On Bott-Morse Foliations and their Poisson Structures in Dimension 3

TL;DR

This paper demonstrates that 3-dimensional Bott-Morse foliations can be equipped with a compatible singular Poisson structure of rank 2, explicitly describing the bivectors and symplectic forms associated with singularities.

Contribution

It introduces a construction of singular Poisson structures for 3D Bott-Morse foliations, detailing the bivectors and symplectic forms for various singular components.

Findings

Existence of a linear, singular Poisson structure of rank 2 for 3D Bott-Morse foliations

Explicit formulas for Poisson bivectors near singularities

Computation of symplectic forms on characteristic distributions

Abstract

We show that a Bott-Morse foliation in dimension 3 admits a linear, singular, Poisson structure of rank 2 with Bott-Morse singularities. We provide the Poisson bivectors for each type of singular component, and compute the symplectic forms of the characteristic distribution.