On the chamber decomposition of a Hamiltonian torus action

Abstract

For a compact connected Hamiltonian T-space (with T a torus), the components of the set of (relative) regular values of the momentum map are open convex polytopes, whose closures (called the chambers) form a polyhedral decomposition of the momentum polytope. In a previous paper we showed that there is a canonical affine stratification of the momentum polytope with the property that, while varying through a stratum, the fibers of the momentum map do not change as T-spaces. In this paper we show that the strata of this stratification are the relative interiors of the faces of the chambers. In doing so, we also give a proof of the fact that these faces form a polyhedral complex, which (it seems) was missing from the literature. We further extend the above to Hamiltonian T-spaces whose momentum map is proper as map into a convex set and, in the compact Kähler case, we point out a description of the above strata in terms of the TC-orbit closure polytopes.

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BibTeX RIS

BibTeX

@misc{mol2026chamber,
  title = {{On the chamber decomposition of a Hamiltonian torus action}},
  author = {Maarten Mol},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.30146},
  eprint = {2609.30146},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.30146},
  note = {CC BY-NC-SA 4.0},
}

RIS

TY  - UNPB
TI  - On the chamber decomposition of a Hamiltonian torus action
AU  - Maarten Mol
PY  - 2026
DA  - 2026/09/24
PB  - arXiv
DO  - 10.48550/arxiv.2609.30146
UR  - https://arxiv.org/abs/2609.30146
AB  - For a compact connected Hamiltonian T-space (with T a torus), the components of the set of (relative) regular values of the momentum map are open convex polytopes, whose closures (called the chambers) form a polyhedral decomposition of the momentum polytope. In a previous paper we showed that there is a canonical affine stratification of the momentum polytope with the property that, while varying through a stratum, the fibers of the momentum map do not change as T-spaces. In this paper we show that the strata of this stratification are the relative interiors of the faces of the chambers. In doing so, we also give a proof of the fact that these faces form a polyhedral complex, which (it seems) was missing from the literature. We further extend the above to Hamiltonian T-spaces whose momentum map is proper as map into a convex set and, in the compact Kähler case, we point out a description of the above strata in terms of the T_C-orbit closure polytopes.
ER  -

Abstract

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