{
  "abstract": "LeBrun's Kähler realization $g_m$ of the Taub--NUT metric on $\\mathbb{C}^2$ is complete, Ricci-flat and not flat. Loi, Zedda and Zuddas proved that no multiple $αg_m$ admits a Kähler immersion into a finite- or infinite-dimensional complex projective space when $m>α/2$, and conjectured that the same holds for every $m>0$. We prove the conjecture. The restriction of the Kähler potential to the axis $z_2=0$ is governed by the Lambert $W$ function, so $\\exp(αΦ_m)$ has a finite radius of convergence as a power series in $|z_1|^2$ although it is real analytic on the whole half-line; the Vivanti--Pringsheim theorem forbids nonnegative Taylor coefficients, and Calabi's criterion fails. We state the mechanism, which Arezzo, Loi, Placini and Zedda recently used for radial metrics, as a general obstruction to Kähler immersions. In statistical terms the axis restriction of $g_m$ would be a natural exponential family with mean domain $(0,\\infty)$ and variance function $μ/(1+2mμ)$; the argument gives an elementary proof of the known fact, due to Bar-Lev, Bshouty and Enis, that no such family exists with variance function $μ/(1+cμ)$ for any $c>0$. The result confirms onemore case of the conjecture of Loi, Salis and Zuddas that Ricci-flat projectively induced Kähler metrics are flat. The analytic core of the proof has been machine-checked in Lean~4.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21950",
  "author_total": 1,
  "authors": [
    {
      "name": "Shaosai Huang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/154s2genxfrg5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21950",
  "fulltext_url": null,
  "id": "154s2genxfrg5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21950",
  "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.DG"
    ],
    [
      "openalex",
      "2608|Geometry and complex manifolds"
    ]
  ],
  "openalex": "W7213895169",
  "pdf_url": "https://arxiv.org/pdf/2609.21950",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/154s2genxfrg5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21950",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21950"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The Taub-NUT Metric Is Not Projectively Induced",
  "title_text": "The Taub-NUT Metric Is Not Projectively Induced",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/154s2genxfrg5",
  "venue": "arXiv",
  "versions": []
}