{
  "abstract": "When $X_G=\\mathbb{C}^n/\\!/G$ is a GIT quotient of a vector space, replacing $G$ with a maximal torus $T \\subset G$ produces a closely related toric variety $X_T:=\\mathbb{C}^N/\\!/T$, the \\emph{Abelianization} of $X_G$. Abelian/non-Abelian correspondences connect structures on $X_G$ with structures on $X_T$. In this paper, we study the interaction between the toric moment map $μ$ of $X_T$ and $X_G$, when $X_G=\\mathrm{Gr}(r,n)$. We do this by looking at the restriction of $μ$ to an intermediate space $Z/T \\subset X_T$, where $Z/G=X_G$. We show that the generic fibres of this map are co-Lagrangians that project to Lagrangians on $X_G$. We completely describe the image in the case where $X_G=\\mathrm{Gr}(2,n)$, and show that in the general $\\mathrm{Gr}(r,n)$ case the image can be sliced so that each piece is determined by a finite set of points, related to matroid circuits.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.42123",
  "author_total": 3,
  "authors": [
    {
      "name": "Elana Kalashnikov"
    },
    {
      "name": "Ruxandra Moraru"
    },
    {
      "name": "Francisco Villacis"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/64txvtey173wj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.42123",
  "fulltext_url": null,
  "id": "64txvtey173wj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.42123",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.SG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.42123",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/64txvtey173wj.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.42123",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.42123"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.SG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Abelianization and moment maps",
  "title_text": "Abelianization and moment maps",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/64txvtey173wj",
  "venue": "arXiv",
  "versions": []
}