Polynomial degeneration and the Poisson geometry of truncated polynomials

We develop a formalism for studying geometric structures that degenerate to polynomial order along a hypersurface $W \subset M$.

We then demonstrate it in the study of Poisson geometry, where it leads to methods for constructing generically symplectic Poisson structures with non-trivial symplectic variation along their degeneracy locus.

This is in contrast to $b/\log$-symplectic and $b^k$-symplectic structures, where this variation always vanishes.

Our main insight is that the higher residue data along the hypersurface is controlled by a group of transverse diffeomorphisms, which in our case is the group $G_k$ of degree-$k$ truncated polynomials.

We show that the symplectic variation of our Poisson structures is determined by the obstruction to lifting a $G_k$-representation of the fundamental group $\pi_1(W)$ to $G_{k+1}$, and we construct maps from a $G_k$-character variety into the moduli space of Poisson structures, with the variation detecting the non-triviality of the resulting families.