{
  "abstract": "In this paper, we introduce Siu's curvature operator \\(A^E_{p,q}\\) for vector-bundle-valued differential forms on Kähler manifolds. When $p=n$, this operator reduces to the classical Akizuki--Nakano curvature operator. We first characterize the semipositivity of \\(A^E_{p,q}\\) in terms of an optimal \\(L^2\\)-estimate condition for the \\(\\bar\\partial\\)-operator, and then prove an Ohsawa--Takegoshi-type extension theorem for \\(E\\)-valued \\((p,q)\\)-forms under the curvature condition \\(A^E_{p,q+1}\\geq0\\), using a new twisted basic estimate adapted to this setting. As an application, we prove the local freeness of the higher direct image sheaf \\(R^q s_*(Ω^p_{X/ B_m}\\otimes E)\\) under the curvature conditions $A^E_{p,q+1}\\geq0$ and $A^E_{p,q}\\geq0$, where $s: X \\to B_m:=\\{t\\in\\mathbb C^m:\\ |t|<1\\}$ is a proper holomorphic submersion from a Kähler manifold $X$, and $E$ is a Hermitian holomorphic vector bundle.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.23094",
  "author_total": 1,
  "authors": [
    {
      "name": "Gang Huang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ba4c00ynas295.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.23094",
  "fulltext_url": null,
  "id": "ba4c00ynas295",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.23094",
  "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.CV"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2608|Geometry and complex manifolds"
    ]
  ],
  "openalex": "W7171532153",
  "pdf_url": "https://arxiv.org/pdf/2607.23094",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ba4c00ynas295.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2607.23094",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.23094"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Siu's curvature positivity and $L^2$ extension theorems for $(p,q)$-forms",
  "title_text": "Siu's curvature positivity and L^2 extension theorems for (p,q)-forms",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/ba4c00ynas295",
  "venue": "arXiv",
  "versions": []
}