Conic sheaves on subanalytic sites and Laplace transform

Conic Sheaves on Subanalytic Sites and Laplace Transform

LUCAPRELLI(*)

ABSTRACT- LetEbe andimensional complex vector space and letEbe its dual. We construct the conic sheaves O^t_ER‡and O^w_ER‡ of tempered and Whitney holo- morphic functions respectively and we give a sheaf theoretical interpretation of the Laplace isomorphisms of [10] which give the isomorphisms in the derived categoryO^{t^}_ER‡[n]' O^t_E

R‡andO^{w^}_ER‡[n]' O^w_E R‡. Introduction.

Classical sheaf theory is not well suited to the study of various objects in Analysis, which are not defined by local properties. To overcome this problem Kashiwara and Schapira in [11] developed the theory of ind- sheaves on a locally compact space and defined the six Grothendieck op- erations in this framework. For a real analytic manifoldX, they defined the subanalytic site X_sa as the site generated by the category of subanalytic open subsets and whose coverings are the locally finite coverings in X.

Moreover they proved the equivalence between the categories of ind-R- constructible sheaves and sheaves onXsa. They constructed the sheaf of tempered distributions as a sheaf onX_sa, and whenXis acomplex manifold they introduced the sheaf of tempered holomorphic functions. Thanks to the results of [16] we have a direct construction of the six Grothendieck operations on Mod(kXsa), without using the theory of ind-sheaves. So we will work directly with sheaves on subanalytic sites.

LetXbe a real analytic manifold endowed with an action ofR^‡. Our first goal is to define conic sheaves onX_saand then, whenXis avector bundle, to extend the construction of the Fourier-Sato transform for classical sheaves

(*) Indirizzo dell'A.: UniversitaÁ di Padova, Dipartimento di Matematica Pura ed Applicata, via Trieste 63, 35121 Padova, Italy.

E-mail: address: [email protected]

to conic sheaves on subanalytic sites. In order to do that we have to choose a suitable definition of conic sheaf: indeed there are several definitions, which are equivalent in the classical case but not in the framework of subanalytic sheaves. We choose the one which satisfies some desirable properties (as the equivalence with sheaves on the conic topology associated to the action and the isomorphism of conic sheaves with limits of conicR-constructible sheaves) and for which the Fourier-Sato isomorphism applies.

Let Ebe a complex vector space. As an application we construct the conic sheaves O^t_E

R‡ and O^w_E

R‡ of tempered and Whitney holomorphic functions respectively and we prove a sheaf theoretical interpretation of the Laplace isomorphisms of [10] which induce isomorphisms in the de- rived categoryO^{t^}_ER‡[n]' O^t_E

R‡ andO^{w^}_ER‡[n]' O^w_E

R‡. In more details, the contents of this paper are as follows.

In Section 1 we first recall the results of [11] and [16] on sheaves on subanalytic sites. Then we consider the category of conic sheaves on subanalytic sites. Letkbe afield and letXbe a real analytic manifold with an actionmof R^‡. LetU be an open subset of X. We say that U isR^‡- connected if its intersections with the orbits ofmare connected. We denote R^‡Uthe conic open set associated toU(i.e.R^‡Uˆm(U;R^‡)):A sheafF onX_sais said to be conic ifG(R^‡U;F)! G(U;F) for eachR^‡-connected relatively compact open subanalytic subsetUofX. We call Mod_R^‡(kXsa) Mod(kXsa) the category of conic sheaves. This definition is different from the classical one. Let us consider the projectionp:XR^‡!X. One can define the subcategory Mod^m(k_Xsa) of Mod(k_Xsa) consisting of sheaves sa- tisfyingm ¹F'p ¹F. The categories Mod^m(k_Xsa) and Mod_R^‡(k_Xsa) are not equivalent in general. The category of conic subanalytic sheaves has many good properties, for example it is equivalent to the category of sheaves on the conic subanalytic siteX_sa;R^‡(i.e. the category of open conic subanalytic subsets of X with the topology induced from X_sa). This equivalence is strictly related to the geometry of subanalytic open subsets ofX. WhenE is a vector bundle, one can define the Fourier-Sato transform which gives an equivalence between conic sheaves on Esa and conic sheaves on E_sa, whereE denotes the dual vector bundle.

In Section 2 we study the conic sheaves of tempered and Whitney holomorphic functions. LetEbe areal vector space, and letE,!Pbe its projective compactification. We define the conic sheaves of tempered distributionsDb^t_ER‡ and WhitneyC¹-functionsC^1;w_E

R‡. IfU is an open sub- analytic cone, the sections G(U;Db^t_E

R‡) are distributions which are tem- pered on the boundary ofUand at infinity, and the sectionsG(U;C^1;w_E

R‡) a re

WhitneyC¹-functions onU with rapid decay at infinity. IfFis aconicR- constructible sheaf onEwe have

RHom(F;Db^t_ER‡)'THom(F;DbE);

RHom(F;C^1;w_E

R‡)'D⁰F^WC¹_E;

where THom(;Db_E) a nd ^WC¹_E are the functors introduced in [10].

WhenEis anndimensional complex vector space we define the sheaves O^t_ER‡andO^w_ER‡ of tempered and Whitney holomorphic functions taking the solutions of the Cauchy-Riemann system with values in tempered dis- tributions and Whitney C¹-functions respectively. We show that these sheaves are invariant by the Laplace transform. In fact the Laplace isomorphisms of [10] induce isomorphisms in the derived category O^{t^}_ER‡[n]' O^t_E

R‡ and O^{w^}_ER‡[n]' O^w_E

R‡, where ^denotes the extension of the Fourier-Sato transform to conic sheaves onEsa. Moreover these iso- morphisms are compatible with the action of the Weyl algebra.

1. Conic sheaves on subanalytic sites.

In the following X will be a real analytic manifold andkafield. Ref- erences are made to [11, 16] for an introduction to sheaves on subanalytic sites. We refer to [2, 12] for the theory of subanalytic sets and to [4, 19] for the more general theory of o-minimal structures.

1.1 ±Review on sheaves on subanalytic sites.

Denote by Op(Xsa) the category of subanalytic open subsets ofX. One endows Op(Xsa) with the following topology:SOp(Xsa) is acovering of U2Op(X_sa) if for any compactKofXthere exists afinite subsetS₀S such thatK\S

V2S0V ˆK\U. We will callX_sa the subanalytic site.

Let Mod(k_Xsa) denote the category of sheaves onX_sa. Then Mod(k_Xsa) is a Grothendieck category, i.e. it admits a generator and small inductive limits, and small filtrant inductive limits are exact. In particular as a Grothendieck category, Mod(k_Xsa) has enough injective objects.

Let Mod_R-c(k_X) be the abelian category ofR-constructible sheaves on X, and consider its subcategory Mod^c_R-c(k_X) consisting of sheaves whose support is compact.

We denote by r:X!Xsa the natural morphism of sites. We have functors

The functorsr ¹andr are the functors of inverse image and direct image respectively. The sheafr_!Fis the sheaf associated to the presheaf Op_sa(X)3U7!F(U). In particular, for U2Op(X) one has r_!kU '

lim

VUƒƒƒ!rk_V, where V 2Op_sa(X). Let us summarize the properties of these functors:

the functorr is fully faithful and left exact, the restriction ofr to Mod_R-c(k_X) is exact,

the functorr ¹ is exact,

the functorr_!is fully faithful and exact,

(r ¹;r) and (r_!;r ¹) are pairs of adjoint functors.

For eachF2Mod(k_Xsa), there exists asmall filtrant inductive system fF_ig, withF_i2Mod^c_R-c(k_X), such thatF'lim

!i rF_i. Moreover letfF_igbe a filtrant inductive system ofk_Xsa-modules and letG2Mod^c_R-c(k_X). One has the isomorphism

Hom(rG;lim

!i F_i)'lim

!i Hom(rG;F_i):

(1:1)

Let X;Y be two real analytic manifolds, and let f :X!Y be areal analytic map. We have a commutative diagram

(1:2)

We get external operationsf ¹,fandf_!!, where the notationf_!!follows from the fact thatf!!r 6'rf!in general. Iff is proper on supp(F) then fF'f!!F, in this casef!!commutes withr. While functorsf ¹andare exact, the functorsHom,f andf!! are left exact and admit right derived functors. In particular the functorRf_!!admits a right adjoint, denoted byf^!, and we get the usual isomorphisms between Grothendieck operations (projection formula, base change formula, KuÈnneth formula, etc.) in the framework of subanalytic sites.

We refer to [16] for adetailed exposition.

Finally we recall the relations between the six Grothendieck operations and the functorsr ¹,Rr andr_!.

the functorr ¹commutes with,f ¹ andRf!!, the functorRrcommutes withRHom,Rfandf^!, the functorr_!commutes withandf ¹,

iff is atopological submersion (i.e. it is locally isomorphic to apro- jection YRⁿ !Y), then f^!'f ¹f^!k_Y commutes with r ¹ and Rf_!!

commutes withr_!.

1.2 ±Conic subanalytic sheaves.

LetXbe a real analytic manifold endowed with an analytic actionmof R^‡. We have a diagram

wherei(x)ˆ(x;1) andpdenotes the projection. We havemiˆpiˆid.

DEFINITION1.2.1. (i) We say that a subset S of X is R^‡-connected if S\b is connected or empty for each orbit b ofm.

(ii) Let S be a subset of X. We setR^‡Sˆm(S;R^‡):

(iii) Let S be a subset of X. Then S is conic if SˆR^‡S. i.e. S is invariant under the action ofR^‡.

IfU2Op(X), thenR^‡U2Op(X) becausemis open.

DEFINITION 1.2.2. A sheaf of k-modules F on X_sa is conic if the restriction morphismG(R^‡U;F)!G(U;F)is an isomorphism for each R^‡-connected U2Op^c(X_sa)withR^‡U2Op(X_sa).

(i) We denote by Mod_R^‡(k_Xsa) the full subcategory of Mod(k_Xsa) consisting of conic sheaves.

(ii) We denote by D^b_R‡(kXsa), the full subcategory of D^b(kXsa) con- sisting of objects F such that H^jF belongs toMod_R^‡(kXsa)for all j2Z.

REMARK 1.2.3. Let X be a real analytic manifold endowed with a subanalytic actionmofR^‡. As in classical sheaf theory one can define the subcategory Mod^m(k_Xsa) of Mod(k_Xsa) consisting of sheaves satisfying

m ¹F'p ¹F. The categories Mod^m(kXsa) and Mod_R^‡(kXsa) are not equiva- lent in general.

In fact let XˆR, set X^‡ˆ fx2R;x>0g and let m be the natural action of R^‡ (i.e. m(x;t)ˆtx). The sheaf r_!k_X^‡ 'lim

n2Nƒƒ!rk₍¹

n;n) belongs to Mod^m(k_Xsa) but not to Mod_R^‡(k_Xsa). Indeed, it is easy to check that G(X^‡;r_!k_X^‡)ˆ0 whileG((a;b);r_!k_X^‡)'k, 05a5b. Moreover, sincek_X^‡is conic and the functors m ¹ and p ¹ commute with r_! we have r_!k_X^‡2 Mod^m(kXsa).

DEFINITION 1.2.4. We denote by Op(X_R^‡) (resp. Op(X_sa;R^‡)) the full subcategory ofOp(X)consisting of conic (resp. conic subanalytic) subsets, i.e. U2Op(X_R^‡)(resp. U2Op(X_sa;R^‡)) if U2Op(X)(resp. U 2Op(X_sa)) and it is invariant by the action ofR^‡.

We denote by X_R^‡(resp. X_sa;R^‡) the categoryOp(X_R^‡)(resp.Op(X_sa;R^‡)) endowed with the topology induced by X (resp. Xsa).

Let h:X!X_R^‡, h_sa:X_sa!X_sa;R^‡ and r_R^‡ :X_R^‡!X_sa;R^‡ be the natural morphisms of sites. We have a commutative diagram of sites

(1:3)

We need to introduce the subcategory of coherent conic sheaves.

DEFINITION 1.2.5. Let U2Op(X_R^‡). Then U is said to be relatively quasi-compact if, for any coveringfU_ig_i2Iof X_R^‡, there exists JI finite such that US

i2JU_i:We write UX_R^‡.

We will denote by Op^c(X_R^‡) (resp. Op^c(X_sa;R^‡)) the subcategory of Op(X_R^‡) consisting of relatively quasi-compact (resp. relatively quasi- compact subanalytic) open subsets.

One can check easily that ifU2Op^c(X), thenR^‡U2Op^c(X_R^‡).

DEFINITION 1.2.6. Let F2Mod(kX_R‡) and consider the family Op(X_sa;R^‡).

(i) F is X_sa;R^‡-finite if there exists an epimorphism G!!F, with G' _i2IkUi, I finite and U_i2Op^c(X_sa;R^‡).

(ii) F is X_sa;R^‡-pseudo-coherent if for any morphism c:G!F, where G is X_sa;R^‡-finite,kercis X_sa;R^‡-finite.

(iii) F is X_sa;R^‡-coherent if it is both X_sa;R^‡-finite and X_sa;R^‡-pseudo- coherent.

We will denote byCoh(X_sa;R^‡)the subcategory ofMod(k_XR‡)consisting of X_sa;R^‡-coherent objects.

Replacing Op^c(X_sa) with Op^c(X_sa;R^‡), we can adapt the results of [11, 16]

and we get the following result (see [5] for a detailed proof).

THEOREM 1.2.7. (i) Let G2Coh(X_sa;R^‡) and let fFig be a filtrant inductive systeminMod(k_Xsa;R‡). Then we have an isomorphism

lim

!i Hom_kXsa;R‡(r_R^‡G;F_i)! Hom_kXsa;R‡(r_R^‡G;lim

!i F_i):

Moreover the functor of direct imager_R^‡associated to the morphismr_R^‡ in(1.3)is fully faithful and exact onCoh(X_sa;R^‡).

(ii) Let F2Mod(k_Xsa;R‡). There exists a small filtrant inductive system fF_ig_i2IinCoh(X_sa;R^‡)such that F'lim

!i r_R^‡F_i.

NOTATION1.2.8. Sincer_R^‡is fully faithful and exact onCoh(X_sa;R^‡), we can identifyCoh(X_sa;R^‡)with its image inMod(kX_sa;R‡). When there is no risk of confusion we will write F instead ofr_R^‡F, for F2Coh(X_sa;R^‡).

Let us consider the category Mod_R^‡(k_Xsa) of conic sheaves onX_sa. The restriction ofh_sa induces afunctor denoted byeh_sa and we obtain a dia- gram

(1:4)

Now assume the hypothesis below:

…i†everyU2Op^c(X_sa) has a finite covering consisting of R^‡-connected subanalytic open subsets;

…ii†for any U2Op^c(X_sa) we haveR^‡U2Op(X_sa);

…iii†for any x2Xthe set R^‡xis contractible;

…iv†there exists acoveringfV_ng_n2N of X_sa such that Vn isR^‡-connected andVnVn‡1 for eachn:

8>

>>

>>

>>

<

>>

>>

>>

>: (1:5)

LetU2Op(Xsa) such thatR^‡Uis still subanalytic. LetF2Mod(kX_sa;R‡).

LetWbe the natural map fromG(R^‡U;F) toG(U;h_sa¹F) defined by G(R^‡U;F) ! G(R^‡U;h_sah_sa¹F)

' G(R^‡U;h_sa¹F)

! G(U;h_sa¹F):

(1:6)

PROPOSITION 1.2.9. Let F2Mod(kX_sa;R‡). Let U2Op(Xsa), assume that U is R^‡-connected and that R^‡U is still subanalytic. Then the morphismWdefined by(1.6)is an isomorphism.

PROOF. (i) Assume that U2Op^c(X_sa) is R^‡-connected. Let F2 Mod(kX_sa;R‡), thenFˆlim

!i r_R^‡Fi, withFi2Coh(X_sa;R^‡). We have the chain of isomorphisms

Homk_Xsa(kU;h_sa¹lim

!i r_R^‡Fi)'Homk_Xsa(kU;lim

!i rh ¹Fi) 'lim

!i Hom_kX(k_U;h ¹F_i) 'lim

!i Hom_kX

R‡(k_R^‡_U;F_i) 'Homk_Xsa;R‡(k_R^‡_U;lim

!i r_R^‡Fi);

where the first isomorphism follows since h_sa¹r_R^‡'rh ¹ and the third one follows from the equivalence between conic sheaves on X and sheaves on X_R^‡. In the fourth isomorphism we used the fact that R^‡U2Op^c(X_sa;R^‡).

(ii) Let U2Op(Xsa) be R^‡-connected. Let fVng_n2N2Cov(Xsa) be a covering ofXas in (1.5) (iv) and setUn ˆU\Vn. We have

G(U;h_sa¹F)'lim

n

G(U_n;h_sa¹F)'lim

n

G(R^‡U_n;F)'G(R^‡U;F):

(1:7)

p THEOREM1.2.10. The functorseh_saandh_sa¹in(1.4)are equivalences of categories inverse to each others.

PROOF. (i) Let F2Mod_R^‡(kXsa), and let U2Op^c(Xsa) be R^‡-con- nected. We have

G(U;F)'G(R^‡U;F)'G(R^‡U;eh_saF)'G(U;h_sa¹eh_saF):

The third isomorphism follows from Proposition 1.2.9. Then (1.5) (i) implies h_sa¹eh_sa'id.

(ii) For anyU2Op^c(X_sa;R^‡) we have:

G(U;h_sah_sa¹F)'G(U;h_sa¹F)'G(U;F)

where the second isomorphisms follows from Proposition 1.2.9. This implies

h_sah_sa¹'id. p

NOTATION 1.2.11. Since h_sa¹ is fully faithful and exact we will often identify Coh(X_sa;R^‡) with its image in Mod_R^‡(k_Xsa). Hence, for F2Coh(X_sa;R^‡) we shall often write F instead of h_sa¹F.

Thanks to Theorem 1.2.7 we can give another description of the cate- gory of conic sheaves.

THEOREM1.2.12. Let F2Mod_R^‡(k_Xsa). Then there exists a small fil- trant systemfFiginCoh(X_sa;R^‡)such that F'lim

!i rh ¹Fi.

Assume (1.5). Injective and quasi-injective objects of Mod(k_Xsa) are not contained in Mod_R^‡(ksa). For this reason we are going to introduce a subcategory which is useful when we try to find acyclic resolutions.

LEMMA1.2.13. Assume that X satisfies(1.5).Then the following prop- erty is satisfied:

each finite covering of anR^‡-connected U2Op^c(X_sa) has a finite refinement fV_igⁿ_iˆ1such that each ordered union S_j

iˆ1V_iisR^‡-connected for each j 2 f1;. . . ;ng:

8<

(1:8) :

PROOF. Let U2Op^c(Xsa) be R^‡-connected. Then each finite cove- ring of U admits a finite refinement consisting of R^‡-connected open subanalytic subsets. LetfU_igⁿ_iˆ1 be afinite covering ofU,U_i2Op^c(X_sa) R^‡-connected for eachi. We will construct arefinement satisfying (1.8).

For kˆ1;. . .;n set Vk11:ˆUk and Vk1i:ˆUs(i)\R^‡(Uk\Us(i)) for

iˆ2;. . .;n and s(i)ˆi 1 if ik, s(i)ˆi if i>k. Then set

U_k2:ˆS_n

iˆ1V_k1iandV_k2i:ˆU_s(i)\R^‡(U_k2\U_s(i)). Forjˆ1;. . .;ndefine recursively U_kjˆS_j

`ˆ1

S_n

iˆ1V_k`i and V_kji ˆU_s(i)\R^‡(U_kj\U_s(i)). Re- mark thatS_j

pˆ1

S_n

`ˆ1

S_n

iˆ1V_p`iˆS_j

pˆ1R^‡U_p\U. By Lemma1.2.14 below all the setsV_kji areR^‡-connected andfV_kjig_i;k;j is arefinement offU_ig_i satisfying (1.8) (with the lexicographic order). p

LEMMA1.2.14. Assume that X satisfies(1.5)(iii). Let U;V;W be open and R^‡-connected. Then U[(V\R^‡(U\V))[(W\R^‡(U\W)) is R^‡-connected.

PROOF. In what follows, when we writeR^‡xwe suppose thatR^‡x'R.

IfR^‡xˆxeverything becomes obvious.

(i) First remark that U\V (resp. U\W, V\W) is R^‡-connected.

Indeed, letx12U\R^‡x,x22V\R^‡xfor somex2X. Thenx1ˆm(x;a), x₂ˆm(x;b). Every path inR^‡xconnectingx₁ andx₂containsm(x;[a;b]).

SinceUandV areR^‡-connected thenU\V m(x;[a;b]).

(ii) Now let us prove that U[(V\R^‡(U\V)) isR^‡-connected. Let x1;x22U[(V\R^‡(U\V))\R^‡x for some x2X. Then x1ˆm(x;a), x2ˆm(x;b). We want to prove thatm(x;[a;b])U[(V\R^‡(U\V)). If x₁;x₂2Uit follows sinceUisR^‡-connected and ifx₁;x₂2V\R^‡(U\V) it follows from (i). So we may assume thatx₁2Uandx₂2V\R^‡(U\V).

SinceUisR^‡-connected andx₂2R^‡x₁, there existsyˆm(x;c)2U\V.

Then m(x;[a;c])U. In the same way m(x;[b;c])V\R^‡(U\V) a nd hencem(x;[a;c][[b;c])U[(V\R^‡(U\V)).

(iii) Let us show thatU[(V\R^‡(U\V))[(W\R^‡(U\W)) isR^‡- connected. Let x₁;x₂2U[(V\R^‡(U\V))[(W\R^‡(U\W))\R^‡x for some x2X. Then x₁ˆm(x;a), x₂ˆm(x;b). We want to prove that m(x;[a;b])U[(V\R^‡(U\V))[(W\R^‡(U\W)). By (i) and (ii) we may reduce to the case x12V, x22W. As in (ii), there exist y1ˆm(x;c)2U\V and y2ˆm(x;d)2U\W. Then m(x;[c;d])2U, m(x;[a;c])V\R^‡(U\V) a nd m(x;[b;d])W\R^‡(U\W). Hence m(x;[c;d][[a;c][[b;d])2U[(V\R^‡(U\V))[(W\R^‡(U\W)) and

the result follows. p

DEFINITION1.2.15. A sheaf F2Mod(kXsa)isR^‡-quasi-injective if for each R^‡-connected U2Op^c(X_sa) the restriction morphism G(X;F)! G(U;F)is surjective.

Remark that the functorh_sa¹sends injective objects of Mod(kX_sa;R‡) to R^‡-quasi-injective objects sinceG(U;h_sa¹F)'G(R^‡U;F) ifU2Op^c(Xsa) isR^‡-connected. Moreover the category ofR^‡-quasi-injective objects is cogenerating in Mod(k_Xsa) since injective objects are cogenerating in Mod(k_Xsa). Once we have (1.8) and (1.5) (iv) it is easy to prove Propositions 1.2.16 and 1.2.17 below in the same way as the corresponding classical results for c-soft sheaves (see Propositions 2.5.8, 2.5.10 and Corollary 2.5.9 of [8]).

PROPOSITION1.2.16. Let0!F⁰!F!F⁰⁰!0be an exact sequence inMod(kXsa)and assume that F⁰isR^‡-quasi-injective. Let U2Op(Xsa)be R^‡-connected. Then the sequence

0!G(U;F⁰)!G(U;F)!G(U;F⁰⁰)!0 is exact.

PROPOSITION1.2.17. Let F⁰;F beR^‡-quasi-injective and consider the exact sequence 0!F⁰!F!F⁰⁰!0 in Mod(kXsa). Then F⁰⁰ is R^‡- quasi-injective.

It follows from the preceding results that

PROPOSITION 1.2.18. R^‡-quasi-injective objects are injective with respect to the functorG(U;), with U2Op(X_sa)andR^‡-connected.

COROLLARY1.2.19. R^‡-quasi-injective objects areh_sa-injective.

THEOREM 1.2.20. The categories D^b(k_Xsa;R‡) and D^b_R‡(k_Xsa) are equi- valent.

PROOF. In order to prove this statement, it is enough to show thath_sa¹ is fully faithful. LetF2D^b(k_Xsa;R‡) and letF⁰be an injective complex quasi- isomorphic to F. Sinceh_sa¹ sends injective objects to R^‡-quasi-injective objects which areh_sa-injective, we haveRh_sah_sa¹F'h_sah_sa¹F⁰'F⁰'F.

This impliesRh_sah_sa¹'id, henceh_sa¹is fully faithful. p COROLLARY 1.2.21. Let F 2D^b_R‡(k_Xsa) and let U 2Op(X_sa) be R^‡- connected. ThenRG(R^‡U;F)! RG(U;F).

Hence for eachF2D^b_R‡(kXsa) we haveF'h_sa¹F⁰withF⁰2D^b(kX_sa;R‡).

REMARK 1.2.22. Thanks to these results, in order to prove that a morphismF!GinD^b_R‡(k_Xsa) is an isomorphism, it is enough to check that RG(U;F)! RG(U;G) for eachU2Op(X_sa;R^‡).

REMARK 1.2.23. It is easy to check that the six Grothendieck opera- tions, except the functor of proper direct image, preserve conic subanalytic sheaves. We refer to [17] for a detailed exposition.

1.3 ±Conic sheaves on vector bundles

LetE!^t Zbe areal vector bundle, with dimensionnover a real ana- lytic manifoldZ. ThenR^‡ acts naturally onEby multiplication on the fi- bers. We identifyZwith the zero-section ofEand denote byi:Z,!Ethe embedding. We setE_ ˆEnZandt_ : _E!Zdenotes the projection.

LEMMA1.3.1. The categoryOp(Esa)satisfies(1.5).

PROOF. Let us prove (1.5) (i). LetfV_ig_i2Nbe alocally finite covering of Z with V_i2Op^c(Z_sa) such that t ¹(V_i)'R^mRⁿ and let fU_ig be are- finement of fVig with Ui2Op^c(Zsa) a nd UiVi for eachi. Then U is covered by afinite number oft ¹(U_i) a ndU\t ¹(U_i) is relatively compact in t ¹(V_i) for each i. We may reduce to the case E'R^mRⁿ. Let us consider the morphism of manifolds

W:R^mS^{n 1}R !R^mRⁿ (z;w;r)7!(z;ri(w));

where i:S^{n 1},!Rⁿ denotes the embedding. Then W is proper and sub- analytic. The subset W ¹(U) is subanalytic and relatively compact in R^mS^{n 1}R.

(a) By a result of [20], W ¹(UnZ) admits a finite coverfW_jg_j2J such that the intersections of each W_j with the fibers of p:R^mS^{n 1} R!R^mS^{n 1} are contractible or empty. Then W(W_j) is an open sub- analytic relatively compactR^‡-connected subset ofR^mRⁿfor eachj. In this way we obtain a finite covering ofUnZconsisting ofR^‡-connected subanalytic open subsets.

(b) Let p2p(W ¹(U\Z)). Then p ¹(p)\U is adisjoint union of in- tervals. Let us consider the interval (m(p);M(p)), m(p)5M(p)2R con- taining 0. Set W_Zˆ f(p;r)2U; m(p)5r5M(p)g. The set W_Z is open subanalytic (it is a consequence of Proposition 1.2, Chapter 6 of [19]), contains W ¹(U\Z) and its intersections with the fibers of p are con- tractible. ThenW(W_Z) is an openR^‡-connected subanalytic neighborhood ofU\Zand it is contained inU.

By (a) there exists a finite coveringfW(W_j)g_j2J of UnZconsisting of R^‡-connected subanalytic open subsets, andW(W_Z)[S

j2JW(W_j)ˆU.

By Proposition 8.3.8 of [8] the category Op(Esa) also satisfies (1.5) (ii).

Moreover (1.5) (iii) and (iv) are clearly satisfied. p

Now let us considerEendowed with the conic topology. In this situa- tion, an objectU2Op(E_R^‡) is the union ofU_ 2Op(E__R^‡) a ndUZ2Op(Z) such thatt_ ¹(U_Z)U. If_ U;V2Op(E_R^‡), thenUVifU_ZV_ZinZ and U_ V_ in E__R^‡ (this means that p(U)_ p(V_) in E=R_ ^‡, where p: _E!E=R_ ^‡denotes the projection).

Applying Theorem 1.2.20 we have the following

THEOREM1.3.2. The categories D^b_R‡(kEsa)and D^b(kE_sa;R‡)are equiva- lent.

Consider the subcategory Mod^cb_R-c;R^‡(k_E) of Mod_R-c;R^‡(k_E) consisting of sheaves whose support is compact on the base (i.e.t(supp(F)) is compact in Z). The restriction ofh ¹ to Coh(E_sa;R^‡) gives rise (see [17] for more de- tails) to an equivalence of categories

h ¹:Coh(E_sa;R^‡)! Mod^cb_R-c;R^‡(k_E):

(1:9)

As aconsequence of Theorem 1.2.12 one has the following

THEOREM 1.3.3. Let F2Mod_R^‡(kEsa). Then there exists a small fil- trant systemfF_iginMod^cb_R-c;R^‡(k_E)such that F'lim

!i rF_i.

We end this section with this result, which will be useful in the next section.

LEMMA1.3.4. Let F2D^b_R‡(kEsa). Then:

(i) RtF'i ¹F.

(ii) Rt!!F'i^!F.

PROOF. (i) The adjunction morphism definesRtF'i ¹t ¹RtF!i ¹F.

LetV 2Op^c(Zsa). Then lim

UVƒƒ!R^kG(U;F)' lim

t(U)ˆVUV

ƒƒ!R^kG(U;F)'R^kG(t ¹(V);F)'R^kG(V;RtF);

where U2Op(E_sa) a ndR^‡-connected. The second isomorphism follows from Corollary 1.2.21.

(ii) The adjunction morphism defines i^!F!i^!t^!Rt_!!F'Rt_!!F. Let V 2Op^c(Zsa), and let K be a compact subanalytic R^‡-connected neigh- borhood ofV inE. Thent ¹(V)nKisR^‡-connected and subanalytic, and

R^‡(t ¹(V)nK)ˆt ¹(V)nZ. By Corollary 1.2.21 we have the isomorphism RG(t ¹(V);RGZF)'RG(t ¹(V);RGKF).

It follows from the definition ofRt!!that for anyk2ZandV2Op^c(Zsa) we haveR^kG(V;Rt_!!F)'lim

!K R^kG(t ¹(V);RG_KF), whereKranges through the family of compact subanalyticR^‡-connected neighborhoods ofV inE.

On the other hand for anyk2Zwe have

R^kG(V;i^!F)'R^kHom(ikV;F)'R^kHom(ii ¹t ¹kV;F)'R^kG(t ¹(V);RGZF)

and the result follows. p

1.4 ±Fourier-Sato transformation.

LetE!^t Zbe areal vector bundle, with dimensionnover a real ana- lytic manifoldZandE !^p Zits dual. We identifyZas the zero-section ofE and denote i:Z,!Ethe embedding, we define similarly i:Z,!E. We denote byp1 andp2 the projections fromEZE:

We set

A:ˆ f(x;y)2E

ZE; hx;yi 0g

A⁰:ˆ f(x;y)2E

ZE; hx;yi 0g and we define the functors

CA⁰ ˆRp1RGA⁰p^!₂:D^b_R‡(kE_sa)!D^b_R‡(kEsa) F_A⁰ ˆRp_2!!()_A⁰p₁¹:D^b_R‡(k_Esa)!D^b_R‡(k_Esa) CAˆRp2RGAp₁¹:D^b_R‡(kEsa)!D^b_R‡(kE_sa) F_AˆRp_1!!()_Ap^!₂:D^b_R‡(k_Esa)!D^b_R‡(k_Esa) 8>

>>

>>

><

>>

>>

>>

:

REMARK1.4.1. These functors are well defined, more generally they send subanalytic sheaves to conic subanalytic sheaves.

LEMMA 1.4.2. Let F2D^b_R‡(k_Esa). Then supp((RG_A(p₁ ¹F))_A⁰) is con- tained in Z_ZE.

PROOF. We may reduce to the case F2Mod_R^‡(kEsa). Then Fˆlim

!i rF_i, withF_i2Mod^cb_R-c;R^‡(kE). We have H^k(RG_A(p1 1lim

!i rF_i)_A⁰)'lim

!i H^k(RG_A(p1 1rF_i)_A⁰) 'lim

!i rH^k(RG_A(p₁ ¹F_i)_A⁰) 'lim

!i r(H^k(RGA(p1 1Fi)_A⁰))_ZZE; where the last isomorphism follows from Lemma 3.7.6 of [8]. p

LEMMA1.4.3. Let B and C be two closed subanalytic subsets of E such that B[CˆE, and let F2D^b(kEsa). ThenRGC(FB)'(RGCF)B.

PROOF. We have a natural arrow (G_CF)_B!G_C(F_B), andR(G_CF)_B ' (RG_CF)_B since ()_B is exact. Then we obtain a morphism (RG_CF)_B! RG_C(F_B). It is enough to prove that for any k2Z and for any F2Mod(kEsa) we have (R^kGCF)B! R^kGC(FB). Since both sides commute with filtrant lim!, we may assumeF2Mod^c_R-c(kE). Then the result follows from the corresponding one for classical sheaves. p PROPOSITION1.4.4. The two functors F_A⁰;C_A:D^b_R‡(k_Esa)!D^b_R‡(k_Esa) are isomorphic.

PROOF. We have the chain of isomorphisms:

F_A⁰FˆRp_2!!(p₁ ¹F)_A⁰ 'Rp2!!RG_A((p1 1F)_A⁰) 'Rp_2!!(RG_A(p₁ ¹F))_A⁰ 'Rp₂(RG_A(p₁ ¹F))_A⁰ 'Rp₂RG_A(p₁ ¹F):

The first isomorphism follows from Lemma1.3.4 (ii), the second one from Lemma 1.4.3, the third one from Lemma 1.4.2 and the last one from Lemma

1.3.4 (i). p

DEFINITION1.4.5. Let F2D^b_R‡(k_Esa).

(i) The Fourier-Sato transformis the functor ()^{^}:D^b_R‡(k_Esa)!D^b_R‡(k_Esa)

F^{^}ˆF_A⁰F'C_AF:

(ii) The inverse Fourier-Sato transformis the functor ()^_:D^b_R‡(kE_sa)!D^b_R‡(kEsa)

F^_ˆC_A⁰F'F_AF:

It follows from definition that the functors ^{^}and ^_commute withRr andr ¹. We have quasi-commutative diagrams

This implies that these functors are the extension to conic sub- analytic sheaves of the classical Fourier-Sato and inverse Fourier-Sato transforms.

THEOREM1.4.6. The functors ^{^} and ^_ are equivalence of categories, inverse to each others. In particular we have

Hom_Db

R‡(k_Esa)(F;G)'Hom_Db R‡(k_E

sa)(F^{^};G^{^}):

PROOF. LetF2D^b_R‡(k_Esa). The functors ^{^}and ^_are adjoint functors, then we have a morphismF!F^{^_}. To show that it induces an isomorphism it is enough to check that RG(U;F)!RG(U;F^{^_}) is an isomorphism on a basis for the topology ofE_sa. Hence we may assume thatUisR^‡-connected.

By Corollary 1.2.21 we may suppose thatUis an open subanalytic cone ofE.

we have the chain of isomorphisms:

RHom(k_U;F^{^_})ˆRHom(k_U;C_A⁰F_A⁰F) 'RHom(F_A⁰k_U;F_A⁰F) 'RHom(FA⁰kU;CAF) 'RHom(F_AF_A⁰k_U;F) 'RHom(kU;F);

where the last isomorphism follows from Theorem 3.7.9 of [8] and from the fact that the functors ^{^} and ^_commute withRr. Similarly we can show that forG2D^b_R‡(k_Esa) we have an isomorphismG^{_^}! G. p

REMARK1.4.7. In the complex case we have the same result with

A:ˆ f(x;y)2E

ZE;Rehx;yi 0g;

A⁰:ˆ f(x;y)2E

ZE;Rehx;yi 0g:

REMARK1.4.8. The Fourier-Sato isomorphism can be extended to the case of ind-sheaves (see [11] for complete exposition). On E_R^‡ one can define the category Ind(k_ER‡) of conic ind-sheaves:F2Ind(k_ER‡) if it is a filtrant ind-limit of Fi2Mod^cb(kE_R‡) (i.e. with compact support on the base). With slight modifications to the results of this section one can extend the Fourier-Sato transform to this setting and prove that it induces an equivalence between the categoriesD^b(Ind(k_ER‡)) andD^b(Ind(k_E

R‡)).

2. Laplace transform.

As an application of the preceding constructions we introduce the conic sheaves of tempered and Whitney holomorphic functions in order to give a sheaf theoretical interpretation of the Laplace isomorphisms of [10]. We refer to [6, 9] for the definition of the functors of temperate and formal cohomology and to [10] for the action of the Laplace transform on tem- perate and formal cohomology.

2.1 ±Review on temperate cohomology.

From now on, the base sheaf isC. LetMbe a real analytic manifold.

One denotes byDb_Mthe sheaf of Schwartz's distributions, and byD_Mthe sheaf of finite order differential operators with analytic coefficients. In [6]

the author defined the functor

THom(;DbM):ModR-c(CM)!Mod(DM)

in the following way: letU be a subanalytic subset ofM andZˆMnU.

Then the sheafTHom(C_U;Db_M) is defined by the exact sequence 0!GZDbM! DbM!THom(CU;DbM)!0:

This functor is exact and extends as functor in the derived category, from D^b_R-c(C_M) toD^b(D_M). Moreover the sheafTHom(F;Db_M) is soft for anyR- constructible sheafF.

Let us denote by C¹_M the sheaf ofC¹-functions and let Zbe aclosed subset ofM. One denotes byI¹_M;Zthe sheaf ofC¹-functions onMvanishing up to infinite order onZ.

DEFINITION2.1.1. A Whitney function on a closed subset Z of M is an indexed family Fˆ(F^k)_k2Nⁿconsisting of continuous functions on Z such that8m2N,8k2Nⁿ,jkj m,8x2Z,8e>0there exists a neighborhood U of x such that8y;z2U\Z

F^k(z) X

jj‡kjm

(z y)^j

j! F^j‡k(y)

ed(y;z)^{m jkj}:

We denote by W_M;Z¹ the space of WhitneyC¹-functions on Z. We denote by W¹_M;Zthe sheaf U7!W_U;U\Z¹ .

In [9] the authors defined the functor

w C¹_M:ModR-c(CM)!Mod(DM)

in the following way: letUbe asubanalytic open subset ofMandZˆMnU.

Then C_U ^w C¹_Mˆ I¹_M;Z, a nd C_Z^w C¹_Mˆ W¹_M;Z. This functor is exact and extends as a functor in the derived category, fromD^b_R-c(C_M) to D^b(D_M).

Moreover the sheafF^w C¹_Mis soft for anyR-constructible sheafF.

Now let X be acomplex manifold, X^R the underlying real analytic manifold andX the complex conjugate manifold. The productXX is a complexification ofX^R by the diagonal embeddingX^R,!XX. One de- notes by O_X the sheaf of holomorphic functions and by D_X the sheaf of finite order differential operators with holomorphic coefficients. For

F2D^b_R-c(CX) one sets

THom(F;O_X)ˆRHom_DX(O_X;THom(F;Db_XR));

F^w O_X ˆRHom_D

X(O_X;F^w C¹_XR);

and these functors are called the functors of temperate and formal coho- mology respectively.

2.2 ±The Weyl algebra.

LetEbe complex vector space of finite dimensionn. Let us denote by O(E) the polynomial ring onE. We denote byD(E) the Weyl algebra onE, that is, the ring of differential operators with coefficients inO(E).

The Fourier transform ^:D(E)!D(E) and the inverse Fourier transform _:D(E)!D(E) induce isomorphisms which are defined as follows: let (z₁;. . .;z_n) and (z₁;. . .;z_n) be two systems of coordinates inE andErespectively. We have

z^{^}_i ˆ @zi and @_z^{^}_i ˆz_i: On the other hand we have

z_iˆ@_z^__i and @_ziˆz^__i:

Let us consider the subanalytic siteEsa. We denote byD(Esa) (resp.O(Esa)) the constant sheaf on Esa associated toD(E) (resp.O(E)). We denote by Mod(D(Esa)) the category ofD(Esa)-modules.

2.3 ±The sheavesDb^t_MandC^1;w_M . LetMbe a real analytic manifold.

DEFINITION2.3.1. One denotes byDb^t_M the presheaf of tempered dis- tributions on M_sadefined as follows:

U7!G(M;DbM)=G_MnU(M;DbM):

As aconsequence of the L^/ ojasievicz's inequalities [13], for U;V2Op(Msa) the sequence

0! Db^t_M(U[V)! Db^t_M(U) Db^t_M(V)! Db^t_M(U\V)!0

is exact. For each U2Op(Msa) the restriction morphism G(M;Db^t_M)! G(U;Db^t_M) is surjective and RG(U;Db^t_M) is concentrated in degree zero.

MoreoverDb^t_Mis exact on Mod_R-c(C_M), i.e. it is aquasi-injective object of Mod(C_Msa). We have the following result (see [11], Proposition 7.2.6)

PROPOSITION2.3.2. For each F2Mod_R-c(C_M)one has the isomorphism Hom(F;Db^t_M)'G(M;THom(F;Db_M)):

Now let X be acomplex manifold, X^R the underlying real analytic manifold and X the complex conjugate manifold. One defines the sheaf O^t_X 2D^b(C_Xsa) of tempered holomorphic functions as follows:

O^t_X ˆRHomr_!DX(r_!O_X;Db^t_XR):

The relation with the functor of temperate cohomology is given by the following result

PROPOSITION2.3.3. For each F2D^b_R-c(C_X)one has the isomorphism THom(F;O_X)'r ¹RHom(F;O^t_X):

Let M be a real analytic manifold. As usual we set D⁰()ˆ RHom(;C_M). Remember that an open subsetU ofX is locally cohomo- logically trivial (l.c.t. for short) ifD⁰C_U 'C_U. We consider aslight gen- eralization of the sheaf of WhitneyC¹-functions of [11].

DEFINITION2.3.4. Let F2ModR-c(CM)and let U2Op(Msa). We define the presheafC^1;w_MjF as follows:

U7!G(M;H⁰D⁰C_UF^w C¹_M):

LetU;V2Op(Msa), and consider the exact sequence 0!C_U\V !C_UC_V !C_U[V !0;

applying the functorHom(;CM)ˆH⁰D⁰() we obtain

0!H⁰D⁰C_U[V !H⁰D⁰C_UH⁰D⁰C_V !H⁰D⁰C_U\V;

applying the exact functors F, ^w C¹_M and taking global sections we obtain

0! C^1;w_MjF(U[V)! C^1;w_MjF(U) C^1;w_MjF(V)! C^1;w_MjF(U\V):

This implies thatC^1;w_MjF is asheaf onMsa. Moreover ifU2Op(Msa) is l.c.t.,