{
  "abstract": "We study the numerical geography and tangent geometry of very amply polarized Calabi--Yau threefolds $(X,H)$ through the positivity of the first jet bundle $J^1(H)$. Writing $d=\\int_X H^3$, $c=\\int_Xc_2(X) H$, and $e=\\int_X c_3(X)$, we exploit two different positivity properties of this single bundle. Mixed intersections on $\\mathbb{P}(J^1(H)^*)$ give $e\\ge-5d-c-c^2/(4d)$, while a volume estimate for a perturbed tautological class gives $e\\ge40d\\left[(1-\\frac{c}{10d})^{3/2}-1\\right]$; in particular $e+6c\\ge0$, improving Sun's inequality $e+10c\\ge0$. As consequences, we obtain the uniform Hodge bounds $-4d-80\\le h^{1,1}(X)-h^{2,1}(X)\\le173d/66$, the lower bound $\\mathrm{deg}X^\\vee\\ge78$ for the dual hypersurface, and, in the critical case $X\\subset\\mathbb{P}^6$, the upper bound $d\\le34$. We also prove that, for every $m\\ge2$, the tangent-incidence morphism associated with $|mH|$ is the normalization of the tangent variety, conjecture tangent birationality for complete embeddings $X\\subset\\mathbb{P}^N$ with $N\\ge7$, and verify it for several families.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17513",
  "author_total": 1,
  "authors": [
    {
      "name": "Atsushi Kanazawa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1hm1a7eqpabwa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17513",
  "fulltext_url": null,
  "id": "1hm1a7eqpabwa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17513",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "hep-th"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.CV"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7213413582",
  "pdf_url": "https://arxiv.org/pdf/2609.17513",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1hm1a7eqpabwa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:11Z",
      "native_id": "2609.17513",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17513"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "physics",
      "origin": "arxiv:hep-th",
      "subfield": "high-energy"
    }
  ],
  "title": "Chern bounds and tangent geometry of polarized Calabi-Yau threefolds",
  "title_text": "Chern bounds and tangent geometry of polarized Calabi-Yau threefolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/1hm1a7eqpabwa",
  "venue": "arXiv",
  "versions": []
}