Formal deformations of Poisson structures in low dimensions

Anne Pichereau

Vol. 239 (2009), No. 1, 105–133

DOI: 10.2140/pjm.2009.239.105

Abstract

In this paper, we study formal deformations of Poisson structures, especially for two families of Poisson varieties in dimensions two and three. For these families of Poisson structures, using an explicit basis of the second Poisson cohomology space, we solve the deformation equations at each step and obtain a large family of formal deformations for each Poisson structure that we consider. With the help of an explicit formula, we show that this family contains, modulo equivalence, all possible formal deformations. We show moreover that, when the Poisson structure is generic, all members of the family are nonequivalent.

Keywords

deformations, Poisson structures, Poisson cohomology

Mathematical Subject Classification 2000

Primary: 17B63, 58H15, 17B55

Milestones

Received: 30 May 2008

Accepted: 4 September 2008

Published: 1 January 2009