Constructibility of momentum maps and linear variation for singular symplectic reduced spaces

2025/08/29 by Maarten Mol, Mol, Maarten
Mathematics · #53A15 (Secondary) #53D20 (Primary) 53D17 #58A35 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Symplectic Geometry (math.SG)

paper · pdf · doi:10.48550/arxiv.2508.21284

openalex publication_date 2025/08/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we show that the transverse image of the momentum map of a Hamiltonian Lie group action admits a natural integral affine stratification with the property that over each stratum the momentum map is an equivariantly locally trivial fibration, provided the group is compact and the momentum map is proper. Using this we extend the linear variation theorem of Duistermaat and Heckman to singular values of the momentum map by showing that the cohomology classes of the symplectic forms on the reduced spaces at values within a stratum vary linearly. We also point out an instance of an invariant cycle theorem for momentum maps. Finally, we extend all of the above to Hamiltonian actions of proper quasi-symplectic groupoids.

Citations

• On the classification of multiplicity-free Hamiltonian actions by regular proper symplectic groupoids
• Stratification of the transverse momentum map
• Regular Poisson manifolds of compact types (PMCT 2)
• The Orbit Space and Basic Forms of a Proper Lie Groupoid
• Geometry of orbit spaces of proper Lie groupoids
• String topology for stacks
• Momentum Maps and Morita Equivalence
• Lie Group Valued Moment Maps

Related

• Stratification of the transverse momentum map
• Hamiltonian group actions on exact symplectic manifolds with proper momentum maps are standard
• Proper group actions and symplectic stratified spaces
• Differential Spaces, Vector Fields, and Orbit-Type Stratifications
• Momentum Maps and Morita Equivalence