{
  "abstract": "We determine the positive scalar curvature Kähler cone of every minimal compact Kähler surface of Kodaira dimension $-\\infty$. By the Enriques--Kodaira classification, every such surface is either $\\mathbb P^2$ or a geometrically ruled surface $\\mathbb P(E)\\toΣ_g$, where $E$ is a rank-two holomorphic vector bundle over a compact Riemann surface of genus $g$. On $\\mathbb P^2$, every Kähler class admits a Kähler metric of positive scalar curvature, whereas on $\\mathbb P(E)$, every Kähler class of positive total scalar curvature admits a metric of positive scalar curvature if and only if $g\\le1$ or $E$ is slope-semistable. We further prove that if a smooth compact Kähler surface admits a birational morphism onto a ruled surface $\\mathbb P(E)\\toΣ_g$, where $g\\ge2$ and $E$ is slope-unstable, then it contains a Kähler class of positive total scalar curvature admitting no positive scalar curvature Kähler metric. Finally, for every smooth compact Kähler surface $X$ birational to $Y=\\mathbb P^2$ or $Y=\\mathbb P(E)$, we show that Kähler classes admitting positive scalar curvature metrics on $Y$ are exactly the pushforwards of classes admitting such metrics on $X$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27870",
  "author_total": 1,
  "authors": [
    {
      "name": "Zehao Sha"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ekqzgvd9f4stf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27870",
  "fulltext_url": null,
  "id": "ekqzgvd9f4stf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27870",
  "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.AG"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2608|Geometry and complex manifolds"
    ]
  ],
  "openalex": "W7214216633",
  "pdf_url": "https://arxiv.org/pdf/2609.27870",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ekqzgvd9f4stf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2609.27870",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27870"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Positive scalar curvature Kähler cones on compact complex surfaces",
  "title_text": "Positive scalar curvature Kähler cones on compact complex surfaces",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/ekqzgvd9f4stf",
  "venue": "arXiv",
  "versions": []
}