Piotr P Karwasz

Hermitian Ôa, bÕ-modules and Saito's "higher residue pairings"

Keywords: Mathematics Subject Classification (2010): 32S25, 32S40, 32S50

) the aim of this article is the study of the existence of Ôa, bÕhermitian forms on regular Ôa, bÕ-modules. We show that every regular Ôa, bÕ-module E with a non-degenerate bilinear form can be written in an unique way as a direct sum of Ôa, bÕ-modules Ei that admit either an Ôa, bÕ-hermitian or an Ôa, bÕ-anti-hermitian form or both; all three cases are equally possible with explicit examples.

As an application we extend the result in [Bel01] on the existence for all Ôa, bÕ-modules E associated with the Brieskorn module of a holomorphic function with an isolated singularity, of an Ôa, bÕ-bilinear non degenerate form on E. We show that with a small transformation Belgrade's form can be considered Ôa, bÕ-hermitian and that the result satisfies the axioms of Kyoji Saito's "higher residue pairings".

Introduction

In this article we will study the self-duality properties of Ôa, bÕ-modules and more precisely the conditions under which they admit a nondegenerate hermitian form. As such we wish to provide the reader with a short introduction to the theory of Ôa, bÕ-modules.

The Ôa, bÕ-modules were introduced by D. Barlet in [Bar93] as a formal completion of the Brieskorn module ( [START_REF] Brieskorn | Die Monodromie der isolierten Singularitäten von Hyperflächen[END_REF])

D : Ω n 1 0 df d Ω n¡1
0 associated to a holomorphic function f : C n 1 C with an isolated singularity at the origin, where we denote by Ω p 0 the germs of holomorphic p-forms in 0.

We wish to recall recall briefly the basic results about Ôa, bÕ-modules and refer the reader to the articles [Bar93] and [START_REF] Barlet | Théorie des Ôa, bÕ-modules. II. Extensions[END_REF] for further details. For a complex number λ È C and an Ôa, bÕ-module E, we call monomial of type Ôλ, 0Õ, an element x È E that satifies to the relation ax λbx. The simplest Ôa, bÕ-modules are those generated over CÖÖb×× by a monomial e λ of type Ôλ, 0Õ. These modules are called elementary and noted E λ .

Given an Ôa, bÕ-module E, a sub-CÖÖb××-module F of E closed to the multiplication by a is called sub-Ôa, bÕ-modules Since the quotient of an Ôa, bÕ-module E by a sub-Ôa, bÕ-module F is not necessarily b-torsion free, a sub-Ôa, bÕ-module F of E will be called normal, if EßF is free and hence have an induced Ôa, bÕ-module structure.

The Ôa, bÕ-modules associated to a Brieskorn module are all regular, i.e. they are sub-Ôa, bÕ-modules an Ôa, bÕ-module E satisfying aE bE (a simplepole Ôa, bÕ-module). The composition series of regular Ôa, bÕ-module satisfy the following property: Proposition 2. Let E be a regular Ôa, bÕ-module, then all its composition series are of the form:

0 E 0 ¤ ¤ ¤ E n¡1 E n E, with E i ßE i¡1 elementary Ôa, bÕ-modules E λ .
As proven in [Bar93], the quotients of two composition series of an Ôa, bÕ-module E are not unique, even if we ignore consider the permutations of the quotients.

2 The Ôa, bÕ-modules and their duality

The dual and bi-dual structure on Ôa, bÕ-modules where first introduced in [START_REF] Barlet | Théorie des Ôa, bÕ-modules. II. Extensions[END_REF] and [START_REF] Belgrade | Dualité et spectres des Ôa, bÕ-modules[END_REF] and then expanded in our thesis (cf. [START_REF] Karwasz | Self-adjoint Ôa, bÕ-modules and hermitian forms[END_REF]). We will therefore begin by giving a formal definition of the duality structures we work with.

In the spirit of the category theory we will define an Ôa, bÕ-morphism as an application ϕ : E F between two Ôa, bÕ-modules E and F , which is a morphism of the underlying CÖÖb××-modules and respects the a-structure ϕÔaxÕ aϕÔxÕ, for any element x È E. We will call ϕ an isomorphism (resp. endomorphism) of Ôa, bÕ-modules if it is bijective (resp. E = F ).

2.1 Ôa, bÕ-linear maps and dual Ôa, bÕ-modules

Let E and F be two Ôa, bÕ-modules. As defined by D. Barlet in [START_REF] Barlet | Théorie des Ôa, bÕ-modules. II. Extensions[END_REF], the CÖÖb××-module Hom CÖÖb×× ÔE, F Õ of CÖÖb××-linear maps from E to F has a natural structure of Ôa, bÕ-module provided by an operator Λ that satisfies ÔΛϕÕ ÔxÕ

a F ÔϕÔxÕÕ ¡ ϕ Ôa E xÕ , (2)
where ϕ È Hom CÖÖb×× ÔE, F Õ, x is an element of E and a E and a F are the astructures of E and F respectively. We use for this Ôa, bÕ-module the notation Hom Ôa,bÕ ÔE, F Õ.

For notation's sake we will denote a E , a F and Λ all by the letter a and to avoid the confusion that such a notation could pose we should read the expression a ¤ ϕÔxÕ as ÔΛϕÕ ÔxÕ, whereas the expression a E ÔϕÔxÕÕ will keep the conventional notation aϕÔxÕ.

We will therefore rewrite the equation 2 as: a ¤ ϕÔxÕ aϕÔxÕ ¡ ϕÔaxÕ.

By choosing E 0 for the codomain of the morphisms, we can give the following definition:

Definition 3 (Barlet). Let E be an Ôa, bÕ-module and E 0 the elementary Ôa, bÕ-module of parameter 0, then we call the module Hom Ôa,bÕ ÔE, E 0 Õ the dual Ôa, bÕ-module of E and note it by E ¦ . Remark 4. When considering only the b-structure of E, the CÖÖb××-module E ¦ corresponds exactly to the definition of dual of a CÖÖb××-module, since E 0 CÖÖb××e 0 , with ae 0 0.

The duality functor ¦ is exact (cf. [START_REF] Barlet | Théorie des Ôa, bÕ-modules. II. Extensions[END_REF]).

Conjugate Ôa, bÕ-module

In [Bel01] R. Belgrade uses another definition of dual Ôa, bÕ-module which is not equivalent to the one of D. Barlet. In order to be able to express on concept in terms of the other the other, we will introduce an operation that exchanges the signs of both a and b, whose behaviour is similar to that of conjugation in the complex field.

As in the case of the complex field C, the ring of formal series CÖÖb×× also admits a rather natural involution

˘: CÖÖb×× CÖÖb×× SÔbÕ SÔbÕ SÔ¡bÕ,
where SÔbÕ È CÖÖb××. This remark allows us to define the conjugate of an Ôa, bÕ-module in the same way as one defines the conjugate of a complex vector space.

Definition 5. Let E be an Ôa, bÕ-module. We call Ôa, bÕ-conjugate of E and note it Ȇ the set E itself, endowed with an a-and b-structure given by:

a ¤ Ȇ v ¡a ¤ E v b ¤ Ȇ v ¡b ¤ E v,
where ¤ Ȇ and ¤ E denote the Ôa, bÕ-structure of Ȇ and E respectively.

Since we change signs of both a and b, the formula ab ¡ ba b 2 is still satisfied.

Remark 6. An Ôa, bÕ-module is not necessarily isomorphic to its conjugate. We can take, for example, the Ôa, bÕ-module of rank 2 generated by two elements x and y that satisfy:

ax λbx ay λby Ô1 αbÕ x, where λ and α È C and α 0. Its conjugate satisfies ax λbx ay λby Ô1 ¡ αbÕ x, and the classification of rank 2 regular Ôa, bÕ-modules, given in [Bar93] implies that the two modules are not isomorphic.

One can see immediately that for an Ôa, bÕ-module E the conjugate of the conjugate ¡ Ȇ© ˘is the Ôa, bÕ-module itself. On the other hand let E and F be Ôa, bÕ-modules and ϕ a morphism between E and F . Since ϕÔ¡axÕ ¡aϕÔxÕ and ϕÔ¡bxÕ ¡bϕÔxÕ, for all x È E the application ϕ is also a morphism between the conjugates Ȇ and F . We call conjugation functor the functor that associates to every Ôa, bÕ-module its conjugate and to every morphism, the morphism itself. Such a functor is exact.

For an Ôa, bÕ-module module E we will be interested especially in a particular kind of conjugate, the conjugate of the dual, which we call adjoint Ôa, bÕ-module and note with Ȇ¦ .

Bilinear forms and tensor product

In order to define Hom Ôa,bÕ ÔE, F Õ we used the equivalent object for its underlying b-structure. We can proceed in a similar way to obtain the concept of Ôa, bÕ-bilinear maps: Definition 7. Let E, F and G be two Ôa, bÕ-modules. An Ôa, bÕ-bilinear map on E ¢ F is a CÖÖb××-linear map Φ, Φ : E ¢ F G, that satisfies the following property:

aΦÔx, yÕ ΦÔax, yÕ ΦÔx, ayÕ.

Remark 8. If Φ is an Ôa, bÕ-bilinear map on E ¢ F with values in G and v is an element of E:

Φ v : ΦÔv, ¤Õ : w ΦÔv, wÕ w È F is not necessarily an Ôa, bÕ-morphism. However the map π : v Φ v is an Ôa, bÕ-morphism between E and Hom Ôa,bÕ ÔF, GÕ. We have in fact:

πÔavÕÔxÕ Φ av ÔxÕ aΦ v ÔxÕ ¡ Φ v ÔaxÕ a ¤ Φ v ÔxÕ aπÔvÕ.
Inherently linked to the concept of Ôa, bÕ-bilinear maps is that of tensor products, that allows a more practical manipulation of these objects. Definition 9. Let E and F be two Ôa, bÕ-modules. We call Ôa, bÕ-tensor product of E and F and write it as E Ôa,bÕ F the CÖÖb××-module E CÖÖb×× F endowed with an a-structure defined as follows:

a Ôv wÕ ÔavÕ w v ÔawÕ for every v È E and w È F .

The a-structure we gave on E Ôa,bÕ F is well defined. We have in fact:

a Ôbv wÕ aÔbvÕ w bv aÔwÕ baÔvÕ w b 2 v w v baÔwÕ aÔvÕ bw v aÔbwÕ a Ôv bwÕ , for each v È E, w È F and it satisfies ab ¡ ba b 2 :

a Ôbv wÕ ¡ ba Ôv wÕ baÔvÕ w b 2 v w bv aÔwÕ ¡ baÔvÕ w ¡ bv aÔwÕ b 2 Ôv wÕ .

We can easily verify that the tensor product defined satisfies the usual universal property: there exists a bilinear map Φ : E ¢ F E Ôa,bÕ F, such that for every bilinear map Ψ on E ¢F with values in a third Ôa, bÕ-module G, there exists an unique Ôa, bÕ-morphism Ψ from E Ôa,bÕ F into G that makes the following diagram commutative:

E ¢ F Φ Ψ / / G E Ôa,bÕ F. Ψ : : u u u u u
We can take as Φ the natural application Φ : E ¢ F E Ôa,bÕ F Ôv, wÕ v Ôa,bÕ w and define Ψ as:

Ψ : E Ôa,bÕ F G v Ôa,bÕ w ΨÔv, wÕ
The unicity of Ψ follows directly from the universal property of the tensor product of CÖÖb××-modules. We need only to verify that the map is a-linear. We will do it on the generators v Ôa,bÕ w of E Ôa,bÕ F , for v È E and w È F :

Ψ aÔv Ôa,bÕ wÕ ¨ Ψ ÔavÕ Ôa,bÕ w v Ôa,bÕ ÔawÕ ¨ ΨÔav, wÕ ΨÔv, awÕ aΨÔv, wÕ a ΨÔv Ôa,bÕ wÕ.

Basing ourselves on the properties of the tensor product of CÖÖb××-modules, we can derive in a similar manner the other properties of the equivalent object in the theory Ôa, bÕ-modules.

Lemma 10. Let E, F and G be three Ôa, bÕ-modules, then the tensor product verifies the following properties:

(i) E Ôa,bÕ F F Ôa,bÕ E, (ii) E Ôa,bÕ F ¨ Ôa,bÕ G E Ôa,bÕ F Ôa,bÕ G ¨, (iii) E Ôa,bÕ F ¨¦ E ¦ Ôa,bÕ F ¦ , (iv) E Ôa,bÕ F ¨˘ Ȇ Ôa,bÕ F , (v)
The Ôa, bÕ-morphism

Φ : E E Ôa,bÕ E 0 v v Ôa,bÕ e 0
where e 0 is a generator of the elementary Ôa, bÕ-module E 0 , is an isomorphism.

(vi) We have the following isomorphism of Ôa, bÕ-modules:

E ¦ Ôa,bÕ F Hom Ôa,bÕ E, F Ôa,bÕ E 0 φ Ôa,bÕ y Φ : x y Ôa,bÕ ϕÔxÕ ¨,
where ϕ È E ¦ , x È E and y È F .

Remark 11. In [START_REF] Belgrade | Dualité et spectres des Ôa, bÕ-modules[END_REF], R. Belgrade defines the concept of δ-dual of an

Ôa, bÕ-module E:

Definition 12 (Belgrade). Let E be an Ôa, bÕ-module and δ È C, then we call the δ-dual of E the set Hom CÖÖb×× ÔE, E δ Õ with the Ôa, bÕ-structure defined as follow:

Öa ¤ ϕ×ÔxÕ ϕÔaxÕ ¡ aϕÔxÕ (3) Öb ¤ ϕ×ÔxÕ ¡bϕÔxÕ ϕÔ¡bxÕ (4)
From property (v) and (vi) of the previous lemma we obtain the isomorphism E ¦ Ôa,bÕ F Hom Ôa,bÕ ÔE, F Õ, which in turn let us find an alternative description of the δ-dual of an Ôa, bÕ-module. In fact from definition 12 it is easy to show that the δ-dual of an Ôa, bÕ-module in Belgrade's terminology is the module

Hom Ôa,bÕ Ě, E δ ¨,
which in turn can be rewritten as Ȇ¦ Ôa,bÕ E δ .

We will call an Ôa, bÕ-bilinear application on E ¢ F with values in G, an Ôa, bÕ-bilinear form if G E 0 . In the rest of this section we will deal with the existence of nondegenerate hermitian forms on Ôa, bÕ-modules. We will need therefore the following definitions.

Definition 13. Let E and F be two Ôa, bÕ-modules and Φ a bilinear form on E ¢ F . We say that Φ is nondegenerate, if the Ôa, bÕ-morphism v ΦÔv, ¤Õ is an isomorphism of E with F ¦ . Definition 14. Let E be an Ôa, bÕ-module. A sesquilinear form on E is a bilinear form on E ¢ Ě. Remark 15. Since a nondegenerate sequilinear form on an Ôa, bÕ-module E induces an isomorphism to its adjoint Ȇ¦ it follows that not all Ôa, bÕ-modules are self-adjoint (e.g. E λ with λ 0 is not) not every Ôa, bÕ-module admits a nondegenerate sesquilinear form.

Consider now a sesquilinear form Φ on E. By applying to it the conjugate functor we obtain a bilinear map Φ on Ě ¢ E with values in Ě0 . If we fix an isomorphism of Ě0 with E 0 , we can consider Φ as a sequilinear form on Ě. Under this assumption, we define Ôa, bÕ-hermitian and anti-Ôa, bÕ-hermitian forms as: Definition 16. Let E be an Ôa, bÕ-module. An Ôa, bÕ-sesquilinear form H on E is called Ôa, bÕ-hermitian (respectively anti-Ôa, bÕ-hermitian) if it satisfies:

HÔv, wÕ ȞÔw, vÕ, ( respectively
HÔv, wÕ ¡ HÔw, vÕ © .
where v È E, w È Ȇ and H is the sesquilinear form on Ě defined above.

We have already shown that in order to admit a nondegenerate sesquilinear form, an Ôa, bÕ-module must be self-adjoint. We will refine the concept of selfadjoint by defining:

Definition 17. Let E be a self-adjoint Ôa, bÕ-module. We say that E is hermitian (resp. anti-hermitian), if it admits a nondegenerate hermitian (resp. anti-hermitian) form.

Let E be an Ôa, bÕ-module endowed with an hermitian form and let Φ :

E Ȇ¦ be the linear form associated to the hermitian form via the remark 8. We can translate the hermitian property into the identity between Φ and its adjoint Φ¦ : E Ȇ¦ . In fact while ΦÔvÕ, for v È E is the linear map: ϕ : w ΦÔv, wÕ, w È Ě, the adjoint map Φ¦ sends an element v È E E ¦¦ to the map:

ϕ : w v ΦÔw, ¤Õ ¨ ΦÔw, vÕ.
We will use this formulation extensively in the following section.

Note moreover that given an isomorphism Φ from an Ôa, bÕ-module E and its δ-dual Ȇ¦ Ôa,bÕ E δ is equivalent to specifying an isomorphism between E Ôa,bÕ E ¡δß2 and Ȇ¦ Ôa,bÕ E δ Ôa,bÕ E ¡δß2 Ȇ¦ Ôa,bÕ E δß2 .

Since we have

E Ôa,bÕ E ¡δß2 ¨¦ Ȇ¦ Ôa,bÕ Ě¦ ¡δß2 Ȇ¦ Ôa,bÕ E δß2 ,
we can identify an isomorphism of E with its δ-dual with an hermitian form on E Ôa,bÕ E ¡δß2 .

Existence of hermitian forms

We will analyze in this section the existence of nondegenerate hermitian forms on regular Ôa, bÕ-modules. We will proceed in two steps: in the first two subsections we will reduce ourselves to a subclass of Ôa, bÕ-modules called indecomposable Ôa, bÕ-modules and show that every regular Ôa, bÕ-module can be decomposed into the direct sum of indecomposable ones and that this decomposition is unique (theorem 24).

In the last subsection we will show that a self-adjoint Ôa, bÕ-module which is indecomposable admits at least an hermitian or anti-hermitian form. The result is optimal since there are examples that admit only an hermitian or only an anti-hermitian form (theorem 30).

3.1 Indecomposable Ôa, bÕ-modules Definition 18. Let E be an Ôa, bÕ-module. We say that E is indecomposable if it cannot be written as direct sum F G of non zero Ôa, bÕ-modules.

Whenever we decompose an Ôa, bÕ-module E into a direct sum of two Ôa, bÕ-modules E F G the rank of the components is strictly less than the rank of E, hence by proceeding by induction for every Ôa, bÕ-module E we can find a decomposition into a sum of indecomposable Ôa, bÕ-modules:

E r i 1 F i ,
where r È N and F i are indecomposable sub-Ôa, bÕ-modules.

We are interested in the question whether the isomorphism classes of the F i are unique and do not depend upon the decomposition. We will need to this purpose an introductory result: Proposition 19. Let E be a regular and indecomposable Ôa, bÕ-module. Then every endomorphism of E is either bijective or nilpotent.

The proof of this proposition will need several steps beginning with a definition: Definition 20. Let E be a regular Ôa, bÕ-module and λ È C. We define

V λ ô F i F i E, F i E λ µ to be the sum of all sub-Ôa, bÕ-modules of E isomorphic to E λ .
The object V λ is clearly a sub-Ôa, bÕ-module. We will use V λ as an induction step in the proof of proposition 19, by choosing a λ such that V λ is normal:

Proposition 21. Let E be a regular Ôa, bÕ-module, λ È C and: λ min inf j Øλ j x È E, ax Ôλ jÕbxÙ be the minimal λ j such that E contains a monome of type Ôλ j, 0Õ.

Then V λmin is a normal sub-Ôa, bÕ-module of E isomorphic as Ôa, bÕ-module to the direct sum of a finite number of copies of E λmin .

Proof. We will use two facts. First, for every W ü

E λmin sub-Ôa, bÕ-module of E, W is normal in E.
Let in fact Øe i Ù be a basis of W with 1 i p the rank of W . Suppose by absurd that there exist some x È W which is in bE, but not in bW .

By eventually translating x by an element of bW , we can assume x p i 1 α i e i , α i È C. We can easily verify that ax λ min bx but now if x by we must have ay Ôλ min ¡ 1Õby, and since y È E it contradicts the minimality of λ min .

On the other hand we can show that V λmin is a direct sum of E λmin . In fact let W be the largest (inclusionwise) direct sum of copies of E λmin included in V λmin . We remark that since W is normal, for any sub-Ôa, bÕ-module F isomorphic to E λmin only one of two cases is possible: either

W F Ø0Ù or F W.
If W F Ø0Ù, let e be the generator of F and SÔbÕb n e È W with SÔ0Õ 0, then SÔbÕe È W by normality and e S ¡1 ÔbÕSÔbÕe È W . We have therefore

F W .
If W contains every sub-Ôa, bÕ-module isomorphic to E λmin , then it is equal to V λmin . Otherwise there is an F such that W F Ø0Ù, hence W F is still in V λmin , which contradicts the maximality of W .

We will now use the sub-Ôa, bÕ-module V λmin to show the following proposition Proposition 22. Let E be a regular Ôa, bÕ-module and ϕ an Ôa, bÕ-morphism between E and itself. Then ϕ is bijective if and only if ϕ is injective.

Proof. To show that bijectivity follows from injectivity, we will proceed by induction on the rank of the module.

If E is of rank 1 the statement of the proof is satisfied: in fact E must be isomorphic to one of the E λ and the only b-linear morphisms from a E λ to itself that are also a-linear are those that send the generator e to αe, α È C. They are all bijective for α 0.

Let now E be of rank n 1. We can find a λ min (cf. [Bar93]) that verifies the minimality property of the previous proposition. Hence the module V λmin is normal and isomorphic to a direct sum of copies of E λmin .

Let Øe i Ù be a basis of V λmin composed of monomials of type Ôλ min , 0Õ and let x another monomial of type Ôλ min , 0Õ. We want to show that x is a linear combination of the elements of the basis, with coefficients in C CÖÖb××.

From the definition of V λmin follows that x È V λmin . Suppose now that x i S i ÔbÕe i and let us apply a to both sides. We obtain:

ax ô i λ min S i ÔbÕbe i S ½ i ÔbÕb 2 e i ¨ λ min bx ô i S ½ i ÔbÕb 2 e i
and since x is a monomial of type Ôλ min , 0Õ, we must have S ½ i ÔbÕ 0 for all i and therefore x

ô i S i Ô0Õe i ,
as we wanted.

Let ϕ : E E be an injective endomorphism of E and Øe i Ù a basis of V λmin . Every ϕÔe i Õ is a monomial of type Ôλ min , 0Õ and therefore is an element of V λmin .

The restriction of ϕ to V λmin is therefore an endomorphism of V λmin :

ϕ V λ min : V λmin V λmin .
Moreover since the coefficients of the ϕÔe i Õ in our base are complex constants, ϕ V λ min behaves as a linear application between finite dimensional spaces: in particular if it is injective, it is also surjective.

In order to apply our induction hypothesis let us consider the following commutative diagram:

0 / / V λmin / / ϕ E / / / / ϕ EßV λmin / / φ 0 0 / / V λmin / / E / / / / EßV λmin / / 0
where φ is the Ôa, bÕ-linear morphism induced on the quotient. As we showed the first downward arrow is bijective The third arrow φ is injective: suppose in fact that we have two different classes with representatives x and y È E that map to the same class modulo V λmin . Then ϕÔx ¡ yÕ is in V λmin . From the bijectivity of ϕ V λ min we can find an element v È V λmin such that ϕÔx ¡ yÕ ϕÔvÕ which in turn implies x ¡ y v by the injectivity of ϕ, which contradicts the fact that x and y are in distinct classes modulo V λmin .

Since the rank of EßV λmin is strictly inferior to the rank of E, we can apply the induction hypothesis to show that φ is also bijective.

It follows from a basic result of homological algebra that the second arrow is bijective if it is injective.

We can now consider endomorphisms that are not necessarily injective. Once again the structure of Ôa, bÕ-modules does not differ essentially from that of finite vector spaces over C: Lemma 23. Let E be a regular Ôa, bÕ-module and ϕ an endomorphism of E.

Then E splits into the direct sum of two ϕ-stable sub-Ôa, bÕ-modules F and N , with ϕ bijective on F and nilpotent on N .

Proof. Consider the sequence of normal sub-Ôa, bÕ-modules

K n Ker ϕ n , n È N.
Since two normal sub-Ôa, bÕ-modules F G are equal if and only if they have the same rank, the sequence of K n stabilizes beginning with a certain integer m: K m K m 1 .

On the other hand if we consider the sequence I n Im ϕ n , let us look at the restriction of ϕ to I m :

ϕ Im : I m I m 1 I m .
This restriction is injective: if y ϕ m ÔxÕ È Ker ϕ, then x È K m 1 which is equal to K m . Hence ϕ m ÔxÕ y 0. From the previous proposition we deduce that this restriction is in fact bijective, which means that I m 1 ϕÔI m Õ I m .

We can now take F I m and N K m . They are clearly stable by ϕ. We will show that E F N .

We have in fact Ker ϕ F Ø0Ù, since the restriction of ϕ to I m is injective. À fortiori, since K Ker ϕ we have F N Ø0Ù.

Let's take an element x È E. Since I m I 2m we can find an element y È E such that ϕ m ÔxÕ ϕ 2m ÔyÕ and call k the element x ¡ϕ m ÔyÕ. Thus we can write x as a sum:

x ϕ m ÔyÕ k of an element ϕ m ÔyÕ È I m and an element k È K m , which implies that:

E N F.
The restriction of ϕ to N is nilpotent, since ϕ m N 0, while we already showed that the restriction to I m F is bijective.

We have now all the elements necessary to prove proposition 19:

Proof. Let E be a regular indecomposable Ôa, bÕ-module and ϕ an endomorphism of E. Then by lemma 23 E splits into a sum E N F of two Ôa, bÕ-modules, with ϕ nilpotent on N and bijective on F . But E is indecomposable, therefore either N 0 and ϕ is bijective or F 0 and ϕ is nilpotent.

Krull-Schmidt theorem

This subsection will be devoted to the proof of a version of the Krull-Schmidt theorem for the theory of Ôa, bÕ-modules. The principal argument of the proof will be proposition 19 from the previous subsection.

Theorem 24 (Krull-Schmidt for Ôa, bÕ-modules). Suppose that we have two decompositions into direct sum of a regular Ôa, bÕ-module E:

E m i 1 E i E n i 1 F i
where m, n È N and all E i and F i are indecomposable Ôa, bÕ-modules. Then m n and up to a reindexing of the modules E i is isomorphic to G i for all 1 i n.

For the proof of this theorem we need a couple of lemmas: Lemma 25. Let E be a regular indecomposable Ôa, bÕ-module and ϕ an automorphism of E. Suppose moreover that ϕ ϕ 1 ϕ 2 . Then at least one of ϕ 1 , ϕ 2 is an isomorphism.

Proof. Be applying ϕ ¡1 to both terms, we can assume without loss of generality that ϕ Id is the identity automorphism.

The two endomorphisms ϕ 1 and ϕ 2 commute. In fact:

ϕ 1 ϕ 2 ¡ ϕ 2 ϕ 1 ϕ 1 Ôϕ 1 ϕ 2 Õ ¡ Ôϕ 2 ϕ 1 Õϕ 1 ϕ 1 ¡ ϕ 1 0.
By lemma 19 the ϕ i can be either nilpotent or isomorphisms. If they were both nilpotent, their sum would be nilpotent, which is absurd. Hence the result.

Remark 26. By subsequently applying the previous lemma, we can extend the result to the sum of more than two endomorphisms.

Lemma If m 1, then E is indecomposable and we must have n 1 and E 1 F 1 .

In the general case consider the morphisms

q i π i ¥ p 1 ,
where the π i s are the projections on F i and the p j s are the projections on E j . Let consider the sum:

ô i p 1 ¥ q i p 1 ¥ ô i π i ¥ p 1 p 1 ¥ p 1 p 1 ,
is the identity on the component E 1 . By the lemma 3.2, there is an i such that p 1 ¥ q i E1 : E 1 E 1 is an isomorphism. Suppose, without loss of generality, it is p 1 ¥ q 1 , then by the lemma 27 q 1 E1 π 1 : E 1 F 1 is an isomorphism.

In order to apply the induction hypothesis, let note G m i 2 F i . We want to show that E 1 G is equal to E F 1 G. Since π 1 is an isomorphism of E 1 onto F 1 and its kernel is G we must have

E 1 G Ø0Ù : if x È E 1 G, then π 1 ÔxÕ 0, but π 1 restricted to E 1 is injective, so x 0.
On the other hand every element of E can be written as v w with v È F 1 and w È G. If y È E 1 is such that π 1 ÔyÕ v, then we have: v w y π 1 ÔyÕ ¡ y w, and π 1 ÔyÕ¡y È W by definition of π 1 . We can then conclude that

E E 1 G E 1 m i 2 E i .
We have immediately EßE 1 G m i 2 E i and we can apply the induction hypothesis to G.

We can now focus on finding hermitian isomorphisms of an Ôa, bÕ-module E with its adjoint Ȇ¦ . The Krull-Schmidt theorem will be useful to show the following decomposition: Proposition 28. Let E be a regular self-adjoint Ôa, bÕ-module. Then E is isomorphic to:

E r i 1 F αi i ¨ s i 1 G i Ǧ¦ i ¨ βi
where r and s as well as the α i and β i are positive integers. The F i are selfadjoint Ôa, bÕ-modules and the G i are non self-adjoint Ôa, bÕ-modules. The isomorphism classes of the F i , G i and Ǧ¦ i are all disjoint.

Proof. Consider a decomposition of E into indecomposable Ôa, bÕ-modules

E ô i E i .
Since E is self-adjoint we have another decomposition given by

E ̦ ô i ̦ i .
The Krull-Schmidt theorem assures us that the factors are unique up to a permutation. So we can divide the E i into two groups.

In the first group we find the self-adjoint components F i with a certain multiplicity.

In the second one we find the non self-adjoint components G i with the respective multiplicity. Since the two decompositions i E i and i ̦ i must contains the same modules up to a permutation, the multiplicity of the G i and the Ǧ¦ i must be equal.

Remark 29. From the definition above we can immediately see that the non self-adjoint part of the decomposition always admits a hermitian nondegenerate form In fact if we consider the module G i Ǧ¦ i , a hermitian form can be given by:

Φ : G i Ǧ¦ i ÔG i Ǧ¦ i Õ ¦ ˇ Ǧ¦ i G i
Ôx, yÕ Ôy, xÕ.

13

If the multiplicity of a self-adjoint term F i is pair, we fall into the same situation. The case of an unpair multiplicity of a self-adjoint component is far more interesting and we will study it in the next subsection.

Hermitian forms on indecomposable Ôa, bÕ-modules

As already noted in the previous subsection, the situation of an indecomposable self-adjoint Ôa, bÕ-module concerning hermitian forms is far less regular and the existence is not always guaranteed. We have in fact the following theorem: Theorem 30. Let E be a regular indecomposable self-adjoint Ôa, bÕ-module and E Ø0Ù. Then it admits a hermitian nondegenerate form or an anti-hermitian one.

Proof. Let Φ : E ̦ be any isomorphism of E with its dual and pose M Φ ¡1 Φ¦ . Consider now the two endomorphisms of E given by:

Id M
and Id ¡ M they commute and can be either isomorphisms or nilpotent, since E is indecomposable. But if they were both nilpotent, their sum 2Id would be nilpotent too, which is absurd.

If Id M is an isomorphism, so is S Φ Φ¦ , which is associated to a nondegenerate hermitian form. The bijectivity of Id ¡ M on the other hand gives us an isomorphism A Φ ¡ Φ¦ , which comes from an anti-hermitian form.

Note that all the cases of the previous theorem are equally possible.

Example 31. The simplest example of a regular self-adjoint and indecomposable Ôa, bÕ-module which admits only a hermitian form is the elementary Ôa, bÕ-module E 0 with the isomorphism that sends the generator e to its adjoint ě¦ .

Example 32. In order to obtain only an anti-hermitian form, we can consider for a given λ, µ È C the Ôa, bÕ-module E of rank 4, generated by Øe 1 , e 2 , e 3 , e 4 Ù which verifies:

ae 1 λbe 1 ae 2 µbe 2 e 1 ae 3 ¡µbe 3 e 1 (5)
ae 4 ¡λbe 4 e 2 ¡ e 3 whose adjoint basis satisfies:

a ¤ ě¦ 4 λbě ¦ 4 a ¤ ě¦ 3 µbě ¦ 3 ¡ ě¦ 4 a ¤ ě¦ 2 ¡µbě ¦ 2 ě¦ 4 a ¤ ě¦ 1 ¡λbě ¦ 1 ě¦ 3 ě¦ 2 .
It is easy to show by calculation that the only isomorphism between E and Ȇ¦ is, up to mutliplication by a complex number, the one that sends e 1 , e 2 , e 3 and e 4 into ě4 , ¡ě 3 , ě2 and ¡ě 1 respectively. This is isomorphism is anti-hermitian and since there are no other isomorphisms E is also indecomposable.

Example 33. The regular Ôa, bÕ-module E 0 E 0 admits both an hermitian and anti-hermitian form.

Duality of geometric Ôa, bÕ-modules

In the study of the Brieskorn lattice K. Saito introduced the concept of "higher residue pairings" (cf. [START_REF] Saito | The higher residue pairings K ÔkÕ F for a family of hypersurface singular points[END_REF]), which can be defined using a set of axiomatic properties.

Using the theory of Ôa, bÕ-modules R. Belgrade showed the existence of a duality isomorphism between an Ôa, bÕ-module associated to a germ of a holomorphic function in C n 1 with an isolated singularity at the origin and its Ôn 1Õ-dual. In this section we'll prove (as already noticed by R. Belgrade in [START_REF] Belgrade | Dualité et spectres des Ôa, bÕ-modules[END_REF]) that the concept of "higher residue pairings" and self-adjoint Ôa, bÕ-module are linked.

In this section D will always denote the Brieskorn module associated to a holomorphic function in C n 1 with an isolated singularity, while E will denote its b-adic completion considered as an Ôa, bÕ-module.

The following theorem of R. Belgrade gives a relationship between E and its Ôn 1Õ-dual.

Theorem 34 (Belgrade). Let E be the Ôa, bÕ-module associated to a germ of holomorphic function f : C n 1 C, then there is a natural isomorphism between E and its Ôn 1Õ-dual:

∆ : E Ȇ¦ Ôa,bÕ E n 1
We can obtain from this isomorphism a series ∆ k : E ¢ E C of bilinear forms defined as follow:

Ö∆ÔyÕ× ÔxÕ Ôn 1Õ! ô k 0 ∆ k Ôx, yÕb k e n 1
with x, y È E.

5 "Higher residue pairings" of K. Saito K. Saito introduced in [Sai83] a series of pairings on the Brieskorn lattice D which are called "higher residue pairings":

K ÔkÕ : D ¢ D C k È N
which are characterized by the following properties:

(i) K ÔkÕ Ôω 1 , ω 2 Õ K Ôk 1Õ Ôbω 1 , ω 2 Õ ¡K Ôk 1Õ Ôω 1 , bω 2 Õ. (ii) K ÔkÕ Ôaω 1 , ω 2 Õ ¡ K ÔkÕ Ôω 1 , aω 2 Õ Ôn kÕK Ôk¡1Õ Ôω 1 , ω 2 Õ.
(iii) K Ô0Õ satisfies K Ô0Õ ÔD, bDÕ K Ô0Õ ÔbD, DÕ 0 and induces Grothendieck's residue on the quotient DßbD.

(iv) K ÔkÕ are Ô¡1Õ k -symmetric.

Remark 35. We notice that from the properties (i) and (iii) above we can deduce that K ÔkÕ ÔD, b k 1 DÕ K ÔkÕ Ôb k 1 D, DÕ 0, so we can consider the pairings K ÔkÕ as being defined on Dßb k 1 D.

In the following section we'll show the following result:

Proposition 36. The ∆ k verify the properties (i)-(iii) of the "higher residue pairings" of K. Saito.

The prove will be performed by steps.

6 Proof of the proposition 6.1 Proof of (i)

We use the b-linearity of ∆ÔyÕ to obtain:

ô k Ôn 1Õ!∆ k Ôbx, yÕb k e n
Ôn 1Õ! ô k ∆ k Ôx, byÕb k e n 1 ∆ÔbyÕÔxÕ ¡b∆ÔyÕÔxÕ Ôn 1Õ! ô k ¡∆ k Ôx, yÕb k 1 e n 1 ,
which implies ∆ k Ôbx, yÕ ¡∆ k 1 Ôx, byÕ.

Proof of (ii)

Since ∆ is an isomorphism we have ∆ÔayÕ a ¤ ̦ E δ Ö∆ÔyÕ× and:

Ôn 1Õ! ô k ∆ k Ôx, ayÕb k e n 1 ∆ÔayÕÔxÕ a ¤ Ö∆ÔyÕ×ÔxÕ ∆ÔyÕÔaxÕ ¡ aÖ∆ÔyÕÔxÕ× Ôn 1Õ! ô k ∆ k Ôax, yÕb k e n 1 ¡ ∆ k Ôx, yÕab k e n 1
The definition of Ôa, bÕ-module and E n 1 (ae n 1 Ôn 1Õbe n 1 ) gives the following relation

ab k e n 1 b k ae n 1 kb k 1 e n 1 Ôn k 1Õb k 1 e n 1
hence follows:

∆ k Ôax, yÕ ¡ ∆ k Ôx, ayÕ Ôn kÕ∆ k¡1 Ôx, yÕ

Grothendieck's residue

We have to show now that the pairing ∆ 0 induces Grothendieck's residue on

DßbD Ω n 1 ßdf Ω n .
Proof of (iv): From the definition of ∆ 0 and the b-linearity of ∆ it's easy to see that ∆ 0 ÔD, bDÕ ∆ 0 ÔbD, DÕ 0. We can hence consider ∆ 0 as a pairing on DßbD. Grothendieck's residue is defined as follows:

ResÔg, hÕ : lim

εj 0,j ÷ f ß zj εj gh dz f ß z 1 ¤ ¤ ¤ f ß z n 1 where g, h È O and dz dz 1 . . . dz n 1 .
The morphism ∆ is defined as composed morphism of six Ôa, bÕ-modules morphism ( [START_REF] Belgrade | Dualité et spectres des Ôa, bÕ-modules[END_REF]) as showed by the following graph:

E α / / F 1 β / / F 2 F 3 γ ÀA A A A A A A δ ~~~~~~~Ẽ n 1 F 5 ζ o o F 4 η o o
These morphisms pass to the quotient by the action of b in order to give a decomposition of the morphism ∆ 0 :

EßbE α / / F 1 ßbF 1 β / / F 2 ßbF 2 F 3 ßbF 3 γ d d J J J J J J J J J δ { { w w w w w w w w w w Ô Ȇ¦ En 1Õ bÔ Ȇ¦ En 1Õ F 5 ßbF 5 ζ o o F 4 ßbF 4 η o o
We have to verify that the image of Ög dz× by ∆ 0 is ResÔg, ¤Õ, where g dz is an element of Ω n 1 . We'll accomplish this in many steps using the decomposition above.

(i)

Step 1: E, F 1 and F 2 . We have the following isomorphisms:

F 1 bF 1 Ω n 1 df Ω n F 2 bF 2 Db n 1 Ô ¯ ¡ df ÕDb n ,
the morphism α coincides with the identity on Ω n 1 ßdf Ω n and β is induced by the inclusion i : Ω n 1

Db n 1 . We deduce that β¥ αÔÖg dz×Õ ÖiÔg dzÕ×. Let write T È Db n 1,0 the current iÔg dzÕ.

(ii)

Step 2: path between F 2 and F 3 By using the description of the lemma 3.4.2 of [START_REF] Belgrade | Dualité et spectres des Ôa, bÕ-modules[END_REF] we see that:

F 3 bF 3 KerÔDb 0,n 1 df Db 1,n 1 Õ ¯ KerÔDb 0,n df Db 1,n Õ
and the isomorphism γ is induced by the inclusion Db 0,n 1 Db n 1 . In order to find S : γ¡1 ÔTÕ we have to solve the following system:

T df α n,0 ¯ α n,0 df α n¡1,1 ¤ ¤ ¤ ¤ ¤ ¤ ¯ α 1,n¡1 df α 0,n ¯ α 0,n S
where the α p,q È Db p,q . There is a solution to this system of equations since the complex ÔDb ,q ; df Õ is acyclic in degree 0 for all q in 0, . . . , n 1 and the solution verifies ÖS× ÖT× where Ö¤× is the equivalence class in

F 2 ßbF 2 . Ô ¯ ¡ df Õ n ô k 0 α k,n¡k ¯ α 0,n ¡ df α n,0 S ¡ T
We can compute this solution explicitly. Let be Ôp, qÕ È N 2 and ϕ p,q a C test form with compact support and of type Ôp, qÕ. The action of T over ϕ 0,n 1 is given by: T, φ 0,n 1 ÷ φ 0,n 1 g dz then the following current verifies T df α n,0 :

α n,0 , φ 1,n 1 lim ε1 0 ÷ 1f ǫ1 φ 1,n 1 g dz 2 . . . dz n 1 1 f in fact: df α n,0 , φ 0,n 1 lim ε1 0 ÷ 1 f ǫ1 φ 0,n 1 df g dz 2 . . . dz n 1 1 f ÷ φ 0,n 1 g dz
and thanks to the Stokes' theorem:

¯ α n,0 , φ 1,n ¡ α n,0 , ¯ φ 1,n lim ε1 0 ¡ ÷ 1 f ǫ1 ¯ φ 1,n g dz 2 . . . dz n 1 1 f lim ε1 0 ÷ 1 f ǫ1 φ 1,n g dz 2 . . . dz n 1 1 f
We'll remark that the currents α n,0 k defined below for 1 k n 1 also satisfy df α n,0 k T :

α n,0 k , φ 1,n 1 lim ε k 0 ÷ k f ǫ k Ô¡1Õ k 1 φ 1,n 1 g dz 1 . . . Ý dz k . . . dz n 1 k f and that Ö ¯ α n,0 × Ö ¯ α n,0 k × in F 2 ßbF 2 : in fact Ô ¯ ¡ df ÕÔα n,0 ¡ α n,0 k Õ ¯ α n,0 ¡ ¯ α n,0 k .
For all k È 0, . . . , n and 1 i 1 . . . i k 1 n 1 let us define:

α n¡k,k i1,...,i k 1 1 Ôk 1Õ! lim ǫi q 0 1 q k 1 ÷ i 1 f ǫi 1 iq f ǫi q Ô¡1Õ Ô q iqÕ 1 g é l i1,...,i k 1 dz l i1 f . . . i k 1 f and let α n¡k,k : α n¡k,k
1,2,...,k 1 . A simple computation gives us:

df α n¡k,k i1,...,i k 1 , ϕ k,n¡k 1 1 k 1 k 1 ô q 1 ¯ α n¡k 1,k¡1 i1,..., Ô iq,...,i k 1 , ϕ k,n¡k 1
using this formula, we can prove by induction on k that the class of the current α n¡k,k i1,...,i k 1 doesn't depend upon the i q s. This gives us df α n¡k,k ¯ α n¡k 1,k¡1 .

In particular ¯ α 0,n acts upon the test function ϕ n 1,0 in the following way:

¯ α 0,n , φ n 1,0 1 Ôn 1Õ! lim ǫ k 0 k ÷ k f ǫ k k φ n 1,0 g 1 f . . . n 1 f (iii)
Step 3 from F 3 ßbF 3 to ÔDßbDÕ ¦ : let notice that S is a current of type Ô0, n 1Õ with support in the origin.

We have the following isomorphisms:

F 4 bF 4 Ker ¡ H n 1 0 ÔX, OÕ df H n 1 0 ÔX, Ω 1 Õ ©
and the isomorphism between F 3 ßbF 3 and F 4 ßbF 4 is the natural one, and

F 5 bF 5 ¢ Ω n 1 df Ω n ª ¦
From steps 1-3 we deduce that ∆ 0 induces Grothendieck's residue.

Property (iv)

We will that the isomorphism given by R. Belgrade can be easily transformed into one that verifies the property. Let ∆ : E Ȇ¦ Ôa,bÕ E n 1 be Belgrade's isomorphism. By tensoring with E Ôn 1Õß2Õ we can show that, the isomorphisms between E and ̦ Ôa,bÕ E n 1 are in bijection with the isomorphisms between E Ôa,bÕ E ¡Ôn 1Õß2 and it adjoint, through the map that sends an isomorphism Φ to Φ Ôa,bÕ Id E ¡Ôn 1Õß2 .

By an easy calculation we can prove the following lemma: Lemma 37. Let ∆ : E Ȇ¦ E n 1 be an isomorphism and

∆ÔyÕÔxÕ Ôn 1Õ! ô k ∆ k Ôx, yÕb k e n 1
for each x and y È E. Then the ∆ k satisfy Saito's condition (iv) if and only if the isomorphism ∆ Ôa,bÕ Id E ¡Ôn 1Õß2 is hermitian.

Proof. ∆ Ôa,bÕ Id E ¡Ôn 1Õß2 is self-adjoint iff we have:

∆ Ôa,bÕ Id E ¡Ôn 1Õß2 y e ¡Ôn 1Õß2 By combining the previous equivalence with the results on the existence of hermitian forms, we can extend Belgrade's result: Theorem 38. Let E be a regular Ôa, bÕ-module associated to a holomorphic function from C n 1 to C with an isolated singularity. Then there exists an isomorphism Φ :

E Ȇ¦ Ôa,bÕ E n 1 with ΦÔyÕÔxÕ Ôn 1Õ! ô k Φ k Ôx, yÕb k e n 1 ,
for all x and y such that the sequence of C-bilinear forms Φ k satisfies all four properties of Saito's "higher residue pairings".

Proof. Let ∆ be Belgrade's isomorphism and ∆ k defined as at the beginning of this section. Consider the isomorphism ∆¦ Ôa,bÕ Id En 1 : E Ȇ¦ Ôa,bÕ E n 1 and let Φ ∆ ∆¦ Ôa,bÕ Id En 1 ¨ß2.

20

It is easy to see that the Φ k satisfy properties (i) and (ii). Moreover since ∆ 0 is symmetric (Grothendieck's residue) and ∆¦ Ôa,bÕ Id En 1 induces the transposed of ∆ 0 on EßbE, we have Φ 0 ∆ 0 t ∆ 0 ¨ß2 ∆ 0 .

We have also Φ Ôa,bÕ Id E ¡Ôn 1Õß2 Φ Ôa,bÕ Id E ¡Ôn 1Õ ¨¦ Φ¦ Ôa,bÕ Id E Ôn 1Õß2

Φ Ôa,bÕ Id E ¡Ôn 1Õß2 , therefore the Φ k satisfy Saito's property (iv).

We just have to show that Φ Ôa,bÕ Id E ¡Ôn 1Õß2 is an isomorphism. Since there exists an isomorphism between E Ôa,bÕ E ¡Ôn 1Õß2 and its adjoint ∆ Ôa,bÕ Id E ¡Ôn 1Õß2 , we can apply proposition 22 and reduce ourselves to prove the injectivity of Φ Ôa,bÕ Id E ¡Ôn 1Õß2 . But if Φ Ôa,bÕ Id E ¡Ôn 1Õß2 were not injective Φ would induce a degenerate form on EßbE, which is absurd.

The existence of a hermitian form on E Ôa,bÕ E ¡Ôn 1Õß2 gives us an interesting restriction on the kind of Ôa, bÕ-module associated with Brieskorn lattices: Corollary 39. Let E be a regular Ôa, bÕ-module associated to a holomorphic function from C n 1 to C with an isolated singularity. Then E Ôa,bÕ E ¡Ôn 1Õß2 is a hermitian Ôa, bÕ-module.

¦

  Institut Élie Cartan Nancy UMR 7502 Nancy-Université, B.P. 70230, 54506 Vandoeuvre-lès-Nancy, France E-mail: piotr.karwasz@iecn.u-nancy.fr Definition 1. Let CÖÖb×× be the ring of formal series in the variable b. An Ôa, bÕ-module is an algebraic structure composed by a free CÖÖb××-module E of finite rank and a C-linear application a : E E that satisfies the commutation relation ab ¡ ba b 2 , (1) where b : E E is the multiplication by the element b È CÖÖb××.

¨ x e ¡Ôn 1Õß2 ¨ ô k S k b k e 0 ∆¨ x e ¡Ôn 1Õß2 ¨ ô k S k b k e 0 ∆ÔyÕÔxÕ ô k S

 0k Ôa,bÕ Id E ¡Ôn 1Õß2 x e ¡Ôn 1Õß2 ¨ y e ¡Ôn 1Õß2 ¨ ô k S k Ô¡bÕ k e 0 .for all x and y È E. On the other hand we have: ∆ Ôa,bÕ Id E ¡Ôn 1Õß2 y e ¡Ôn 1Õß2 k b k e n 1 .

  27. Let E and F be indecomposable regular Ôa, bÕ-modules and α :

E

F and β : F E two Ôa, bÕ-linear morphisms. Suppose that β ¥ α is an isomorphism, then α and β are also isomorphisms. Proof. Let prove that F Im α Ker β. If αÔxÕ È Ker β, we have β ¥ αÔxÕ 0, hence x 0 and therefore Im α Ker β Ø0Ù. Consider now an element x È F and let y α ¥ Ôβ ¥ αÕ ¡1 ¥ βÔxÕ. We have βÔx ¡ yÕ βÔxÕ ¡ βÔyÕ βÔxÕ ¡ Ôβ ¥ αÕ ¥ Ôβ ¥ αÕ ¡1 ¥ βÔxÕ βÔxÕ ¡ βÔxÕ 0. We can thus write x as sum of an element y of Im α and an element x ¡ y of Ker β. This implies F Im α Ker β. Now since β ¥ α is injective, so must be α and Im α can not be 0. But F is indecomposable therefore we must have Im α F and Ker β 0. It follows that α is bijective and β Ôβ ¥ αÕ ¥ α ¡1 must be also bijective. Proof of Krull-Schmidt theorem for Ôa, bÕ-modules. We will show this theorem by induction on m.

  1 Ö∆ÔyÕ× ÔbxÕ b Ö∆ÔyÕ× ÔxÕ ô Ôn 1Õ!∆ k Ôx, yÕb k 1 e n 1 which gives us ∆ k Ôx, yÕ ∆ k 1 Ôbx, yÕ. And similarly by using the b-linearity

	of ∆ and the adjoint morphism, we obtain:
	∆ÔbyÕÔxÕ	∆¦ ÔxÕÔbyÕ ¡b ∆¦ ÔxÕÔyÕ ¡b∆ÔyÕÔxÕ,
	and therefore

k

Acknowledgements

I would like to thank Daniel Barlet for his guidance during my Ph.D. thesis, Michel Meo for his help on the complex analysis topics.


Hervé Leblond

David Kremer

Dumitru Mihalache

Ultrashort spatiotemporal optical solitons in quadratic nonlinear media: generation of line and lump solitons from few-cycle input pulses

Keywords: numbers: 42.65.Tg, 42.65.Re, 05.45.Yv

By using a powerful reductive perturbation technique, or a multiscale analysis, a generic Kadomtsev-Petviashvili evolution equation governing the propagation of femtosecond spatiotemporal optical solitons in quadratic nonlinear media beyond the slowly-varying envelope approximation is put forward. Direct numerical simulations show the formation, from adequately chosen few-cycle input pulses, of both stable line solitons (in the case of a quadratic medium with normal dispersion) and of stable lumps (for a quadratic medium with anomalous dispersion). Besides, a typical example of the decay of the perturbed unstable line soliton into stable lumps for a quadratic nonlinear medium with anomalous dispersion is also given.

I. INTRODUCTION

Over the past two decades, the development of colliding-pulse mode-locked lasers and selfstarted solid-state lasers has been crowned with the generation of few-cycle pulses (FCPs) for wavelengths from the visible to the near infrared (for a comprehensive review see Ref. [1]). Presently, there is still a great deal of interest in the study of propagation characteristics of these FCPs in both linear and nonlinear media. Thus the comprehensive study of intense optical pulses has opened the door to a series of applications in various fields such as light matter interaction, high-order harmonic generation, extreme nonlinear optics [START_REF] Wegener | Extreme Nonlinear Optics[END_REF], and attosecond physics [START_REF] Scrinzi | [END_REF]4]. The theoretical studies of the physics of FCPs concentrated on three classes of main dynamical models: (i) the quantum approach [5][6][7][8], (ii) the refinements within the framework of the slowly varying envelope approximation (SVEA) of the nonlinear Schrödinger-type envelope equations [9][10][11][12], and non-SVEA models [13][14][15][16][17]. In media with cubic optical nonlinearity (Kerr media) the physics of (1+1)-dimensional FCPs can be adequately described beyond the SVEA by using different dynamical models, such as the modified Korteweg-de Vries (mKdV) [13], sine-Gordon (sG) [14,15], or mKdV-sG equations [16,17]. The above mentioned partial differential evolution equations admit breather solutions, which are suitable for describing the physics of few-optical-cycle solitons. Moreover, a non-integrable generalized Kadomtsev-Petviashvili (KP) equation [18] (a two-dimensional version of the mKdV model) was also put forward for describing the (2+1)-dimensional few-optical-cycle spatiotemporal soliton propagation in cubic nonlinear media beyond the SVEA [19,20].

Notice that of particular interest for the physics of the (1+1)-dimensional FCPs is the mKdV-sG equation; thus by using a system of two-level atoms, it has been shown in Ref. [16] that the propagation of ultrashort pulses in Kerr optical media is fairly well described by a generic mKdV-sG equation. It is worthy to mention that this quite general model was also derived and studied in Refs. [21,22]. However, another model equation for describing the physics of FCPs, known as the short-pulse equation (SPE) has been introduced, too [23][24][25][26] and some vectorial versions of SPE have been also investigated [27][28][29]. Other kind of SPE containing additional dispersion term has been introduced in Ref. [30]. A multi-dimensional version of the SPE was also put forward [31] and self-focusing and pulse compression have been studied, too [32]. Notice that the SPE model has been considered again in its vectorial version and the pulse self-compression and FCP soliton propagation have been investigated in detail [33]. The main dynamical (1+1)-dimensional FCP models mentioned above have been revisited recently [34]; it was thus proved that the generic dynamical model based on the mKdV-sG partial differential equation was able to retrieve the results reported so far in the literature, and so demonstrating its remarkable mathematical capabilities in describing the physics of (1+1)-dimensional FCP optical solitons [34].

As concerning the extension of these studies to other relevant physical settings, it was shown recently that a FCP launched in a quadratically nonlinear medium may result in the formation of a (1+1)-dimensional half-cycle soliton (with a single hump) and without any oscilatting tails [35]. It was proved that the FCP soliton propagation in a quadratic medium can be adequately described by a KdV equation and not by mKdV equation. Notice that in Ref. [35] it was considered a quadratic nonlinearity for a single wave (frequency), and that no effective third order nonlinearity was involved (due to cascaded second order nonlinearities). This is in sharp contrast with the nonlinear propagation of standard quadratic solitons (alias two-color solitons) within the SVEA where two different waves (frequencies) are involved, namely a fundamental frequency and a second harmonic [36]- [43].

As mentioned above, there are only a few works devoted to the study of (2+1)-dimensional few-optical-cycle solitons, where an additional spatial transverse dimension is incorporated into the model; see e.g., Refs. [15,19,20], where the propagation of few-cycle optical pulses in a collection of two-level atoms was investigated beyond the traditional SVEA. The numerical simulations have shown that in certain conditions, a femtosecond pulse can evolve into a stable few-optical cycle spatiotemporal soliton [15,19,20].

The aim of this paper is to derive a generic partial differential equation describing the dynamics of (2+1)-dimensional spatiotemporal solitons in quadratic nonlinear media beyond the SVEA model equations. Our study rely on the previous work [35] where the theory of half-cycle (1+1)-dimensional optical solitons in quadratic nonlinear media was put forward. This paper is organized as follows. In Sec. II we derive the generic KP equation governing the propagation of femtosecond spatiotemporal solitons in quadratic nonlinear media beyond the SVEA. To this aim we use the powerful reductive perturbation technique up to sixth order in a small parameter ε. We have arrived at the conclusion that when the resonance frequency is well above the inverse of the typical pulse width, which is of the order of a few femtoseconds, the long-wave approximation leads to generic KP I and KP II evolution equations. These evolution equations and their known analytical solutions, such as line solitons and lumps are briefly discussed in Sec. III. Direct numerical simulations of the governing equations are given in Sec. IV, where it is shown the generation of both stable line solitons (for the KP II equation) and of stable lumps (for the KP I equation), and a typical example of the decay of the perturbed unstable line soliton of the KP I equation into stable lumps is also provided. Section V presents our conclusions.

II. DERIVATION OF THE KP EQUATION

We consider the same classical and quantum models developed in Ref. [35], but with an additional transverse spatial dimension (we call it the y-axis). In the quantum mechanical approach, the medium composed by two-level atoms is treated using the density-matrix formalism and the dynamics of electromagnetic field is governed by the Maxwell-Bloch equations. Thus the evolution of the electric field component E along the polarization direction of the wave (the x-axis) is described by the (2+1)-dimensional Maxwell equations, which reduce to

∂ 2 y + ∂ 2 z E = 1 c 2 ∂ 2 t (E + 4πP ) , (1)
where P is the polarization density.

a. Classical model. First we consider a classical model of an elastically bounded electron oscillating along the polarization direction of the wave. The wave itself propagates in the z-direction. The evolution of the position x of an atomic electron is described by an anharmonic oscillator (here damping is neglected)

d 2 x dt 2 + Ω 2 x + ax 2 = -e m E. (2)
The parameter a measures the strength of the quadratic nonlinearity. The polarization density is P = -Nex, where N is the density of atoms. Here Ω is the resonance frequency, (-e) is the electron charge, m is its mass, and E is the electric field component along the x-axis.

b. Quantum model. We consider a set of two-level atoms with the Hamiltonian

H 0 =   ω a 0 0 ω b   , (3)
where Ω = ω b -ω a > 0 is the frequency of the transition. The evolution of the electric field E is described by the wave equation (1). The light propagation is coupled with the medium by means of a dipolar electric momentum µ directed along the same direction x as the electric field, according to

H = H 0 -µE, (4)
and the polarization density P along the x-direction is

P = NTr (ρµ) , ( 5
)
where N is the volume density of atoms, and ρ is the density matrix. The density-matrix evolution equation (Schrödinger equation) is written as

i ∂ t ρ = [H, ρ] + R, (6)
where R is a phenomenological relaxation term. In the case of cubic nonlinearity, it has been shown that it was negligible [14], and it will be neglected here as well.

In the absence of permanent dipolar momentum,

µ =   0 µ µ * 0   (7)
is off-diagonal, and the quadratic nonlinearity is zero.

The quadratic nonlinearity can be phenomenologically accounted for as follows: it corresponds to a deformation of the electronic cloud induced by the field E, hence to a dependency of the energy of the excited level b with respect to E, i.e., to a Stark effect. The nonlinearity is quadratic if this dependency is linear, as follows

ω b -→ ω b -αE. (8)
This contribution can be included phenomenologically in the Maxwell-Bloch equations by replacing the free Hamiltonian H 0 with

H 0 -αE   0 0 0 1   . (9)
c. Scaling. Transparency implies that the characteristic frequency ω w of the considered radiation (in the optical range) strongly differs from the resonance frequency Ω of the atoms, hence it can be much higher or much lower. We consider here the latter case, i.e., we assume that ω w is much smaller than Ω. This motivates the introduction of the slow variables

τ = ε t - z V , ζ = ε 3 z, η = ε 2 y, (10)
ε being a small parameter. The delayed time τ involves propagation at some speed V to be determined. It is assumed to vary slowly in time, according to the assumption ω w ≪ Ω. The pulse shape described by the variable τ is expected to evolve slowly in time, the corresponding scale being that of variable ζ. The transverse spatial variable y has an intermediate scale as usual in KP-type expansions [44].

A weak amplitude assumption is needed in order that the nonlinear effects arise at the same propagation distance scale as the dispersion does. Next we use the reductive perturbation method as developed in Ref. [44]. To this aim we expand the electric field E as power series of a small parameter ε:

E = ε 2 E 2 + ε 3 E 3 + ε 4 E 4 + . . . , (11)
as in the standard KdV-type expansions [44]. The polarization density P is expanded in the same way.

d. Order by order resolution

The computation is exactly the same as in [35] up to order ε 4 in the Schrödinger equation (6) or classical dynamical equation [START_REF] Wegener | Extreme Nonlinear Optics[END_REF], and up to order ε 5 in the wave equation (1). The fourth order polarization density P 4 has the same expression as in [35]. Then the wave equation ( 1) at order ε 6 yields the evolution equation

∂ ζ ∂ τ E 2 = A∂ 4 τ E 2 + B∂ 2 τ (E 2 ) 2 + V 2 ∂ 2 η E 2 , (12)
which is a generic KP equation.

The coefficients A and B in the above (2+1)-dimensional evolution equation are

A = 4πV Ne 2 2mc 2 Ω 4 , B = -4πV aNe 3 2m 2 c 2 Ω 6 , (13)
in the classical model, and respectively,

A = 4πN|µ| 2 nc Ω 3 , B = -4πNα|µ| 2 nc 2 Ω 2 , ( 14
)
in the corresponding quantum model.

Notice that the dispersion coefficient A and the nonlinear coefficient B can be written in a general form as

A = 1 6 d 3 k dω 3 ω=0 = 1 2c d 2 n dω 2 ω=0 , (15)
B = -2π nc χ (2) (2ω; ω, ω) ω=0 , ( 16
)
where n is the refractive index of the medium and ω is the wave pulsation (for a detailed discussion of the corresponding expressions of the dispersion and nonlinear coefficients in the case of cubic nonlinear media see, e.g., Ref. [14]). The velocity is obviously V = c/n.

III. TWO KP EQUATIONS: KP I AND KP II

The KP equation ( 12) is reduced to

∂ Z ∂ T u = ∂ 4 T u + ∂ 2 T u 2 + σ∂ 2 Y u, (17)
where σ = sgn(A), by the change of variables

ζ = Z, τ = A 1/3 T, η = |A| 1/6 V 2 Y, E 2 = A 1/3 B u. (18)
For our starting models (both the classical and quantum ones), the dispersion coefficient A is positive. It is the case of the normal dispersion. Hence σ = +1 and Eq. ( 17) is the so-called KP II equation. Both KP I and KP II are completely integrable by means of the inverse scattering transform (IST) method [START_REF] Ablowitz | Solitons and the Inverse Scattering Transform[END_REF], however the mathematical properties of the solutions differ. KP II admits stable line solitons, but not stable localized soliton solutions.

The line soliton has the analytical expression

E 2 = 6p 2 A B sech 2 p τ + aη + a 2 V 2 + 4p 2 A ζ (19)
where a and p are arbitrary constants and A, B have the expressions given in the previous Section.

If Eq. ( 12) is generalized to other situations using the expressions ( 15)-( 16) for the dispersion and nonlinear coefficients, and if we assume an anomalous dispersion, i.e., A < 0, then σ = -1 and Eq. ( 17) becomes the so-called KP I equation, for which the above line soliton is not stable, but which admits stable lumps [START_REF] Satsuma | [END_REF]47] as

E 2 = 12A B -(t ′ + p r y ′ ) 2 + p 2 i y ′ 2 + 3/p 2 i (t ′ + p r y ′ ) 2 + p 2 i y ′ 2 + 3/p 2 i 2 , (20)
with

t ′ = τ + p 2 r + p 2 i Aζ, (21)
y ′ = η -2A V -2p r Aζ, (22)
where p r and p i are arbitrary real constants. The analytical lump solution written above is plotted in Fig. 1.

The size of the lump can be evaluated as follows. Assume for the sake of simplicity that the lump exactly propagates along z, i.e., that p r = 0. Then Eq. ( 20) has the form

E 2 ∝ t ′ 2 3/p 2 i + y ′ 2 3/p 4 i + 1 -2 , (23)
which gives the normalized transverse width and duration as w ′ y = √ 3/p 2 i and w ′ t = √ 3/p i , respectively, while the scaled peak amplitude of the lump is E 2max = 4Ap 2 i /B. Using Eqs. ( 21)-( 22) and the scaling (10), we get the corresponding width and duration in physical units, as

w y = √ 3 p 2 i ε 2 V -2A , w t = √ 3 p i ε , (24)
while the peak amplitude is E m = ε 2 E 2max . Using expressions ( 15) and ( 16) of the coefficients A and B, we obtain

w y = w 2 t c √ -3nn ′′ , E m = -3nn ′′ πχ (2) w 2 t , (25)
where n ′′ and χ (2) are shortcuts for d 2 n/dω 2 | ω=0 and χ (2) (2ω; ω, ω) ω=0 , respectively. Hence the transverse width of the lump decreases as the pulse duration does and the peak amplitude increases. However, the initial assumptions and scaling (10) imply that the field varies much slower in the y-direction than longitudinally, hence the width of the lump should, in principle, be large with respect to its length. If this assumption is not satisfied, the validity of the KP model is questionable.

IV. NUMERICAL SIMULATIONS OF KP I AND KP II EQUATIONS

The KP equation ( 17) is solved by means of the fourth order Runge Kutta exponential time differencing (RK4ETD) scheme [48]. It involves one integration with respect to τ .

The inverse derivative is computed by means of a Fourier transform, which implies that the integration constant is fixed so that the mean value of the inverse derivative is zero, but also that the linear term is replaced with zero, i.e., the mean value of the function ∂ 2 y u is set to zero. For low frequencies, the coefficients of the RK4ETD scheme are computed by means of series expansions, to avoid catastrophic consequences of limited numerical accuracy.

A. KP II

For a normal dispersion (n ′′ > 0), as in the case of the two-level model above, Eq. ( 12) is KP II. Then, starting from an input in the form of a Gaussian plane wave packet

u = u 0 exp - (T -T 1 ) 2 w 2 t cos 2π λ (T -T 1 ) (26)
where

T 1 = -λ 2 exp - Y 2 w 2 y ( 27
)
accounts for a transverse initial perturbation, we have checked numerically (see Fig. 2) that (i) The FCP input transforms into a half-cycle pulse, line soliton solution of KP II equation, whose temporal profile is the sech-square shaped KdV soliton, plus the accompanying dispersing-diffracting waves, and that (ii) The initial perturbation vanishes and the line soliton becomes straight again during propagation.

According to the stability analysis [START_REF] Ablowitz | Solitons and the Inverse Scattering Transform[END_REF], the stabilization of the perturbed line-soliton occurs if the perturbation is not too large. Hence the essential requirement to evidentiate the above features is the one-dimensional problem of the formation of a KdV soliton from a Gaussian input. In principle, this problem is solved by the IST: a wide pulse with high enough intensity is expected to evolve into a finite number of solitons, plus some dispersive waves, called 'radiation' in the frame of the IST method. This has been addressed numerically in [35]: it has been seen that, for a FCP-type input, the amount of radiation was quite important, and the exact number and size of solitons might depend on the carrier-envelope phase.

Numerically, the finite size of the computation box and periodic boundary conditions prevent the dispersive wave from being totally spread out. Hence, in order to evidence numerically the phenomenon mentioned above, we need to reduce the energy of the dispersive part to the lowest level possible, and produce a single soliton. Very roughly, one can say that each positive half-cycle oscillation with high enough amplitude results in one soliton.

The initial carrier envelope phase should be close to zero, the main oscillation have a large enough amplitude, and the pulse be short enough so that only one peak is emitted. On the other hand, the length of the KdV soliton ( 19) is 2/p, while a half-cycle of the pulse (26) obviously has duration λ/2. On the other hand, the amplitude of the KdV soliton ( 19) is 6p 2 A/B, to be compared to the amplitude u 0 of pulse (26). In a first approximation, the half-cycle matches the KdV soliton if both the durations and amplitude are the same, hence if

u 0 = 96A λ 2 B . ( 28
)
Numerical computation confirms that relation (28) between the wavelength and amplitude of the Gaussian pulse is roughly optimal.

Recall that our numerical computations are performed using the normalized form (17) of the KP equation, which corresponds to A = B = 1. We observed that, for the particular values of λ and w t pertaining to Fig. 2, for u 0 5, the soliton emerges from the dispersive wave, while, at optimum (28) (u 0 ≃ 11), the soliton amplitude is about three times the amplitude of the dispersive wave. A stable plane-wave half-cycle quadratic optical soliton can thus be formed from a FCP input. Further, the quadratic nonlinearity may restore the spatial coherence of the soliton.

B. KP I

Let us now assume a quadratic nonlinear medium with anomalous dispersion (n ′′ < 0), in which the KP equation ( 12) is valid. It reduce to Eq. ( 17) with σ = -1, i.e., KP I equation.

We consider an initial data of the form

u = u 0 exp - T 2 w 2 t - Y 2 w 2 y cos 2π λ T , (29)
with arbitrary values of the initial amplitude u 0 , width w y , and duration w t of the pulse. The evolution of the input FCP into a lump is shown on Fig. 3; notice the remaining dispersive waves. The fact that they seem to be confined in the Y -direction is a numerical artifact, due to the fact that absorbing boundary conditions have been added in the Y -direction only. Hence the diffracting part of the dispersive waves rapidly decreases as if the width of the numerical box were infinite, while the dispersive waves which propagate along the Z-axis remain confined in the numerical box (with length 160), and hence decays slower as it would happen for a single pulse.

Figure 4 shows the decay of a perturbed unstable line soliton of KP I equation into lumps.

The input is given by Eq. ( 26), with the transverse perturbation

T 1 = -λ 2 cos 2π L Y , (30)
where the wavelength L of the transverse perturbation is chosen so that 2L = 28.3 is the width of the numerical box, in accordance with the periodic boundary conditions used in our numerical computation. This transversely perturbed line-soliton, in contrast with the case of KP II for normal dispersion shown in Fig. 2, does not recover its initial straight line shape and transverse coherence, but breaks up into localized lumps. Notice that the conditions on the parameters needed to obtain a result comparable to that shown in Fig. are much less restrictive than the conditions required for the numerical observation of line solitons; see Fig. 2 and the detailed discussion given above on the optimal conditions of generation of line solitons of KP II equation. Hence spatiotemporal quadratic solitons may form from a FCP input, their transverse focusing follows from any transverse perturbation of the incident plane wave.

V. CONCLUSION

In conclusion, we have introduced a model beyond the slowly varying envelope approximation of the nonlinear Schrödinger-type evolution equations, for describing the propagation of (2+1)-dimensional spatiotemporal ultrashort optical solitons in quadratic nonlinear media. Our approach is based on the Maxwell-Bloch equations for an ensemble of two level atoms. We have used the multiscale approach up to the sixth-order in a certain small perturbation parameter and as a result of this powerful reductive perturbation method, two generic partial differential evolutions equations were put forward, namely, (i) the Kadomtsev-Petviashvili I equation (for anomalous dispersion), and (ii) the Kadomtsev-Petviashvili II equation (for normal dispersion). Direct numerical simulations of these governing partial differential equations show the generation of both stable line solitons (for the Kadomtsev-Petviashvili II equation) and of stable lumps (for the Kadomtsev-Petviashvili I equation). Moreover, a typical example of the decay of the perturbed unstable line solitons of the Kadomtsev-Petviashvili I equation into stable lumps is also given.

The present sudy is restricted to (2+1) dimensions, however, due to the known properties of the Kadomtsev-Petviashvili equation in (3+1) dimensions [49,50], analogous behavior is expected in three dimensions, i.e., stability of the spatial coherence of a plane wave fewcycle quadratic soliton for normal dispersion, and spontaneous formation of spatio-temporal few-cycle 'light bullets' in the case of anomalous dispersion.

FIG. 1 :

 1 FIG. 1: (Color online) The lump solution of the KP I equation. Here the parameters are A = -1, B = -1, V = 1, p r = -1, and p i = 2.

FIG. 2 :

 2 FIG. 2: (Color online) Shape and profile of the line soliton. a-b: input data, c-d: after propagation Z = 15 (exactly 14.95). Parameters are λ = 3, u 0 = 10.7, w t = 1.4, and w y = 3.87.

FIG. 3 :

 3 FIG. 3: (Color online) Shape and profile of the lump. a-b: input data, c-d: after propagation Z = 15 (exactly 14.95). Parameters are λ = 3, u 0 = 6.12, w t = 1.2, and w y = 6.36.

4

 4

FIG. 4 :

 4 FIG. 4: (Color online) Generation of lumps from a perturbed unstable line soliton. a : input data, b: after propagation Z = 1.45. Parameters are λ = 3, u 0 = 10.7, w t = 1.4, and L = 14.14,


Kol Éh

È A Coulibaly-Pasquier

ONSAGER-MACHLUP FUNCTIONAL FOR UNIFORMLY ELLIPTIC TIME-INHOMOGENEOUS DIFFUSION

. We finish by a application to small ball probability for weighted sup norm, for inhomogeneous diffusion.

Introduction

Let M be a n-dimensional manifold, and a inhomogeneous uniformly elliptic operator L t over M . It is always possible to put a time dependent family of metrics g(t) over M such that (1.1)

L t = 1 2 ∆ t + Z(t),
where ∆ t is a Laplace Beltrami operator for a metric g(t) and Z(t, .) is a time dependent vector field over M . Let X t (x 0 ) a L t -diffusion process on M starting at point x 0 , an example of such a diffusion could be the g(t)-MB introduced in [START_REF] Coulibaly | Brownian motion with respect to time-changing riemannian metrics[END_REF], when the family of metrics g(t) come from the Ricci flow.

Let d(t, x, y) be the Riemannian distance on M according to the metric g(t). Consider a smooth curve ϕ : [0, T ] → M , such that ϕ(0) = x 0 , we are interested in the asymptotic equivalent as ǫ goes to zero of the following probability P x 0 [∀t ∈ [0, T ] d(t, X t , ϕ(t)) ≤ ǫ]. This asymptotic will depend on the product of two terms. The first one is a decreasing function of ǫ that does not depend on the curve and the geometries (except the dimension), and a second term that depends on the geometries around the curve ϕ. This second term is expressed as a certain Lagrangian, and maximizing this term could be interpreted as finding the most probable path for the diffusion. This term is historically called the Onsager-Machlup functional of the diffusion X t . This computation will be made using the same technique as in the paper of Takahashi and Watanabe [START_REF] Takahashi | The probability functionals (Onsager-Machlup functions) of diffusion processes[END_REF]. We will also use the non singular drift introduced by Hara. Using this drift, Hara and Takahashi in [START_REF] Hara | Lagrangian for pinned diffusion process[END_REF] have made a substantial simplification of the previous proof of Onsager-Machlup 1 functional. We propose a time dependent parallel transport along a curve according to a family of metrics and use it to make the computation of the Onsager-Machlup functional in the inhomogeneous case. This is the main result of the paper, here in the following theorem Theorem 1.1. Let X t (x 0 ) be a L t diffusion process starting at point x 0 , where L t = 1 2 ∆ t + Z(t, .), then we have the following asymptotic:

P x 0 [∀t ∈ [0, T ] d(t, X t , ϕ(t)) ≤ ǫ] ∼ ǫ↓0 C exp{- λ 1 T ǫ 2 } exp{- T 0 H(t, ϕ, φ)},
where H is a time dependent function on the tangent bundle defined for v ∈ T x M as:

H(t, x, v) = 1 2 Z(t, x) -v 2 g(t) + 1 2 div g(t) (Z)(t, x) - 1 12 R g(t) (x) + 1 4 trace g(t) ( ġ(t)).
Here C , λ 1 are explicit constants, div g(t) and R g(t) are respectively the divergence operator and the scalar curvature with respect to the metric g(t).

The paper will be organized as follows: In the first section we will give a parallel transport along a curve according to a family of metrics. We will use this parallel transport to get a Fermi coordinate around the smooth curve ϕ. We will also give some local development of certain tensor that we need in the sequel.

To be self contained, in the second section we will quickly expose some probabilistic lemmas all of them are clearly exposed in the paper of Capitaine [START_REF] Capitaine | On the Onsager-Machlup functional for elliptic diffusion processes[END_REF], so we will keep the same notation. In this paper [START_REF] Capitaine | On the Onsager-Machlup functional for elliptic diffusion processes[END_REF] the case of different norms are investigated. In the literature the case of non smooth functions ϕ are also investigated. But we will not discuss these cases in this paper.

In the three last sections we will expose the proof of the theorem and give some related results. The case of g(t)-BM, when the family of metrics g(t) comes from the Ricci flow will be investigated as an application of the theorem. The resulting Lagrangian gives a notion of "space-time distance" which presents many similarities with the L 0 -distance used in the theory of Ricci flow.

Parallel transport along a curve, and Fermi coordinate

Let ϕ : [0, T ] -→ M be smooth curve. And consider that the manifold M is endowed with a C 1 -family of metrics g(t) t∈[0,T ] . This family of metrics produces a time dependent family of Levi-Civita connexions that we will write ∇ t .

If A is a bilinear form over a vector space E, v, w ∈ E and we have a scalar product ., . g(t) over E then we define

A #g(t) (v) ∈ E such that A #g(t) v, w = A(v, w).
Proposition 2.1. Let (e 1 , e 2 , ..., e n ) be an orthonormal basis of T ϕ(0) M for the metric g(0). Let τ t e i be the solutions of the following first order equation on T M above the curve ϕ:

   ∇ t φ(t) τ t e i = - 1 2 ġ(t) #g(t) (τ t e i )
τ 0 e i = e i .

Then (τ t e 1 , τ t e 2 , ..., τ t e n ) is an orthonormal basis of T ϕ(t) M for the metric g(t).

Proof. We have just to compute quantity like: + ġ(t)(τ t e i , τ t e j ) = 0.

d dt τ t e i ,
We are now able to write the Fermi coordinate around a curve. Let ϕ : [0, T ] -→ M be a smooth curve and let τ be the parallel transport above ϕ in the sense of (2.1), where we have fix a g(0) orthonormal basis (e 1 , ..., e n ) of T ϕ(0) M . Consider the map

Ψ : U ⊂ [0, T ] × R n -→ V ⊂ [0, T ] × M (t, v 1 , ..., v n ) -→ (t, exp t ϕ(t) (τ t n 1 v i e i )).
Where exp t x means the exponential map for the metric g(t). The map Ψ is clearly a diffeomorphism on some neighborhood U of [0, T ] × 0, and let V = Ψ(U ). Remark that for each fixed t, the map Ψ(t, .) is the normal coordinate for the metric g(t) around the point ϕ(t).

Let X t (x 0 ) be a L t -diffusion that starts at the point x 0 , where L t is a time dependent operator as is (1.1). Using this Fermi coordinate, the timedependent norm in the theorem 1.1 will be translate in term of Euclidean one, but the generator will be changed by the pull back by Ψ of the operator L t . The generator of (t, X t ) is ∂ t + 1 2 ∆ t + Z(t, .) and we will compute the generator of Ψ -1 (t, X t ), or more precisely its local development.

Ψ * (∂ t + 1 2 ∆ t + Z(t, .)) = ∂ ∂ t + 1 2 ∆t + Z(t, .).
The second term in the right hand side is computed in [START_REF] Coulibaly | Brownian motion with respect to time-changing riemannian metrics[END_REF] as :

∆t = g ij (Ψ(t, .)) ∂ ∂x i ∂ ∂x j - 1 2 g kl (Ψ(t, .))Γ i kl (Ψ(t, .)) ∂ ∂x i ,
where (x t 1 , ..., x t n ), g ij (Ψ(t, .)), g ij (Ψ(t, .)), and Γ i kl (Ψ(t, .)) are respectively the normal coordinate at the point ϕ(t) for the metric g(t) with respect to the vector basis (τ t e 1 , ..., τ t e 1 ), the coefficient of metric g(t) in this basis, its inverse, and the Christoffel symbols of the Levi-Civita connexion of the metric g(t) in this basis. Clearly we have

Z(t, .) = n i=1 Z i (t, .) ∂ ∂x i where Z i (t, .) = Z(Ψ(t, .)), ∂ ∂x t i | Ψ(t,.) g(t) . For a point (t, x) ∈ [0, T ] × M in the neighborhood V of {(t, ϕ(t)), t ∈ [0, T ]} in which Ψ induce a diffeomorphism, we write Ψ -1 (t, x) = (t, x t 1 , ..., x t 1 ) ∈ [0, T ] × R n . We have to compute ∂ ∂t | (t,x) = n i=1 a i (t, x) ∂ ∂x i +a 0 (t, x) ∂ ∂t , clearly we have a i (t 0 , x) = ∂ ∂t |t 0 (x t i )| Ψ(t 0 ,x)
. We will compute this term using the equality :

∂ ∂t (exp g(t) (ϕ(t), n i=1 τ t e i x t i )) = 0,
where, for v ∈ T x M , exp g(t) (x, v) is the exponential map for the metric g(t) at the point x. The three propositions below will be used to compute

a i (t, x) = ∂ ∂t (x t i )| Ψ(t,x) . Proposition 2.2. Let v ∈ T x M then ∂ ∂t |t 0 exp g(t) (x, v) = O( v 2 g(t 0 ) )
Proof. Let x i (t, s) be the coordinate of the geodesic exp g(t) (x, s.v) in the normal coordinate system centered at ϕ(t 0 ) with respect to the metric g(t 0 ), we will write ẋ(t, s) for ∂ ∂s x(t, s). The usual equation for the geodesic gives:

∂ ∂t |t 0 x i (t, s) = - ∂ ∂t |t 0 s 0 du u 0 dl Γ i jk (t, x(t, l)) ẋj (t, l) ẋk (t, l) = - s 0 du u 0 dl ∂ ∂t |t 0 Γ i jk (t, x(t 0 , l)) ẋj (t 0 , l) ẋk (t 0 , l) + dΓ i jk (t 0 , .), ∂ ∂t |t 0 x(t, l) ẋj (t 0 , l) ẋk (t 0 , l) + 2Γ i jk (t 0 , x(t 0 , l)) ∂ ∂t |t 0 ( ẋj (t, l)) ẋk (t 0 , l) . Note that ẋ(t 0 , s) 2 g(t 0 ) = v 2 g(t 0 ) , Γ i jk (t 0 , x) = O( x g(t 0 ) ), also | ∂ ∂t Γ i jk (t, .) | and dΓ i jk (t, .) are bounded by some constant C, in a neighborhood V of {(t, ϕ(t)), t ∈ [0, T ]}, hence : ∂ ∂t |t 0 x(t, s) := ∂ ∂t |t 0 (x 1 (t, s), ..., x n (t, s)) = O( v 2 g(t 0 ) ) + s 0 dl O( v 2 g(t 0 ) ) ∂ ∂t |t 0 x(t, l) + s 0 du u 0 dl O( v 2 g(t 0 ) ) ∂ ∂t |t 0 x(t, l).
By Gronwall's Lemma we deduce that :

∂ ∂t |t 0 x(t, 1) = O( v 2 g(t 0 ) )
Proposition 2.3. Let (x 1 (t), ..., x n (t)) be the coordinate of exp g(t 0 ) (ϕ(t), n i=1 τ t e i x t i ) in the normal coordinate system at the point ϕ(t 0 ) for the metric g(t 0 ), and ∂ i are the associated vectors field (it is a short notation for ∂ ∂

x t 0 i ). Then ∂ ∂t |t 0 x i (t) = ∂ ∂t |t 0 x t i - 1 2 ∂ ∂t |t 0 (g(t)) ϕ(t 0 ) (∂ i , n j=1 x t 0 j ∂ j ) + ∂ ∂t |t 0 ϕ(t), ∂ i g(t 0 ) + O( x t 0 2 ).
Proof. As in the proof of the above proposition, we write the geodesic exp g(t 0 ) (ϕ(t), s. n i=1 τ t e i x t i ) in normal coordinate, it usually satisfy :

               ẍi (t, s) = - jk Γ i jk (t 0 , x(t, s)) ẋj (t, s) ẋk (t, s) ẋi (t, 0) = n l=1 τ t e l x t l , ∂ i| ϕ(t) g(t 0 ) x i (t, 0) = ϕ(t) i
We have :

∂ ∂t |t 0 x i (t, s) = - s 0 du u 0 dl jk ∂ ∂t |t 0 Γ i jk (t 0 , x(t, l)) ẋj (t, l) ẋk (t, l) + s ∂ ∂t |t 0 ẋi (t, o) + ∂ ∂t |t 0 x i (t, o).
After the same computation as before, we could write this equality in a matrix form, using again the Gronwall's lemma we deduce that ∂ ∂t |t 0 x i (t, s) is bounded for s ∈ [0, 1], so the first term in the previous computation is a O( x t 0 2 ). Hence we have to) .

∂ ∂t |t 0 x i (t, 1) = O( x t 0 2 g(t 0 ) ) + ∂ ∂t |t 0 n l=1 τ t e l x t l , ∂ i | ϕ(t) g(t 0 ) + ∂ ∂t |t 0 ϕ(t), ∂ i | ϕ(t 0 ) g(
Remark that ∂ i | ϕ(t 0 ) = τ t 0 e i (only for t 0 ), so:

∂ ∂t |t 0 x i (t, 1) = O( x t 0 2 g(t 0 ) ) + ∂ ∂t |t 0 x t l δ l i + n l=1 x t 0 l ∂ ∂t |t 0 τ t e l , ∂ i | ϕ(t) g(t 0 ) + ∂ ∂t |t 0 ϕ(t), ∂ i | ϕ(t 0 ) g(to)
.

By the construction of the parallel transport τ we have :

∂ ∂t |t 0 τ t e l , ∂ i | ϕ(t) g(t 0 ) = ∇ t 0 τ t e l , ∂ i | ϕ(t 0 ) g(t 0 ) + τ t 0 e l , ∇ t 0 ∂ i g(t 0 ) = - 1 2 ∂ ∂t |t 0 (g(t))(τ t 0 e l , ∂ i | ϕ(t 0 ) ) = - 1 2 ∂ ∂t |t 0 (g(t))(∂ l | ϕ(t 0 ) , ∂ i | ϕ(t 0 ) )
and the last term of the right hand side of the first equality vanishes because ∂ i comes from a normal coordinate for the metric g(t 0 ). And the result follows.

Proposition 2.4.

∂ ∂t |t 0 x t j = 1 2 ∂ ∂t |t 0 (g(t)) ϕ(t 0 ) (∂ i , n j=1 x t 0 j ∂ j ) - ∂ ∂t |t 0 ϕ(t), ∂ i g(t 0 ) + O( x t 0 2 )
.

Proof. Recall that:

∂ ∂t (exp g(t) (ϕ(t), n i=1 τ t e i x t i )) = 0,
and the two propositions above compute the first term of the previous equation.

We get the Taylor series of the generator :

∂ ∂ t + Lt := Ψ * (∂ t + 1 2 ∆ t + Z(t, .)) |(t,x) = ∂ ∂ t + n i,j=1 g ij (t, x) ∂ ∂x i ∂ ∂x j + n i=1 bi (t, x) ∂ ∂x i = ∂ ∂ t + n i,j=1 1 2 ġ(t) ∂ ∂x t i , ∂ ∂x t j x j -φ(t) i ∂ ∂x i - 1 2 n k,l,i=1 g kl (t, x)Γ i kl (t, x) ∂ ∂x i + 1 2 n i,j=1 g ij (t, x) ∂ ∂x i ∂ ∂x j + n i=1 Z i (t, x) ∂ ∂x i + O( x 2 ),
where g ij (t, x) are the metric g(t) in the normal coordinate (x t 1 , ..., x t n ) evaluated at the point Ψ(t, x), also Γ k ij (t, x) are the Christoffel symbols in this coordinate at the point Ψ(t, x), φi (t) and Z i (t, x) are the coordinate of the corresponding vector in this normal coordinate.

Remark 2.5. We have no time dependence such as O( . g(t) ) because all the metrics are equivalent on U .

Transfer to R n , and probabilistic lemma

Let X(t) be a L t diffusion, let T = inf {t ∈ [0, T ], s.t. (t, X(t)) / ∈ V } and we define X(t) a process in R n such that (t ∧ T , X(t)) = Ψ -1 (t ∧ T , X(t ∧ T ). Then for a small ǫ we have :

P x 0 [ sup t∈[0,T ] d(t, X(t), ϕ(t)) ≤ ǫ] = P 0 [ sup t∈[0,T ] X(t) ≤ ǫ].

Clearly (t, X(t)) is a ∂

∂t + Lt diffusion, so for a R n -valued Brownian motion B, X(t) is a solution of the following Itô stochastic differential equation :

       d Xi (t) = n j=1 √ g ij (t, X(t)) d Bj t + bi (t, X(t)) dt X(0) = 0
Where √ g ij (t, x) is the square root of the metric g(t) in the coordinate (x t 1 , ..., x t 1 ) at the point Ψ(t, x) (we take the same notation for Γ) and bi (t, x) = -φi (t) -1 2 kl g kl (t, x)Γ i kl (t, x)

+ 1 2 ġ(t) | ϕ(t) ∂ ∂x t i , n j=1 x j ∂ ∂x t j + Z i (t, x) + O( x 2 ).
We will quickly describe the Hara Besselizing drift. We have the following equality which essentially comes from the fact that the coordinate is normal, and Gauss Lemma. For all i ∈ [1.

.n] we have :

m j=1 g ij (t, x)x j = x i , m j=1 √ g ij (t, x)x j = x i .
Let us define the Hara drift :

γ i (t, x) = 1 2 n j=1 ∂g ij ∂x j (t, x).
It satisfies the following useful equation :

i=1 (1 -g ii (t, x)) = 2 n j γ j (t, x)x j .
Let write σij (t, .) = √ g ij (t, x) the unique square root of the metric g(t) in the normal coordinate. We recall the equation of X(t) :

(3.1) d X(t) = σ(t, X(t)) d Bt + b(t, X(t)) dt X(0) = 0
We now define the process Y (t) as a solution of the following Itô equation

(3.2) dY (t) = σij (t, Y (t)) d Bi t + γ(t, Y (t)) dt Y (0) = 0.
Using Itô formula and the definition of the vector field γ we get :

d Y (t) 2 = 2 n k=1 Y k (t)d Bk t + n dt. Let B(t) = n k=1 t 0 Y k (s)
Y (s) d Bk s , using Lévy's Theorem, it is a one dimensional Brownian motion in the filtration generated by B and d Y (t) 2 = 2 Y (t) dB t + n dt, so Y (t) is a n dimensional Bessel process. Let us define a Girsanov's transform, such that after a change of probability, the process Y (t) become equal in law to the process X(t). Let (3.1). By the uniqueness in law we have :

N t = t 0 σ-1 (t, Y (t))( b(t, Y (t)) -γ(t, Y (t))), d Bt , M t = exp(N t - 1 2 N t ) Q = M T .P, by Girsanov's Theorem, (Y, Q) is a solution of
(3.3) P 0 [ sup t∈[0,T ] X(t) ≤ ǫ] = Q[ sup t∈[0,T ] Y (t) ≤ ǫ] = E P [M T ; sup t∈[0,T ] Y (t) ≤ ǫ] = E P [M T | sup t∈[0,T ] Y (t) ≤ ǫ]P[ sup t∈[0,T ] Y (t) ≤ ǫ]. The term P[sup t∈[0,T ] Y (t) ≤ ǫ],
is easily controled by a stopping time argument. So the problem of finding the Onsager Machlup functional become a study of the comportment of conditioned exponential martingale, as in the paper [START_REF] Takahashi | The probability functionals (Onsager-Machlup functions) of diffusion processes[END_REF]. Let us rewrite the last term in equation (3.3) as :

(3.4)

E P [exp n i,j=1 T 0 √ g ij (t, Y (t))δ j (t, Y (t))d Bi t - 1 2 n i,j=1 T 0 g ij (t, Y (t))δ i (t, Y (t))δ j (t, Y (t))dt | sup t∈[0,T ] Y (t) ≤ ǫ].
Where we write δ i (t, x) = bi (t, x) -γ i (t, x).

Remark 3.1. From the Lemma 1 in [START_REF] Capitaine | On the Onsager-Machlup functional for elliptic diffusion processes[END_REF] it is enough to control the exponential momentum one by one in the following sense.

Let us recall briefly this lemma :

Lemma 3.2 ([5]
, [START_REF] Capitaine | On the Onsager-Machlup functional for elliptic diffusion processes[END_REF]). Let I 1 , ..., I n be n random variables, {A ǫ } 0<ǫ a family of events, and a 1 , ..., a n be real numbers. If, for every real number c and every

1 ≤ i ≤ n, lim sup ǫ→0 E[exp(cI i ) | A ǫ ] ≤ exp(ca i ), then, lim ǫ→0 E(exp( n i=1 I i ) | A ǫ ) = exp( n i=1 a i ).
We recall Cartan's Theorem concerning Taylor series of metric and curvature in normal coordinate. Note that in this case, all the metrics g(t) are equivalent. We have :

g ij (t, x) = δ j i - 1 3 kl R iklj (t, 0)x k x l + O( x 3 ),
where R iklj (t, 0) are the components of the Riemannian curvature tensor, for the metric g(t) in normal coordinate centered at the point ϕ(t). So we deduce the following equality :

g ij (t, x) = δ j i + O( x 2 ), γ i (t, x) = - 1 6 n j=1 R ij (t, 0)x j + O( x 2 ),
where R ij (t, 0) are the component of the Ricci curvature tensor, for the metric g(t) in normal coordinate, at the point ϕ(t). Using the definition of the Christoffel symbol we have,

(3.5) Γ k ij (t, x) = 1 2 ( ∂ ∂ x i g jk (t, x) + ∂ ∂ x j g ik (t, x) - ∂ ∂ x k g ij (t, x)) = - 1 3 n l=1 (R jlik (t, 0) + R iljk (t, 0))x l + O( x 2 ).
So we obtain,

- 1 2 n i,j=1 g ij (t, x)Γ k ij (t, x) = - 1 3 n l=1 R lk (t, 0)x l + O( x 2 ), and thus, (3.6)
δ i (t, x) = -φi (t) + n j=1 1 2 ġij (t, 0) - 1 6 R ij (t, 0) x j + Z i (t, x) + O( x 2 ), = -φi (t) + Z i (t, 0) + n j=1 1 2 ġij (t, 0) - 1 6 R ij (t, 0) + ∂ ∂x j Z i (t, 0) x j + O( x 2 ), where ġij (t, 0) = ġ(t) ∂ ∂x t i | ϕ(t) , ∂ ∂x t j | ϕ(t) .

Proof of the theorem

According to Lemma 3.2 we will separately compute the terms in (3.4). The easiest to compute is the drift term : (4.1)

lim sup ǫ→0 E[exp{- c 2 T 0 g ij (t, Y (t))δ i (t, Y (t))δ j (t, Y (t))dt} | sup t∈[0,T ] Y (t) ≤ ǫ] ≤ lim ǫ→0 exp[- c 2 T 0 δ j i (-φi (t) + Z i (t, 0) 2 + O(ǫ) dt] ≤ exp(- c 2 T 0 δ j i (-φi (t) + Z i (t, 0)) 2 dt),
where we have used in the second inequality the fact that the O(ǫ) is uniform in t according to the uniform equivalence of the family of metrics

{g(t)} t∈[0,T ] .
In order to control the first term in (3.4) we will use the following Theorem [START_REF] Hara | Lagrangian for pinned diffusion process[END_REF].

Theorem 4.1 ([4]

). Let α be a one form on [0, T ] × R n , which does not depend on dt and Y (t) be a diffusion process in R n whose radial part is a Bessel process, and

. 0 Y i dY j -Y j dY i , Y . = 0 ∀i, j.
Then the following estimate holds for the stochastic line integral * d(t,Yt) α (in the sens of Stratonovich integration of a one form along a process):

E[exp( * d(t,Yt) α) | sup t∈[0,T ] Y (t) ≤ ǫ] = O(ǫ).
Remark 4.2. The proof of this Theorem is based on the Stokes theorem witch is transferred to stochastic Stokes theorem using Stratonovich integral, and Kunita-Watanabe theorem on orthogonal martingale theorem.

To use the above Theorem we have to write the first term of (3.4) in term of Stratonovich integral of a one form along a Bessel radial part process. Using equation (3.2) we get :

d Bi t = n j=1 σ-1 ij (t, Y (t)) dY j t - n j=1 σ-1 ij γ j (t, Y (t)) dt, (4.2)
n i,j=1 T 0 √ g ij (t, Y (t))δ j (t, Y (t))d Bi t = n i,j=1 T 0 g ij (t, Y (t))δ j (t, Y (t))dY i t - T 0 g ij (t, Y (t)δ j (t, Y (t))γ i (t, Y (t)) dt = n i,j=1 T 0 g ij (t, Y (t))δ j (t, Y (t)) * dY i t - 1 2 n i,j=1 T 0 d(g ij (t, Y (t))δ j (t, Y (t))), dY i t - T 0 g ij (t, Y (t)δ j (t, Y (t))γ i (t, Y (t)) dt.
Where * d is the Stratonovich differential.

Proposition 4.3. Let the event A ǫ be written as {sup t∈[0,T ] Y (t) ≤ ǫ} The following equalities hold, for all i, j ∈ [1.

.n] and c ∈ R:

i) E[exp(c T 0 n i,j=1 g ij (t, Y (t))δ j (t, Y (t)) * dY i t ) | A ǫ ] = O(ǫ)
ii)

lim sup ǫ→0 E[exp(- c 2 T 0 d(g ij (t, Y (t))δ j (t, Y (t))), dY i t ) | A ǫ ] ≤ exp(- c 2 T 0 δ j i g ij (t, 0){ 1 2 ġij (t, 0) - 1 6 R ij (t, 0) + ∂ ∂x j Z i (t, 0)} dt) iii) lim sup ǫ→0 E[exp(-c T 0 g ij (t, Y (t)δ j (t, Y (t))γ i (t, Y (t)) dt) | A ǫ ] = 1 Proof. i) Let α = c n i,j=1 g ij (t, x)δ j (t, x)dx i in the neighborhood U ⊂ [0, T ] × R n ,
and extend it in all the space. The asymptotic in the proposition is a direct consequence of Theorem 4.1.

ii) Using Itô formula, equation (3.2) leads to

lim sup ǫ→0 E[exp(- c 2 T 0 d(g ij (t, Y (t))δ j (t, Y (t))), dY i t ) | A ǫ ] = lim sup ǫ→0 E[exp(- c 2 T 0 n l=1 ∂ ∂x l (g ij (t, .)δ j (t, .))(Y (t))dY l dY i ) | A ǫ ] = lim sup ǫ→0 E[exp(- c 2 T 0 n l=1 ∂ ∂x l (g ij (t, .)δ j (t, .))(Y (t))g il (t, Y (t)) dt) | A ǫ ] ≤ exp(- c 2 δ j i T 0 g ij (t, 0)( 1 2 ġii (t, 0) - 1 6 R ii (t, 0) + ∂ ∂x i Z i (t, 0)) dt).
In the last computation we used the Taylor expansion that we compute in the last section. iii) In a similar way we have :

(4.3) lim sup ǫ→0 E[exp(-c T 0 g ij (t, Y (t)δ j (t, Y (t))γ i (t, Y (t)) dt) | A ǫ ] = lim sup ǫ→0 E[exp(-c T 0 O( Y (t) ) dt) | A ǫ ] = 1.
Y (t) ≤ ǫ] = exp T 0 {- 1 2 Z(t, ϕ(t)) -φ(t) 2 g(t) - 1 4 (Tr g(t) ( ġ(t))) ϕ(t) + 1 12 R(t, ϕ(t)) - 1 2 div g(t) Z(t, ϕ(t))}dt = exp - T 0 H(t, ϕ(t), φ(t)) dt
The second term in (3.3) is clearly given by the scaling property of Brownian motion:

P 0 [ sup t∈[0,T ] Y (t) ≤ ǫ] = P 0 [τ n 1 (B) > T ǫ 2 ],
where τ n 1 (B) is the hitting time of the ball of radius 1; and B is a n dimensional Brownian motion. With standard argument of stopping time, Dirichlet problem and spectral Theorem we get the following :

P 0 [ sup t∈[0,T ] Y (t) ≤ ǫ] ∼ ǫ→0 C exp(-λ 1 T ǫ 2 ),
where λ 1 is the first eigenvalue of Laplace operator (-1 2 ∆ R n ) in the unit ball in R n with Dirichlet's boundary condition and C is also an explicit constant that only depends on the dimension. (4.5)

Small discussion about the most probable path

In this section we will use Theorem 1.1, and we will give the equation of the "most likely" curve. We keep the same notations as before.

Let ϕ and ψ be two curves in M such that ϕ(0) = ψ(0) and ϕ(T ) = ψ(T ), in the same way as in Theorem 1.1 we have :

lim ǫ→0 P x 0 [sup t∈[0,T ] d(t, X(t), ϕ(t)) ≤ ǫ] P x 0 [sup t∈[0,T ] d(t, X(t), ψ(t)) ≤ ǫ] = exp - T 0 H(t, ϕ(t), φ(t)) dt exp - T 0 H(t, ψ(t), ψ(t)) dt .
Let us compute the equation of the curve which is critical for the functional :

ϕ → T 0 H(t, ϕ(t), φ(t)) dt,
when the end point is also fixed. In the next proposition we will compute the equation of this curve in a particular case of g(t) -Brownian motion (see [START_REF] Coulibaly | Brownian motion with respect to time-changing riemannian metrics[END_REF]), the general case could be easily deduced by the same computation. Proposition 5.1. Let X t be a L t := 1 2 ∆ t diffusion, where ∆ t is the Laplace operator with respect to a family of metric g(t) that come from the Ricci flow (∂ t g(t) = α Ric g(t) ) as in [START_REF] Coulibaly | Brownian motion with respect to time-changing riemannian metrics[END_REF]. Then the critical curve for the functional: E : ϕ → T 0 H(t, ϕ(t), φ(t)) dt satisfy the following second order differential equation:

∇ t ∂t φ(t) + α Ric #g(t) ( φ(t)) + 1 -3α 12 ∇ t R t (ϕ(t)) = 0
Proof. Let ϕ be a critical curve for E and let exp be the exponential map according to some fixed metric, then for all vector field V over ϕ such that V (0) = V (1) = 0, we have:

∂ ∂ s | s=0 E[t → exp ϕ(t) (sV (t))] = 0.
Let us recall that in this situation Z(t, .) = 0, so

H(t, x, v) = 1 2 v 2 g(t) - 1 -3α 12 R g(t) (x).
Let us write the variation of the curve ϕ as ϕ V (t, s) := exp ϕ(t) (sV (t)), and φV (t, s) := ∂ ∂t ϕ V (t, s) then the above equation becomes:

∂ ∂ s | s=0 T 0 1 2
φV (t, s) + ġ(t) φ(t), V (t) , and by the final condition of the vector field V (0) = V (T ) = 0 we have:

T 0 ∂ t φ(t), ∇ t ∂t V (t) g(t) dt = 0.

Hence equation (5.1) becomes, for all V vector field such that V (0) = V (T ) = 0 : T 0 ∇ t ∂t φ(t), V (t) g(t) + α Ric #g(t) ( φ(t)), V (t) + 1 -3α 12 ∇ t R g(t) (ϕ(t)), V (t) g(t) dt = 0.

We conclude that ϕ is a critical value of E if and only if it satisfies:

∇ t ∂t φ(t) + α Ric #g(t) ( φ(t)) + 1 -3α 12 ∇ t R g(t) (ϕ(t)) = 0

Remark 5.2. The choice of α = 1 3 for the speed of the backward Ricci flow, produces a simplification of the expression above and makes the functional E positive for all time.

Remark 5.3. In the similar way let X t be a L t := 1 2 ∆ t diffusion, where ∆ t is the Laplace operator with respect to a family of metric g(t) then the

Proof. Theorem 1. 1

 1 Putting all things together, Lemma 3.2 , (3.3), (3.4), (4.1) and Proposition 4.3, we get : (4.4) lim ǫ→0 E P [M T | sup t∈[0,T ]

  τ t e j g(t) = ∇ t g(t)(τ t e i , τ t e j ) + ∇ t τ t e i , τ t e j g(t) + τ t e i , ∇ t τ t e j g(t)

	+ ġ(t)(τ t e i , τ t e j )
	= -	1 2	ġ(t) #g(t) (τ t e i ), τ t e j g(t) -	1 2	τ t e i , ġ(t) #g(t) (τ t e j ) g(t)

  R g(t) (ϕ(t)), V (t) g(t) dt.Since ∂ t and ∂ s commute, and the connection ∇ t is torsion free we have∇ t ∂s φV (t, s) = ∇ t ∂t ∂ ∂s ϕ V (t, s).So the characterization of the critical curve become for all vector field V such that V (0) = V (T ) = 0 :

					2 g(t) -	1 -3α 12	R g(t) (ϕ V (t, s)) = 0
	=	0	T	φV (t, 0), ∇ t ∂s φV (t, 0) g(t)
	-	1 -3α 12	∇ t R g(t) (ϕ V (t, 0)),	∂ ∂ s | s=0	ϕ V (t, s) g(t) dt
	= ∇ t (5.1) T 0 φ(t), ∇ t ∂s φV (t, s) g(t) -1 -3α 12 T 0 φ(t), ∇ t ∂t V (t) g(t) -1 -3α 12 ∇

t R g(t) (ϕ(t)), V (t) g(t) dt = 0. By directs computations, ∂ t φ(t), V (t) g(t) = ∇ t ∂t φ(t), V (t) g(t) + φ(t), ∇ t ∂t V (t) g(t)

E-critical curve ϕ satisfy:

∇ t (Tr g(t) ġ(t))(ϕ(t)) = 0.

We could also use this formula for the Brownian motion that come from the mean curvature flow as in [START_REF] Coulibaly | Some stochastic process without birth, linked to the mean curvature flow[END_REF], and compute the most probable path for this inhomogeneous diffusion. We could use this result to compute the most probable path for the degenerated diffusion Z(t) (see Remark 2.9 of [START_REF] Coulibaly | Some stochastic process without birth, linked to the mean curvature flow[END_REF]).

6. Small ball properties of inhomogeneous diffusions for weighted sup norm Let X t (x) be a L t = 1 2 ∆ t +Z(t) diffusion, with the same notation as in the introduction. Let f ∈ C 1 ([0, T ]) which we assume to be a positive function on [0, T ], we want to estimate the following probability

when ǫ is closed to 0. Proposition 6.1.

where

Proof. Let g(t) = 1 f 2 (t) g(t), and d(t, ., .) the associated distance. Then the probability that we have to estimate is

Now after a change of time we will transform X the L t diffusion to a L(t) diffusion, in order to use Theorem 1.1. Let

and let X(t) := X δ(t) , then X become a L(t) diffusion, where

We deduce that:

where in the last line we have used Theorem 1.1, also the Lagrangian H in the last equation is related to the diffusion X. After a change of variable we get the proposition.

We directly deduce the following small ball estimate: Corollary 6.2.

ds