Polynomial degeneration and the Poisson geometry of truncated polynomials

Abstract

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.

Updated and streamlined version. See the first version for a more extensive account. This paper includes an expanded version of sections 2 and 7 from arXiv:2311.17045. The remaining sections were expanded in arXiv:2508.20241