{
  "abstract": "To any compact holomorphically symplectic manifold and a nef BBF-isotropic class we associate a quasi-projective variety called the Markman--Markushevich space which parametrises the Lagrangian fibrations with this isotropic class modulo Tate--Shafarevich twists. We give a systematic interpretation of known results about Tate--Shafarevich twists of Lagrangian fibrations using this space, employ it to obtain a conjectural result for the last deformation type for which such a result is not yet proposed, and make some first steps towards its proof. As a byproduct, we generalise the results of Kumar on existence Shioda--Inose partners for abelian surfaces to a slightly more general class of K3 surfaces, and associate a canonical K3 surface with transcendental lattice isometric to $U^2\\oplus\\langle-2\\rangle^2$ to a smooth cubic surface with a choice of a Schläfli double-six. Finally, we construct a noncompact six-manifold with an abelian fibration associated to a cubic surface with a Schläfli double-six, and give its interpretation in terms of the K3 surfaces we introduced.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20044",
  "author_total": 1,
  "authors": [
    {
      "name": "Rodion Déev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fwz26xafkzwa1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20044",
  "fulltext_url": null,
  "id": "fwz26xafkzwa1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20044",
  "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.AG"
    ],
    [
      "arxiv",
      "math.CV"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7221860915",
  "pdf_url": "https://arxiv.org/pdf/2610.20044",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fwz26xafkzwa1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.20044",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20044"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Markman--Markushevich principle and Lagrangian fibrations on O'Grady six-manifolds",
  "title_text": "Markman--Markushevich principle and Lagrangian fibrations on O'Grady six-manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/fwz26xafkzwa1",
  "venue": "arXiv",
  "versions": []
}