{
  "abstract": "Let $X$ be a Stein manifold, and let $K=\\{h\\le0\\}$ be a compact sublevel set of a strictly plurisubharmonic function $h$ defined near $K$. It is known that $K$ is meromorphically convex in $X$ if and only if $dd^ch$ extends to a Hodge form on $X$. We show that if $H_2(X,\\mathbb Z)$ is the direct sum of a free abelian group and a torsion group, for example if it is finitely generated, then any extension of $dd^ch$ to a Kähler form on $X$ can be modified, after replacing $h$ by a positive multiple, into a Hodge form. Thus meromorphic convexity of $K$ is equivalent to the extendability of $dd^ch$ as a Kähler form. We also show that the topological assumption cannot be dropped: we construct a Stein manifold $X$ of dimension $6$ with $H_2(X,\\mathbb Z)\\cong\\mathbb Q$, a sublevel set $K$ as above that is not meromorphically convex although $dd^ch$ extends to a Kähler form, and a totally real torus in $K$ that is isotropic for a Kähler form but is not meromorphically convex. This shows that the Hodge condition in Guedj's characterization of meromorphically convex totally real submanifolds cannot be weakened to a Kähler condition, and gives counterexamples to two criteria stated by Cieliebak and Eliashberg.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38055",
  "author_total": 1,
  "authors": [
    {
      "name": "Rasul Shafikov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8hksdgcpnt8y2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38055",
  "fulltext_url": null,
  "id": "8hksdgcpnt8y2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38055",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38055",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8hksdgcpnt8y2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2610.38055",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38055"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the Hodge Extension of Local Kähler Forms",
  "title_text": "On the Hodge Extension of Local Kähler Forms",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/8hksdgcpnt8y2",
  "venue": "arXiv",
  "versions": []
}