{
  "abstract": "Let $(L,e^{-ϕ})$ be a positive Hermitian holomorphic line bundle over a compact Riemann surface $X$, and let $ω=i\\partial\\overline{\\partial}ϕ$. We obtain explicit pointwise estimates for the Bergman form of the tensor power $mL$. If $\\mathrm{Ric}\\,ω\\leqω$ and the shortest nonconstant closed geodesic has length at least $2π$, then \\[ K_{mϕ}\\geq \\frac{2m-1}{4π}\\,ω, \\] with sharpness holding for $(\\mathbb P^1,\\mathcal O_{\\mathbb P^1}(2))$. We also obtain a local version, depending on an upper curvature bound and the injectivity radius, which recovers the first two terms of the Bergman expansion when the curvature is constant. We also find a higher dimensional version. Under the two-sided bound $-ω\\leq\\mathrm{Ric}\\,ω\\leqω$ and the same closed-geodesic hypothesis, we also prove \\[ K_{mϕ}\\leq \\frac{mω}{2π} \\left(1+\\frac{3}{2m}\\right). \\] The lower estimates use the deformation to the tangent space version of the Ohsawa--Takegoshi theorem established by He, Wang, and the author, whereas the upper bound via Błocki--Zwonek and isoperimetric inequalities.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.03493",
  "author_total": 1,
  "authors": [
    {
      "name": "Johannes Testorf"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7pjymkwdbng1j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.03493",
  "fulltext_url": null,
  "id": "7pjymkwdbng1j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.03493",
  "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.CV"
    ],
    [
      "arxiv",
      "math.DG"
    ],
    [
      "openalex",
      "2608|Geometry and complex manifolds"
    ]
  ],
  "openalex": "W7172528008",
  "pdf_url": "https://arxiv.org/pdf/2608.03493",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7pjymkwdbng1j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2608.03493",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.03493"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Explicit Estimates for the Bergman Kernel Form",
  "title_text": "Explicit Estimates for the Bergman Kernel Form",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/7pjymkwdbng1j",
  "venue": "arXiv",
  "versions": []
}