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Poisson Manifolds of Compact Types

Marius Crainic, Rui Loja Fernandes, David Martínez Torres

TL;DR

This work develops Poisson and Dirac geometry for manifolds of proper type (PMCTs/DMCTs) as a robust generalization of compact Lie theory. It introduces two canonical stratifications, a Weyl resolution that desingularizes PMCTs, and a leaf-space structure that carries an integral affine orbifold structure, with a Weyl group acting as a Coxeter group on the affine leaf-space universal cover. A Duistermaat–Heckman theory is extended to s-proper Poisson manifolds, yielding linear variation of leafwise cohomology and a Weyl integration formula; crucially, the results imply that Poisson manifolds of compact type are regular. The framework unifies local normal forms, canonical stratifications, and Weyl-type resolutions across Poisson and Dirac settings, connects leaf-space geometry to integral affine structures, and culminates in a rich set of open problems and directions for future research.

Abstract

We develop the theory of Poisson and Dirac manifolds of compact types, a broad generalization in Poisson and Dirac geometry of compact Lie algebras and Lie groups. We establish key structural results, including local normal forms, canonical stratifications, and a Weyl type resolution, which provides a way to resolve the singularities of the original structure. These tools allow us to show that the leaf space of such manifolds is an integral affine orbifold and to define their Weyl group. This group is a Coxeter group acting on the orbifold universal cover of the leaf space by integral affine transformations, and one can associate to it Weyl chambers, reflection hyperplanes, etc. We further develop a Duistermaat-Heckman theory for Poisson manifolds of s-proper type, proving the linear variation of cohomology of leafwise symplectic form and establishing a Weyl integration formula. As an application, we show that every Poisson manifold of compact type is necessarily regular. We conclude the paper with a list of open problems.

Poisson Manifolds of Compact Types

TL;DR

This work develops Poisson and Dirac geometry for manifolds of proper type (PMCTs/DMCTs) as a robust generalization of compact Lie theory. It introduces two canonical stratifications, a Weyl resolution that desingularizes PMCTs, and a leaf-space structure that carries an integral affine orbifold structure, with a Weyl group acting as a Coxeter group on the affine leaf-space universal cover. A Duistermaat–Heckman theory is extended to s-proper Poisson manifolds, yielding linear variation of leafwise cohomology and a Weyl integration formula; crucially, the results imply that Poisson manifolds of compact type are regular. The framework unifies local normal forms, canonical stratifications, and Weyl-type resolutions across Poisson and Dirac settings, connects leaf-space geometry to integral affine structures, and culminates in a rich set of open problems and directions for future research.

Abstract

We develop the theory of Poisson and Dirac manifolds of compact types, a broad generalization in Poisson and Dirac geometry of compact Lie algebras and Lie groups. We establish key structural results, including local normal forms, canonical stratifications, and a Weyl type resolution, which provides a way to resolve the singularities of the original structure. These tools allow us to show that the leaf space of such manifolds is an integral affine orbifold and to define their Weyl group. This group is a Coxeter group acting on the orbifold universal cover of the leaf space by integral affine transformations, and one can associate to it Weyl chambers, reflection hyperplanes, etc. We further develop a Duistermaat-Heckman theory for Poisson manifolds of s-proper type, proving the linear variation of cohomology of leafwise symplectic form and establishing a Weyl integration formula. As an application, we show that every Poisson manifold of compact type is necessarily regular. We conclude the paper with a list of open problems.

Paper Structure

This paper contains 60 sections, 91 theorems, 482 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Local Normal Forms
  3. The Hamiltonian normal form
  4. The linear normal form
  5. Local models in the Dirac setting
  6. The Canonical Stratifications
  7. Stratifications and orbit types
  8. General stratifications
  9. The stratifications induced by a proper action
  10. The stratifications induced by a proper Lie groupoid
  11. Stratification of foliations of proper type
  12. The canonical infinitesimal stratification of PMCTs
  13. The canonical stratification of PMCTs
  14. Canonical stratifications of DMCTs
  15. The Weyl Resolution
  16. ...and 45 more sections

Key Result

Theorem 1

Each proper symplectic integration $(\mathcal{G},\Omega)$ of a Poisson manifold $(M,\pi)$ induces an integral affine orbifold structure on its leaf space $B$. Moreover:

Figures (2)

  • Figure 1: Homotopy supported on Moebius band
  • Figure 2: Orbifold cover and Weyl chambers

Theorems & Definitions (209)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 2.1
  • proof : Proof of Theorem \ref{['thm-lf-Poisson-v2']}
  • Remark 2.2: Morita equivalences
  • Theorem 2.3
  • ...and 199 more