{
  "abstract": "In this paper we investigate adapted complex polarizations associated with left-invariant Koszul connections on real Lie groups. Adapted complex polarizations arise as a natural generalization of a complex structure on a manifold. We investigate the existence of such polarizations on the tangent bundle of certain Lie groups with left-invariant connections. We use a practical criterion for the existence of global adapted complex polarizations in terms of the $i\\mathbb R$-completeness of the geodesic flow of the connection. In the Lie case, the Euler-Arnold vector field determines the behaviour of the geodesic flow. By combining what we know about the Euler-Arnold vector field of certain Lie groups and the $i\\mathbb R$-completeness criterion, we provide several examples of left-invariant ac-polarizations defined on the tangent bundle of Lie groups. Along with pseudo-Riemannian metric connections, we discuss biinvariant connections in several cases. Our focus is on solvable Lie groups, because non-solvable Lie groups are better understood. The main result of this paper is that we demonstrate several not locally symmetric but $\\mathcal P$-complete connections.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.10937",
  "author_total": 1,
  "authors": [
    {
      "name": "Balázs Forman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fz43cn639nnyq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.10937",
  "fulltext_url": null,
  "id": "fz43cn639nnyq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.10937",
  "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.DG"
    ],
    [
      "openalex",
      "2608|Geometry and complex manifolds"
    ]
  ],
  "openalex": "W7221120265",
  "pdf_url": "https://arxiv.org/pdf/2610.10937",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fz43cn639nnyq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.10937",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.10937"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Adapted complex polarizations on solvable Lie groups of dimension 3",
  "title_text": "Adapted complex polarizations on solvable Lie groups of dimension 3",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/fz43cn639nnyq",
  "venue": "arXiv",
  "versions": []
}