{
  "abstract": "We study codimension-one holomorphic foliations admitting invariant singular real-analytic Levi-flat varieties. Our main problem is to determine when the local meromorphic first integrals furnished by the Levi-flat geometry can be continued coherently and globalized. We introduce a pencil-singularity condition and a projective compatibility condition for local meromorphic parameters. In the projective plane these yield a characterization of foliations admitting a primitive rational first integral, while in higher-dimensional projective space they give a rational integrability criterion under a generic local-product hypothesis along the dicritical locus. We also obtain semi-global and Stein versions of these results. In a different direction, we study foliations admitting infinitely many invariant Levi-flat varieties and prove a Darboux-type alternative: either the foliation admits a rational first integral, or infinitely many Levi-flats contain a common invariant algebraic curve whose virtual holonomy is abelian. Under additional assumptions on the singularities, this leads to the existence of a closed rational, and in the absence of saddle-nodes a closed logarithmic, defining \\(1\\)-form.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.12497",
  "author_total": 1,
  "authors": [
    {
      "name": "Bruno Scárdua"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9ycsm9p419kyh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.12497",
  "fulltext_url": null,
  "id": "9ycsm9p419kyh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.12497",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CV"
    ],
    [
      "arxiv",
      "math.GT"
    ],
    [
      "openalex",
      "2608|Advanced Differential Equations and Dynamical Systems"
    ]
  ],
  "openalex": "W7221516694",
  "pdf_url": "https://arxiv.org/pdf/2610.12497",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9ycsm9p419kyh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.12497",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.12497"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.GT",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Singular Levi-flats and holomorphic foliations on projective spaces",
  "title_text": "Singular Levi-flats and holomorphic foliations on projective spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/9ycsm9p419kyh",
  "venue": "arXiv",
  "versions": []
}