{
  "abstract": "In 2009, Peterzil and Starchenko proved the following beautiful generalization of Chow's theorem: An entire complex analytic set $X\\subset \\mathbb{C}^n$ that is definable in an o-minimal structure on $\\mathbb{R}$ must be an algebraic set. This result is known nowadays as the o-minimal Chow theorem. In this article, we present some geometric and metric versions of Chow's theorem that generalize the o-minimal Chow's theorem. For instance, we prove that if $X\\subset \\mathbb{C}^n$ is a pure $d$-dimensional entire complex analytic set, then the following statements are equivalent: (1) $X$ is a complex algebraic set; (2) $\\mathcal{H}^{2d+1}(C(X,\\infty))=0$, where $\\mathcal{H}^{k}(A)$ denotes the $k$-dimensional Hausdorff measure of $A$, and $C(X,\\infty)$ denotes the tangent cone at infinity of $X$; (3) $\\mathcal{H}^{2d+2}(C_{\\mathbb{C}}(X,\\infty))=0$, where $C_{\\mathbb{C}}(X,\\infty)$ denotes the complex tangent cone at infinity of $X$; (4) $\\mathcal{H}^{2d}(Z(X,\\infty))=0$, where for $A\\subset \\mathbb{C}^k$, $Z(A,\\infty)=\\{[v]\\in \\mathbb{C}P^{k-1};v\\in C_{\\mathbb{C}}(A,\\infty)\\}$; (5) For any $k\\in\\{d+1,...,n\\}$ and for any projection $π\\colon \\mathbb{C}^{n}\\to \\mathbb{C}^{k}$ such that $π^{-1}(0)\\cap C_{\\mathbb{C}}(X,\\infty)=\\{0\\}$ and $Y=π(X)$, $\\mathcal{H}^{2d}(Z(Y,\\infty))<\\mathcal{H}^{2d}(\\mathbb{C}P^{d})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27810",
  "author_total": 1,
  "authors": [
    {
      "name": "José Edson Sampaio"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ctg2pz6vrvd02.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27810",
  "fulltext_url": null,
  "id": "ctg2pz6vrvd02",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27810",
  "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",
      "math.AG"
    ],
    [
      "arxiv",
      "math.CV"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2608.27810",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ctg2pz6vrvd02.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.27810",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27810"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Metric-geometric Chow theorem",
  "title_text": "Metric-geometric Chow theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/ctg2pz6vrvd02",
  "venue": "arXiv",
  "versions": []
}