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

## TL;DR

This paper demonstrates that the transverse image of a momentum map admits an integral affine stratification, enabling the extension of the linear variation theorem to singular values and generalizing to quasi-symplectic groupoids.

## Contribution

It introduces a natural stratification of the momentum map's image and extends the linear variation theorem to singular values and broader groupoid actions.

## Key findings

- The transverse image of the momentum map admits an integral affine stratification.
- The linear variation theorem extends to singular values of the momentum map.
- An invariant cycle theorem for momentum maps is identified.

## 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.