Poisson geometry of truncated polynomials and hypersurface algebroids

We study symplectic forms on hypersurface algebroids. These are a broad generalization of the $b^{k}$-Poisson structures studied extensively by Miranda, Scott, and collaborators, and their geometry is intimately related to the group of truncated polynomials under composition. They induce Poisson structures that are generically symplectic and drop rank along a codimension $1$ submanifold $W$. However, unlike in the case of $b^{k}$-Poisson structures, the symplectic foliation along $W$ can have non-zero symplectic variation, reflecting the obstruction to extending the order of vanishing of a hypersurface algebroid. In addition to studying the symplectic geometry of these algebroids, in this paper we carry out a detailed study of the Lie algebroid de Rham complex, and develop a method for deforming symplectic forms along paths in a $k$-jet character variety. As a result, we are able to produce a large class of new examples of Poisson structures. Finally, we construct universal hypersurface algebroids and show that in even dimensions they admit canonical Poisson structures.