{
  "abstract": "We develop a direct analytic approach to positivity of relative pluricanonical direct images through a fixed-point Hermitian metric by direct differential curvature calculations. The curvature resolvent formula from fixed-point equation leads to two-sided estimates of entire Nakano curvature by determinant curvature. In this way, we identify generic rank of determinant curvature with birational variation dimension for a projective algebraic fibration between smooth connected projective varieties with general fiber of general type and maximal birational variation, assuming $mK_F$ is globally generated or the $m$-canonical map is birational for general fiber $F$. In the global generation case, Green--Griffiths Conjecture asks for the tautological bigness of the ordinary symmetric powers $\\operatorname{Sym}^{\\ell}f_*ω_{X/Y}^{\\otimes m}$, without passing to reflexive hulls, for $\\ell\\geq P_m(F)=h^0(F,mK_F)$ whenever $m\\geq 1$. We prove it for all $\\ell\\geq1$ at levels $m\\geq 2$, also in the case of birational $m$-canonical map, but construct a counterexample at level $m=1$. Over curves, curvature and boundary estimates give a direct analytic proof of Viehweg's ampleness theorem requested by Fujino. For smooth projective families, we also establish numerical dimension and vanishing results related to the birational variation dimension.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35590",
  "author_total": 2,
  "authors": [
    {
      "name": "Jixiang Fu"
    },
    {
      "name": "Yunyang Xiao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9f3vekqaq1xjq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35590",
  "fulltext_url": null,
  "id": "9f3vekqaq1xjq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35590",
  "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"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7223891484",
  "pdf_url": "https://arxiv.org/pdf/2610.35590",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9f3vekqaq1xjq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2610.35590",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35590"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Positivity of Relative Pluricanonical Direct Images via Fixed-Point Metrics",
  "title_text": "Positivity of Relative Pluricanonical Direct Images via Fixed-Point Metrics",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/9f3vekqaq1xjq",
  "venue": "arXiv",
  "versions": []
}