differential geometry

Polynomial degeneration and the Poisson geometry of truncated polynomials

arXiv:2602.14341

summary

The paper introduces a formalism for geometric structures that degenerate polynomially along a hypersurface and applies it to Poisson geometry, constructing generically symplectic Poisson structures with non‑trivial symplectic variation whose behavior is governed by lifting problems for representations of truncated‑polynomial groups.

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 $b / lo g$ -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 $I ¨ _{1} (W)$ 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

Topics & keywords

#poisson geometry #symplectic degeneration #truncated polynomial groups #character varieties #symplectic variationPoisson structuresb-symplecticb^k-symplecticG_k grouprepresentation liftingmoduli space