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E-structures and almost regular Poisson manifolds

Alfonso Garmendia, Eva Miranda

TL;DR

This work investigates the Poisson geometry of $E$-structures and almost regular Poisson manifolds by placing them in the Lie bi-algebroid and Poisson groupoid framework. It establishes a duality between $E$-symplectic structures and almost regular Poisson structures, and proves that their integrating groupoids carry multiplicative Poisson structures that can be described explicitly near the identity; in key cases such as $b$- and $b^m$-symplectic, the results yield concrete local models and, via desingularization, global realizations. The paper also analyzes the integration problem in various settings (cosymplectic, elliptic, and edge-type structures), providing explicit formulas for $oldsymbol{s}$, $oldsymbol{t}$, and $oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol}}}}}$-level Poisson structures, and connects commutative frames to Darboux–Carathéodory-type normal forms. These contributions give practical tools for symplectic integration of singular Poisson manifolds and illuminate the local-to-global structure of Poisson groupoids arising from $E$-structures and almost regular foliations.

Abstract

In recent years, $b$-symplectic manifolds have become important structures in the study of symplectic geometry, serving as Poisson manifolds that retain symplectic properties away from a hypersurface. Inspired by this rich landscape, $E$-structures were introduced by Nest and Tsygan in \cite{NT2} as a comprehensive framework for exploring generalizations of $b$-structures. This paper initiates a deeper investigation into their Poisson facets, building on foundational work by \cite{MS21}. We also examine the closely related concept of almost regular Poisson manifolds, as studied in \cite{AZ17}, which reveals a natural Poisson groupoid associated with these structures. In this article, we investigate the intricate relationship between $E$-structures and almost regular Poisson structures. Our comparative analysis not only scrutinizes their Poisson properties but also offers explicit formulae for the Poisson structure on the Poisson groupoid associated to the $E$-structures as both Poisson manifolds and singular foliations. In doing so, we reveal an interesting link between the existence of commutative frames and Darboux-Carathéodory-type expressions for the relevant structures.

E-structures and almost regular Poisson manifolds

TL;DR

This work investigates the Poisson geometry of -structures and almost regular Poisson manifolds by placing them in the Lie bi-algebroid and Poisson groupoid framework. It establishes a duality between -symplectic structures and almost regular Poisson structures, and proves that their integrating groupoids carry multiplicative Poisson structures that can be described explicitly near the identity; in key cases such as - and -symplectic, the results yield concrete local models and, via desingularization, global realizations. The paper also analyzes the integration problem in various settings (cosymplectic, elliptic, and edge-type structures), providing explicit formulas for , , and -level Poisson structures, and connects commutative frames to Darboux–Carathéodory-type normal forms. These contributions give practical tools for symplectic integration of singular Poisson manifolds and illuminate the local-to-global structure of Poisson groupoids arising from -structures and almost regular foliations.

Abstract

In recent years, -symplectic manifolds have become important structures in the study of symplectic geometry, serving as Poisson manifolds that retain symplectic properties away from a hypersurface. Inspired by this rich landscape, -structures were introduced by Nest and Tsygan in \cite{NT2} as a comprehensive framework for exploring generalizations of -structures. This paper initiates a deeper investigation into their Poisson facets, building on foundational work by \cite{MS21}. We also examine the closely related concept of almost regular Poisson manifolds, as studied in \cite{AZ17}, which reveals a natural Poisson groupoid associated with these structures. In this article, we investigate the intricate relationship between -structures and almost regular Poisson structures. Our comparative analysis not only scrutinizes their Poisson properties but also offers explicit formulae for the Poisson structure on the Poisson groupoid associated to the -structures as both Poisson manifolds and singular foliations. In doing so, we reveal an interesting link between the existence of commutative frames and Darboux-Carathéodory-type expressions for the relevant structures.

Paper Structure

This paper contains 17 sections, 24 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. The objects
  3. Definitions
  4. Some motivating examples
  5. The groupoid of an almost regular foliation
  6. Almost regular Poisson manifolds and $E$-symplectic manifolds
  7. Relations between $E$-symplectic manifolds and almost regular Poisson manifolds
  8. Review on Poisson groupoids for $E$-structures
  9. The Poisson groupoid integrating the Hamiltonian vector fields for almost regular Poisson manifolds
  10. The Poisson groupoid integrating the Hamiltonian vector fields for a $b^m$-symplectic manifold.
  11. The case of $b$-symplectic manifolds
  12. The symplectic groupoid of $(\mathbb{R}^2,f(x)\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y})$ for $f$ with discrete zeros
  13. The integrating groupoid for a desingularization of $b^{2k}$-symplectic manifolds
  14. On the symplectic integration of almost regular Poisson manifolds
  15. The case of cosymplectic manifolds
  16. ...and 2 more sections

Key Result

Proposition 2.8

Let $M$ be a manifold and $E\subset \mathfrak{X}(M)$ a singular foliation, the following three statements are equivalent.

Theorems & Definitions (67)

  • Example 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Proposition 2.8
  • Definition 2.9
  • ...and 57 more