[La classe modulaire d'une variété de Poisson régulière et la classe de Reeb de son feuilletage symplectique]
Pour une variété de Poisson régulière, il existe une application linéaire naturelle de la 1-cohomologie feuilletée vers la 1-cohomologie de Poisson qui envoie la classe de Reeb du feuilletage symplectique sur la classe modulaire de la structure de Poisson. Nous donnons une interprétation riemannienne de la classe de Reeb ; ce qui permettra d'avoir des critères géométriques pour décider de la nullité ou non de la classe modulaire. Finalement, nous prouvons que la 1-cohomologie feuilletée est un invariant de l'équivalence de Morita.
We show that, for any regular Poisson manifold, there is an injective natural linear map from the first leafwise cohomology space into the first Poisson cohomology space which maps the Reeb class of the symplectic foliation to the modular class of the Poisson manifold. A Riemannian interpretation of the Reeb class will give some geometric criteria which enables one to tell whether the modular class vanishes or not. It also enables one to construct examples of unimodular Poisson manifolds and others which are not unimodular. Finally, we prove that the first leafwise cohomology space is an invariant of Morita equivalence.
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@article{CRMATH_2003__337_1_61_0,
author = {Abouqateb, Abdelhak and Boucetta, Mohamed},
title = {The modular class of a regular {Poisson} manifold and the {Reeb} class of its symplectic foliation},
journal = {Comptes Rendus. Math\'ematique},
pages = {61--66},
publisher = {Elsevier},
volume = {337},
number = {1},
year = {2003},
doi = {10.1016/S1631-073X(03)00254-1},
language = {en},
url = {http://www.numdam.org/articles/10.1016/S1631-073X(03)00254-1/}
}
TY - JOUR AU - Abouqateb, Abdelhak AU - Boucetta, Mohamed TI - The modular class of a regular Poisson manifold and the Reeb class of its symplectic foliation JO - Comptes Rendus. Mathématique PY - 2003 SP - 61 EP - 66 VL - 337 IS - 1 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/S1631-073X(03)00254-1/ DO - 10.1016/S1631-073X(03)00254-1 LA - en ID - CRMATH_2003__337_1_61_0 ER -
%0 Journal Article %A Abouqateb, Abdelhak %A Boucetta, Mohamed %T The modular class of a regular Poisson manifold and the Reeb class of its symplectic foliation %J Comptes Rendus. Mathématique %D 2003 %P 61-66 %V 337 %N 1 %I Elsevier %U http://www.numdam.org/articles/10.1016/S1631-073X(03)00254-1/ %R 10.1016/S1631-073X(03)00254-1 %G en %F CRMATH_2003__337_1_61_0
Abouqateb, Abdelhak; Boucetta, Mohamed. The modular class of a regular Poisson manifold and the Reeb class of its symplectic foliation. Comptes Rendus. Mathématique, Tome 337 (2003) no. 1, pp. 61-66. doi : 10.1016/S1631-073X(03)00254-1. http://www.numdam.org/articles/10.1016/S1631-073X(03)00254-1/
[1] Sur la cohomologie feuilletée, Compositio Math., Volume 49 (1983), pp. 195-215
[2] Holonomy on Poisson Manifolds and the modular class, Israel J. Math., Volume 122 (2001), pp. 221-242
[3] Poisson cohomology of Morita equivalent Poisson manifolds, Internat. Math. Res. Notices, Volume 10 (1992), pp. 199-205
[4] The Godbillon measure of amenable foliations, J. Differential Geom., Volume 23 (1986), pp. 347-365
[5] Groupes de Lie à structure symplectique invariante, Symplectic Geometry, Groupoids and integrable systems, Math. Sci. Res. Inst. Publ., 20, Springer, 1991, pp. 247-266
[6] A metric formula for the Godbillon–Vey invariant for foliations, Proc. Amer. Soc., Volume 38 (1973)
[7] Foliations on Riemannian Manifolds, Springer-Verlag, 1988
[8] Lectures on the Geometry of Poisson Manifolds, Progr. Math., 118, Birkhäuser, Berlin, 1994
[9] The modular automorphism group of a poisson manifold, J. Geom. Phys., Volume 23 (1997), pp. 379-394
[10] The local structure of Poisson manifolds, J. Differential Geometry, Volume 18 (1983), pp. 523-557
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