On a Non Logsymplectic Logarithmic Poisson Structure with Poisson Cohomology Isomorphic to the Associated Logarithmic Poisson Cohomology

The main purpose of this article is to show that there are non logsymplectic Poisson structures whose Poisson cohomology groups are isomorphic to corresponding logarithmic Poisson cohomology groups.


Introduction
Symplectic geometry was discoved in 1780 by Joseph Louis Lagrange when he considered the non constants variable and defined the bracket of two such elements.From symplectic manifold, Poisson defined his brackets as tool for classical dynamics.Charles Gustave Jacobi realized the importance of those bracket and elucidated their algebraic properties.Sophus Lie and others authors began the study of their geometry.Connection of poisson geometry with numbers of areas including harmonic analytic, mechanics of particles and continua; completely integrable systems, justify this recent development.It is interested to recall that number 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.J. Huebschmann in (Huebschmann, J., 1990) study in 1990 Poisson algebra and apply its Lie-Rinehart cohomology in the study of their geometric quantization.A. Polishchuk in (Pichereau, A., 2006) study in 1997 the Poisson brackets in algebraic framework.
In 2002 Ryushi Goto (Goto, R., 2002), with the aim of generalizing the approach of the symplectic, Atiyah class to the construction of the invariants of knots, defined the logsymplectic manifold and study several examples.The notion of logsymplectic manifold is based on the theory of logarithmic differential forms extensively study in (Saito, K., 1980).Logsymplectic manifold is simply a complex manifold X equipped with a symplectic form ω that has simple poles along a hypersurface D ⊂ X.In other words, Poisson structures defined on X − D by any logsymplectic form ω extends to a Poisson bracket on all X whose pfaffian in a reduced defining equation for D. Logsymplectic manifolds can arise when one attempts to compactify symplectic manifolds.Many modulis space in algebraic geometry and gauge theory come equipped with logsymplectic structure.Such Poisson structure can then play an important role in geometric quantization of many classical observable.According to I. Vaisman in (Vaisman, I., 1991), obstruction of quantization of such classical space is measure by Poisson cohomology.But it style very difficult to determine explicit form of Poisson cohomology as we can see in (Pichereau, A., 2006) and (Monnier, Ph., 2002).
In other to propose an alternative method in the computation of such Poisson cohomology, the first author introduce in (Dongho, J., 2012) the notion of logarithmic principal Poisson structure and prove that such Poisson structure induced a Lie-Rinehart structure on the module of logarithmic differential form along a finite generated ideal I, from which he introduce the notion of logarithmic Poisson cohomology, and prove that such logarithmic Poisson cohomology are in general different to the associated Poisson cohomology.It was also prove in (Dongho, J., 2012)  Recently in (Dongho, J. & Yotcha, S. R., 2016), the Differential Point of view of such cohomology has been study and and apply in the prequantization of such Poisson manifold.
More general theory of logarithmic Poisson cohomology and logarithmic Poisson algebra is in preparation in (Dongho, J., et al.).
The main results of this paper are:

Proposition 1 The Poisson cohomology groups of the Poisson algebra
and for all n ≥ 2, Those can be generalize to the case of any algebra with two generator over a non zero characteristic ring.

A-module of Differential Form
It follows that, We deduce the following cochain complex where d i , i = 0, 1, 2 are associated Poisson differential and there are defined by It therefore follows that For a better understanding, we address the cases n = 1 and n = 2 and we end with a generalization.

On the other hand, H(dx) ∈ Der
By definition d 1 is an epimorphism and we have ).Thus, we can write β is a monomorphism and we have In this section, we recall and generalize the methods and results obtained in (Dongho, J., 2012).By using the reasoning above, the associated Poisson differential are: ) .

That is
).We deduce that φ 1 = xu, u ∈ A and consequently β is a monomorphism and we have ker where C 1 [x] = {a 0 + a 1 x; a 0 , a 1 ∈ C} is the vector space of polynomials of degree less than or equal to 1.In other hand, we have At this stage, the calculation of differentials is no longer a secret.We therefore obtains

Calculation of Associated Poisson Cohomological Groups
In this section, we compute all Poisson cohomological groups associated to the above Poisson complex.

P
For any φ ∈ A, we have d 0 φ = 0 if and only if φ = cte ∈ C. We deduce that We deduce that Im d 1 = x n−1 A. In addition, ).Thus φ 1 = xu with u ∈ A and consequently Consider the application β is a monomorphism and we have ker ) By a simple computation, we have ), d 0 (A) are parts of ker d 1 .We deduce that and then, ker since β is a monomorphism, we have and for all n ≥ 2,

Associated Logarithmic Poisson Cohomology
The Poisson 2-form remain π = x n ∂ x ∧ ∂ y and the module of 1-form logarithmic along x, y]dy and the associated logarithmic Hamiltonian map is This Hamiltonian map induced the following complex 0 In particular, for f 1 = g n−1 and a(x) = 0, we have f 2 = ∫ (ng n−1 − x∂xg n−1 )dy.Moreover, for all 0 g 0 (y) + xg 1 (y) + ... + x n−2 g(y), the following equation

haven't solution in
It is a monomorphism of C-modules and it follows from the above description of Ker∂ 1 that In addition, for all g ∈ η(

3 andProposition 2
The logarithmic Poisson cohomology groups of the Poisson structure defined by the logarithmic Poisson 2-form π = x n ∂x ∧ ∂y, which is logarithmic along the ideal I = x n C[x, y] are

Proposition 3
The Poisson cohomology groups of the Poisson algebra(A = C[x, y], {x, y} = x n ) , (n ∈ N * ) x n−1 C[x, y] ⊕ (O A × x n C[x])), there exist u ∈ C[x, y] and v ∈ C[x] such that g = η(x n−1 u) + (0, x n v(x)) = x n−1 (u, −x( ∫ ∂yudy − v)).This element is in ∂ 0 (A) if and only if, there exist a ∈ A such that ∂ 0 (a) = g.This imply that there exist c(x) ∈ C[x] such that a = ∫ udy+c(x) and ∂x(a) = ∂xudy−v(x).But this imply that c(x) = − ∫ v(x)dx and then a = ∫ udy − ∫ v(x)dx and η(x n−1 C[x, y] ⊕ (O A × x n C[x])) ⊂ ∂ 0 (A).In other hand the following equation in u have no solution in C[x, y]Therefore Ker∂ 1 ≃ C n−1 [x] ⊕ η( n−2 ⊕ i=0 x i C[y]) ⊕ ∂ 0 (A) and then H 1 PS (x n ) ≃ C n−1 [x] ⊕ η(This complete the proof of the following proposition Proposition 3 The logarithmic Poisson cohomology groups of the Poisson structure defined by the logarithmic Poisson 2-form π = x n ∂x ∧ ∂y, which is logarithmic along the ideal I = x n C[x, y that when logarithmic Poisson structure are logsymplectic one, the two, Poisson cohomology and logarithmic Poisson cohomology are equivalent.The main objectif of this paper is to prove that there are non logsymplectic Poisson structure with isomorphic Poisson and logarithmic Poisson cohomology.
It follow that The order zero logarithmic Poisson cohomology group is H 0 PS , A is decomposed as follows