Formal deformations of Poisson structures in low dimensions

TL;DR

This paper investigates formal deformations of Poisson structures in low dimensions, providing explicit classifications and showing that, for generic cases, all deformations are non-equivalent.

Contribution

It offers an explicit basis for second Poisson cohomology and solves deformation equations to classify all formal deformations in low-dimensional cases.

Findings

Constructed large families of formal deformations for specific Poisson structures.

Proved that these families encompass all possible formal deformations up to equivalence.

Showed that, in generic cases, all deformations are non-equivalent.

Abstract

In this paper, we study formal deformations of Poisson structures, especially for three 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 which 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 non-equivalent.