{
  "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 $T_\\mathbb{C}$-orbit closure polytopes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.30146",
  "author_total": 1,
  "authors": [
    {
      "name": "Maarten Mol"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cjp5s4dbgse5w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.30146",
  "fulltext_url": null,
  "id": "cjp5s4dbgse5w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.30146",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.SG"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214344933",
  "pdf_url": "https://arxiv.org/pdf/2609.30146",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cjp5s4dbgse5w.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.30146",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.30146"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "On the chamber decomposition of a Hamiltonian torus action",
  "title_text": "On the chamber decomposition of a Hamiltonian torus action",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/cjp5s4dbgse5w",
  "venue": "arXiv",
  "versions": []
}