l)_S:

LetVbe a connected subanalytic open subset ofS_e, for someR>e>0, and assumeV is contractible. Then:

C^m'``lim_!

R>d>0

''G(V;Cⁿ_SdC^{m n}_S )'lim_!

l

G(V;H⁰(M_l)_S)'lim_!

l

H⁰(M_l):

On the other hand:

C^{m n}'``lim_!

R>d>0

''G(S;Cⁿ_SdC^{m n}_S )'lim_!

l

G(S;H⁰(Ml)_S)'lim_!

l

H⁰(Ml):

This entails thatC^m 'C^{m n}and hence,nˆ0, which is a contradiction.

REMARK2.8. Let us consider the irregular holonomicDX-module M ˆ D_X=D_X(z²@_z‡1):

In this case, exp (1=z) is a fundamental solution of the differential operator z²@z‡1 inXnf0g. Arguing as in the proof of Proposition 2.6, we findR>0 with the following property: given an open subanalytic subset V of the sectorSˆS(0;p=4;R), then there existsA>0 such that Re( 1=z)<A, for every z2V, if and only if there exists 0<d<R such that V fz2S;jzj>dg. Moreover, by Proposition 2.7, one has the iso- morphism below:

``lim_!

R>d>0

''CSd! Ihomb_XDX(b_XM;O^t_X)_S:

On the other hand, M. Kashiwara and P. Schapira proved in [8] the following isomorphism onX,

``lim_!

e>0

''C_UeIhom_bXDX(b_XM;O^t_X);

whereUeˆXnBe(e;0);andBe(e;0) denotes the open ball with center at (e;0) and radiuse, for everye>0.

Let us check that

``lim_!

R>d>0

''C_Sd '``lim_!

e>0

''C_Ue\S:

It is enough to prove that, for every 0<d<R, there existse>0 such thatS_dUe\Sand that, for everye>0, there exists 0<d<Rsuch that U_e\SS_d. In fact, for each 0<d<R, ifx‡iy2S_d, thenx²‡y²>d² andx>y>0. It follows that 2x²>d² and hence,x²‡y²>x²>x(d=2), this is, (x d=4)²‡y²>(d=4)². Thus, SdUd=4\S. Conversely, given e>0 andx‡iy2Ue\S, we havex²‡y²>2xeandx>y>0. This gives x>eand so,x²‡y²>e². Therefore,Ue\SSe.

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Manoscritto pervenuto in redazione il 20 febbraio 2006.