1. Introduction

Symplectic geometry was discovered in 1780 by Joseph Louis Lagrange when he considered the orbital element of planet in solar system as non constants variables and defined the brackets of two such elements. From symplectic manifold, Poisson defined his brackets as tool for classical dynamics. Charles Gustave Jacob Jacobie realized the importance of those bracket and elucidated their algebraic property. Sophus Lie and other authors began the study of their geometry. Connection of Poisson geometry with members of areas including harmonic analytic, mechanics of particles and continua; completely integrable systems, justify his recent development. It is interested to recall that numbers of proprieties and results in this theory was developed in the case of differential manifold. Too few authors have worked in the case of singular varieties. Recently, Huebschmann in^[5] study the algebraic point of view; A. Polishchuk in ^[7] study the Poisson brackets in algebraic framework.

In 2002, Ryushi Goto, with the aim of generalised the approach to the symplectic Atiyah class to the construction of the invariants of knots defined the Log Symplectic manifold and studied several examples in ^[2].

In the first section of this paper, we recall the concepts of logarithmic derivation and logarithmic form, as well as free divisor, all of them due to Kyogi Saito ^[1] and the definition of Log Symplectic manifold. We end the section with the notion of Log Poisson brackets on such varieties.

In the second part, we define the concept of Logarithmic multi-vector field. In the aim to define the notion of Log Poisson manifold the section will end with the notion of Log Schuten brackets.

In the third part, we define the notion of Log Poisson manifold and give some main examples. We introduce the Lichenerowicz Log Poisson cohomology and Log Poisson Chern class of a log complex line bundle over Log Poisson manifold; using the concept of contravariant log derivative who generalize the concept of Contravariant derivative given by I. Vaisman in ^[6].

Section fourth is devoted to the formulation of the integrality Log prequantization condition of Log Poisson manifold.

2. Log-Symplectic Poisson Structure

In this section, we recall the notion of Log-symplectic manifold introduce in 2002 by ^[2] and we construct Poisson structure induce by a Log symplectic structure.

2.1. Log symplectic manifold. Recall that; a logarithmic q-forme is a multilineal skew-symmetric function

A logarithmic 0-form are germs of . Sheaf of germs of log form are constructed for singular or normal crossing divisor by ^[1]. The coherent sheaf is the dual of a sheaf of germs of logarithmic vector fields Hence is reflexive, i.e., normal and torsion-free. If is local section of then where The sheaf of smooth differential forms is subsheaf of . The following will be very useful in what follow.

Lemma 2.1. is Lie subalgebra of

Proof. It is enough to prove that is closed with Lie bracket of vector field. Let be two logarithmic vectors field along For all Therefor:

The lemma allows us to define the following mapping by

(2.1)

It is easy to prove that

In particular, for all germs of is -linear map on with values in For all germ of where is locally -module structure on If i.e. -linear, then:

In this paper, A logarithmic form is close if

Definition 2.2. The complex is logarithmic De Rham complex. The corresponding cohomology is called logarithmic De Rham cohomology and denoted

The locally -module structure of could be extended to as follow:

for all and for all and

More generally, for all where is contraction of logarithmic form by logarithmic vector field.

A simple calculation give us

(2.2)

It follow that is Lie homomorphism. In ^[1] it is prove that each germs of have the form where are smooth forms. is called residue form of and denoted . Since is the dual of the there exist an isomorphism

induce an isomorphism

Definition 2.3. The sheaf is called sheaf of logarithmic -vector field.

Definition 2.4. ^[2] A Log symplectic manifold is a triple where is complex manifold, is reduced divisor of and is logarithmic 2-form satisfy:

(2.3)

(2.4)

The singular part of reduced divisor is denoted by Then on the smooth part we have the residue 1 form We also have a log generalization of Darboux's theorem:

Lemma 2.5. ^[2](Log DARBOUX THEOREM).

Let be a log symplectic manifold. There exist holomorphic coordinates of a neighborhood of each smooth point of such that is given by

(2.5)

where We refer to those coordinates as log Darboux coordinates.

In log Darboux coordinates, which is closed. Here, we have an integrable and dimensional leaves on We have another useful lemma:

Lemma 2.6. ^[2] The log symplectic form defines a symplectic structure on each leaf.

2.2. Examples of log symplectic manifold. In litterature, there exist more examples of such manifold we refer the reader to ^[2] for some one.

(1) ^[2] Let be a complex surface with a reduced divisor If the class is anticanonical class then the triple is log symplectic manifold with log simplectic form

(2) On Toda lattice of dimension two the symplectic form

Induced on a structure of log symplectic manifold

2.3. Hamiltonian and log Hamiltonian vector field. The important role played by Hamiltinian vector field and the corresponding Hamiltonian in classical mechanic force us to answer the question that is it possible to define the notion of Hamiltonian vector field on symplectic manifold? The fact that such manifold are not smooth, complicate concept of vector field; seem tangent vector that do not always exist at each point.

2.3.1. General construction of symplectic Poisson structure. Begin by recalling the general concept of Hamiltonian operator. The reader could refer to^[9] for more explanation. Let a Lie algebra with Lie structure (the commutator) and an -module. If is a fixed 2-form, In the space consider the subspace

Introduce on the following operations:

(2.6)

(2.7)

On The following theorem prove that when is closed, is Lie algebra.

Theorem 2.7. ^[9] If then is closed relative to and hence a Lie algebra with respect to it.

By construction, we have the following commutative diagrams.

We denote By definition, there exist such that i.e., for all

We recall that for all Then, for all Therefor, for all

Remark 2.8.

(1) If and for all

then for all If is non degenerated, (it is the case for log symplectic structure),

(2) Let Chose such that

We set

(2.8)

Il such that . Since is claused under addition, i.e., i.e., Therefor, This prove that the operation didn't depend on the choice of and .

Corollary 2.9. If then the operation is Poisson structure on N

Definition 2.10. The operation defined by equation (2.8) is called Poisson symplectic structure when is symplectic form.

2.3.2. Application to the construction of log symplectic Poisson structure. If is a log symplectic structure, then and is non degenerated. Consider

For all We consider the sheaf of algebra generated by We have the following lemma:

Lemma 2.11. is sheaf of -module

Proof. Let Then for all Define by .

Set

The image of by second projection map is It follow from remark [2.8] that for all there exist such that

Let the corresponding operation defined by equation (2.8) is:

(2.9)

Where are such that

From corollary [2.9], it come that the pairs is Poisson algebra.

According to the definition on Poisson manifold, the log symplectic manifold become a Poisson manifold. Since the Poisson structure is coming from a symplectic structure, we will call this log symplectic Poisson manifold.

Definition 2.12. The operation (2.9) is log symplectic Poisson structure and the algebra is called log symplectic Poisson algebra.

2.3.3. Hamiltonian vector field and Hamiltonian operator. In the last section, we have seen that for all there exist an unique such that Therefore, we have the following definitions:

Definition 2.13. (1) The pairs is called log Hamiltonian pairs associated to

(2) In each log Hamiltonian pairs when a is called Hamiltonian vector field and m is his Hamiltonian.

(3) If and is log Hamiltonian pairs, then a is called log Hamiltonian vector field and m is his hamiltonian.

Let be a log Hamiltonian pairs. For all Then Therefore, a is Hamiltonian vector field on M iff there exist such that It is log Hamiltonian vector field on M iff there exist such that.

In the goal of defining the notion of Hamiltonian operator, set the subsheaf of d-image of

In general, if and are at in section one, and if is the d-image of then an linear operator is saying skew-symmetric if for any

(2.10)

Let be a skew-symmetric operator. Consider It's prove in ^[9] that is well define 2-form on

Definition 2.14. ^[9] A skew-symmetric operator is said to be Hamiltonian if:

a) is subalgebra of Lie algebra #

b) is closed in .

Now, giving a log symplectic manifold , and is non degenerated. Therefore, for all which is an element of

Then induce an isomorphism We could state the following lemma.

Lemma 2.15. The operator is an Hamiltonian operator from to

Definition 2.16. The map H in [2.15] is called log Hamiltonian operator.

Remark 2.17.

For all and for

therefore, for all for all Consequently, for al is the Hamiltonian vector field of Hamiltonian and for each is the log Hamiltonian vector field associated to the log Hamiltonian

It follows that the log symplectic Poisson structure is defined by:

(2.11)

2.4. Some properties of log symplectic Poisson structure.

Lemma 2.18. The bracket is logarithmic derivation on each component.

Proof. From lemma 2.5, at each point of there exist holomorphic coordinates such that For all we have by a simple calculate:

(2.12)

and

(2.13)

For Then

Therefore, for all

Corollary 2.19. There exist holomorphic local coordinates of each point of such that

Corollary 2.20. For each log symplectic Poisson structure Poisson there exist a unique such that

Definition 2.21. is log symplectic Poisson bivector.

If A and B are two matrix of the same dimension denote Then the matrix of is

3. On a Generalization of Log Symplectic Poisson Bracket: Log Poisson Structure

3.1. Logarithmic multivector field. It is well knowed that the dual of sheaf is the sheaf of logarithmic vector field. Let is the module of sections of is Lie subalgebra of which play an important role in the notion of Poison Manifold in the point of view of A. Lichenerowicz. In the goal of constructing the notion of Log Poisson Manifold, we shall define the notion of a multi vector field. First of all, we recall that for all and a logarithmic contravariant antisymmetric p-vector field is a p-linear map The set of logarithmic contravariant antisymmetric p-vector field at a point is denoted We denote

is subbundle of tangent bundle in the smooth case. We will called the logarithmic tangent bundle and we denoted the module of his section. Element of is -linear combination of homogeny of degree p; i.e; with

Definition 3.1. Element of are called log p-vector field.

Exemple 3.2.

(1) ^[10] Let be a complex manifold of dimension and a normal crossing divisor of define by where local system of holomorphic coordinates of The set form a basis of From S1 we deduce a basis of In the same way, we construct basis of

(2) On denoted by the closed sphere of radius 1. The bivector is log 2-vector field

we shall remark that if is local coordinates of M and is the corresponding local coordinates of divisor for all then Therefore, we could then state the following lemma.

Lemma 3.3. Let M be a complex manifold of dimension n and D a reduced divisor of M. For each and each local section of there is such that:

(1)

(2) D

(3)

Proof. Let is local holomorphic coordinates of and is the corresponding local coordinates of divisor Set

and

is a generator of

Since there is such that

However, we know that there is family of holomorphic functions such that

We set and

Definition 3.4. Let be a log p-vector field.

(1) if then is called smooth p-vector field associated to

(2) is saying log Euler along D if

3.2. Logarithmic super-algebra of Lie. Always in the main fine the definition of log Poisson manifold using the Lichenerowicz methods, we shall construct on the algebra a Lie structure. Of cause, we shall define the notion of logarithmic Schouten bracket on To do that, we shall define an important notion of interior product of log vector field with logarithmic form. It is the main of the following definition.

Definition 3.5. Let be a logarithmic -form and a log p-vector field. The interior product of with is define as follows:

• If let

• If and where then

(3.1)

When and is finial dimension manifold, the equation 3.1 is an isomorphism between and

One check that

(3.2)

In the goal to construct an hyper Lie structure on the associative, supercomutative algebra we recall the following useful proposition.

Proposition 3.6. Let be a Lie algebra. There is an unique bracket on which extend the Lie bracket on g and such that if then,

(1)

(2)

(3)

in the above section, we have proven that endowed with the Lie-Jacobi structure is Lie algebra; we can then apply the above theorem on it and its became a Gestenhaber algebra. The corresponding hyper Lie structure we be called Log Schouten bracket. Precisely, we can follow the usual prove of Schouten Theorem to prove what follow.

Theorem 3.7. (Log Schouten Bracket Theorem.)

There is a unique bilinear operator such that

(1) It is biderivation of degree -1, that is it is bilinear, and for all

(2) It is determined on and by:

(a)

(b)

(c) is the Jacobi-Lie bracket of logarithmic vector field

(3).

Definition 3.8. The operation of theorem [3.7] is called logarithmic Schouten bracket.

In what follows we will denoted From theorem [3.7] we will deduce the following lemma.

Lemma 3.9. is super-Lie algebra.

3.3. Log Poisson structures. We can now give the definition of logarithmic Poisson structure.

Definition 3.10. A log-Poisson structure is local section of the sheaf such that .

We deduce the definition of log-Poisson manifold.

Definition 3.11. A log-Poisson manifold is a triple where is complex manifold, is reduced free divisor of and is log-Poisson structure along

It follows from definition of that log Euler Poisson structure degenerate on divisor

3.4. Examples of log-Poisson manifold. log-Poisson structure appear in many way. We will give some of them.

(1) log-symplectic manifold. Each log-symplectic manifold is log-Poisson maifold.

(2) Singular Toda lattice.

Let and consider the bracket

where and The log bivector associated is log-Poisson structure on who become a log-Poisson manifold.

4. The Lichenerowicz-log Poisson Cohomology of Log-Poisson Manifold

Let be a Log Poisson manifold. As in the case of Poisson manifold, we introduce hamiltonian vector field on by setting, for any . Follow ^[4] or simply use the definition of sharp map we have The jacobian identity is equivalent to

The map is the anchor of algebroide with the following Lie bracket

(4.1)

Therefore, bracket satisfy the following equality

(4.2)

The existence of this Lie-braked on allowed us to define the following exterior product by

(4.3)

It is clear that Therefore, constituted a chain complex calling log Poisson complex.

Definition 4.1. We call Lichenerowiecz-log Poisson Cohomology (L-log P cohomology) of the cohomology of

It is denoted by or simply and for any

(4.4)

The cohomology class of any element will be denoted by .

By definition of anchor map we have

(4.5)

where is the Lie bracket on

In the Poisson Manifold, induce cohomological homomorphism which in quasi-isomorphism in symplectic case. We are interest to know if or not the result trow in log Poisson manifold.

Let Define on by

(4.6)

It follows that;

since

then

and then

We have prove the following lemma

Lemma 4.2. The homomorphism is chain map,

Hence we deduce

Proposition 4.3. If is the logarithmic de Rham cohomology, then the homomorphism induces a homomorphism in cohomology, also denoted by

5. Log Poisson Chern Class of Complex Line Bundle over a Log Poisson Manifold

Let be a log Poisson manifold. a complex line bundle over the space of global cross section of and the space of the complex linear endomorphism of In order to define the notion of log prequantization, we shall define the notion of log contravariant derivative.

Definition 5.1. A log contravariant derivative on is mapping such that:

(5.1)

for all and a local section of

Equivelently, is log contravariant derivative if for all and .

We say that is hermitian or compatible with hermitian metric on if for all

Remark 5.2. If is the connection on with logarithmic singularities along is log contravariant derivative on

Definition 5.3. The curvature of a log contravariant derivative on is a mapping ; define for all

(5.2)

We have the following proposition

Proposition 5.4. is -bilinear skew- symmetric.

Proof. For all

Let be a section of we have

From the above result and the fact that is complex line bundle, there exists a globelley defined complex bivector field on M with such that for all and

(5.3)

By linearity, the cohomology operator on the log multivector fields on M by setting for any

then Consequently, is a chain complex whose cohomology will be called the complex Lichnerowicz-log Poisson of and will be denoted by or

Proposition 5.5. Let be a complex line bundle over a log Poisson manifold a log contravariant derivative on the curvature of and the complex bivector field on associated to Then

i) define a cohomology class in

ii) does not depend of the log contravariant derivative

iii) In the case where is compatible with a hermitian metric on is purely imaginary.

Proof.

i) Let be a nowhere vanishing local section of Since the complex dimension of the fiber of is 1, we may associate to s as follows. It is clear that for any logarithmic 1-form on is a complex function on and the application is -linear Hence, there exists a unique complex local vecto field on with local real logarithmic vector fields in such that, for all

we have that

Hence for all then

Consequently, wich mean that define a cohomology class in .

ii) Let be another log contravariant derivative on having curvature and corresponding local complex log vector field. We denote by the corresponding complex log bivector field on We obtain i.e;

Now, for any the mapping

Therefore there exists a global defined complex log vector field such that for all

then

i.e;

hence

iii) Assume that is compatible with a hermitian metric on and let be a local orthogonal basis of then for all we have

i.e;

i.e;

i.e;

hence is purely imaginary and because of we conclude that is purely imaginary.

From the property iii) of the theorem, it follows that Therefore, we have the following definitions.

Definition 5.6. is the first real log Poisson-Chern class of

6. Prequantization of Log Poisson Structure

Let be a Log Poisson manifold and a hermitian line bundle over with a log contravariant derivative with curvature . We define a representation of Lie algebra on by associating to each a complex endomorphism of that is defined for any local section of by

(6.1)

It is well know that is Lie algebra where denote the usual communtator on

Proposition 6.1. The representation is a homomorphism of Lie algebra iff

(6.2)

Proof. It is simply computation using (6.1)

Definition 6.2. We say that Log Poisson manifold is prequantizable if there exist an hermitian complex line bundle such that operator make sence on and satisfy (6.2)

Therefore, the prequantization problem of Log Poisson manifold has solution iff there exist a Hermitian line bundle equipped with a log contravariant derivative whose the curvature satisfy

(6.3)

We can now give a prequantization condition of Log Poisson manifold. Let be a Log Poisson manifold with a reduced free divisor of Denote We have the following sequence:

where the last map is induce by the sharp map. This allow us to think about an integrable and real Log Poisson cohomology class.

Proposition 6.3. The Log Poisson manifold is prequantizable if and only if, there exist a logarithmic vector field on and closed logarithmic 2-form on which represents an integral cohomology class of such that the following relation holds on

(6.4)

Proof. Suppose that there exist a logarithmic vector field and closed logarithmic 2-form on such that 6.4 is true on Then, there exists hermitian complex line bundle over equipped with a hermitian connection with logarithmic pole along having as curvature 2-form the purely imaginary closed 2-form . Using we define a log contravariant derivative on as follows: for all and ,

(6.5)

It is easy to see that is Hermitian and his curvature satisfy 6.3.

Conversely, we suppose that is Prequatizable. Then there exist a Hermitian complex line bundle with Hermitian log contravariant derivative whose curvature verify 6.3. Consequently

(6.6)

where is the purely imaginary, -closed, logarithmic bivector associated to Let be the Hermitian connection on with curvature 2-form . So, is real closed logarithmic 2-form on and represents the first real Chern class which is integral, i.e., Now we consider the Hermitian log contravariant derivative on defined by Let be the purely imaginary logarithmic bivector field on M associated to We have Therefore, which means that there exists a purely imaginary logarithmic vector field on such that Then

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