{
  "abstract": "Let $(M,\\nabla)$ be a real analytic Koszul manifold. An adapted complex structure (ac-structure) on a neighborhood $N$ of the zero section in $TM$ is a complex structure on $N$ such that the leaves of the Levi-Civita foliation are holomorphic curves. More generally, a complex polarization $P$ on $N$ is called an ac-polarization if the leaves of the Levi-Civita foliation are tangential to $P$. The bundle of (1,0) tangent vectors of an ac-structure is an ac-polarization. The connection is called entire (resp. polarization complete or simply $\\mathcal P$-complete) if the ac-structure (resp. the ac-polarization) exists on $TM$. Although on a small enough $N$ an ac-structure always exists, entire connections are rear and if a maximal domain of definition $N_{max}$ of an ac-structure exists (different from $TM$), $N_{max}$ is a complicated domain. On the other hand in many cases the associated ac-polarization can be extended to the whole $TM$. The main purpose of the paper is to gain better understanding of this phenomenon using a generalized version of the polar map. As special cases we show that many of those metrics studied by Aslam-Burns-Irvine and Halverscheid-Iannuzzi although are not entire but are $\\mathcal P$-complete.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15673",
  "author_total": 2,
  "authors": [
    {
      "name": "Balázs Forman"
    },
    {
      "name": "Róbert Szőke"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c0jpzwnk8q369.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15673",
  "fulltext_url": null,
  "id": "c0jpzwnk8q369",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15673",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CV"
    ],
    [
      "arxiv",
      "math.DG"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.15673",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c0jpzwnk8q369.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2609.15673",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15673"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Polarization complete left invariant connections",
  "title_text": "Polarization complete left invariant connections",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/c0jpzwnk8q369",
  "venue": "arXiv",
  "versions": []
}