{
  "abstract": "Consider a holomorphic vector bundle $E$ over a compact complex curve $C$ which decomposes as a sum of stable vector bundles. For the projectivization $\\mathbb{P}(E)$, we prove that the existence of a compatible extremal almost Kähler (aK) metric of involutive type in the sense of Lejmi is equivalent to the existence of a Calabi extremal Kähler metric. This result rests on the Yau--Tian--Donaldson correspondence in terms of the moment polytope $Δ$ for $\\mathbb{P}(E)$, proved by the author and Yin in a previous work. The main advantage is that compatible extremal aK metrics of involutive type are solutions to a second-order linear PDE, rather than a fourth-order nonlinear PDE for Calabi's extremal Kähler metrics. As an application, we prove that when $E$ has rank $4$ and $C$ is an elliptic curve or the projective line, $\\mathbb{P}(E)$ is a Calabi dream manifold, i.e. admits an extremal Kähler metric in every Kähler class.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01094",
  "author_total": 1,
  "authors": [
    {
      "name": "Simon Jubert"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3pntb5e8dhnh9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01094",
  "fulltext_url": null,
  "id": "3pntb5e8dhnh9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01094",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2608|Geometry and complex manifolds"
    ]
  ],
  "openalex": "W7206186763",
  "pdf_url": "https://arxiv.org/pdf/2609.01094",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3pntb5e8dhnh9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.01094",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01094"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "New extremal Kähler metrics on projective bundles",
  "title_text": "New extremal Kähler metrics on projective bundles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/3pntb5e8dhnh9",
  "venue": "arXiv",
  "versions": []
}