Stability of higher order singular points of Poisson manifolds and Lie algebroids

Dufour, Jean-Paul; Wade, Aïssa

Annales de l'Institut Fourier, Tome 56 (2006) no. 3, pp. 545-559.

Nous étudions la stabilité des singularités de structures de Poisson lisses et des algèbroïdes de Lie générales. Nous donnons des conditions suffisantes de stabilité reposant sur la première approximation (pas nécessairement linéaire) d’une structure de Poisson ou d’algèbroïde de Lie en un point singulier. Les principaux outils utilisés ici sont la cohomologie de Lichnerowicz-Poisson classique et la cohomologie de déformation introduite récemment par Crainic et Moerdijk. De plus, nous fournissons plusieurs exemples de points singuliers stables d’ordre $k\ge 1$ pour des structures de Poisson et des algèbroïdes de Lie. Finalement, nous appliquons nos résultats aux feuilles pré-symplectiques des variétés de Dirac.

We study the stability of singular points for smooth Poisson structures as well as general Lie algebroids. We give sufficient conditions for stability lying on the first-order approximation (not necessarily linear) of a given Poisson structure or Lie algebroid at a singular point. The main tools used here are the classical Lichnerowicz-Poisson cohomology and the deformation cohomology for Lie algebroids recently introduced by Crainic and Moerdijk. We also provide several examples of stable singular points of order $k\ge 1$ for Poisson structures and Lie algebroids. Finally, we apply our results to pre-symplectic leaves of Dirac manifolds.

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