{
  "abstract": "Let $M$ be a complex surface endowed with a holomorphic projective connection and let $C \\subset M$ be a compact smooth holomorphic curve. Given two singular holomorphic foliations $\\mathcal{F}$ and $\\mathcal{G}$ near $C$, with $C$ not $\\mathcal{G}$-invariant, we attach to every point $p \\in C$ an index $\\mathrm{Ind}(\\mathcal{F},\\mathcal{G},C,p) \\in \\mathbb{C}$ of $\\mathcal{F}$ relative to the reference $\\mathcal{G}$, and we prove that these indices add up to $T\\mathcal{G} \\cdot C$. When $C$ is $\\mathcal{F}$-invariant the index is the Camacho--Sad index corrected by the order of tangency of $\\mathcal{G}$ with $C$, and the index formula reduces to the Camacho--Sad formula. The main applications are two contact formulas. At a tangency point $p$ of $\\mathcal{G}$ with $C$, let $k(\\mathcal{G},C,p)$ be the ratio of the curvatures at $p$ of the leaf of $\\mathcal{G}$ and of the curve; it is a projective invariant, although each curvature separately depends on a choice of metric. If $\\mathcal{G}$ has no singular points on $C$ and only simple tangencies with $C$, then $$ \\sum_{p} \\frac{1}{1 - k(\\mathcal{G},C,p)} = \\frac{2}{3}\\left( C \\cdot C + g - 1 \\right), $$ where $g$ is the genus of $C$: the number of tangencies depends on $\\mathcal{G}$, but this weighted count does not. For a pencil of lines in the projective plane it is the Plücker formula for the class of a plane curve. The second formula asserts that, for a curve in general position with respect to $\\mathcal{F}$ and $\\mathcal{G}$ which is not a geodesic, the total index of the tangencies of $\\mathcal{F}$ with $\\mathcal{G}$ along $C$ equals the number of tangencies of $\\mathcal{G}$ with $C$ minus one third of the number of inflection points of $C$; in particular, it does not depend on $\\mathcal{F}$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22279",
  "author_total": 2,
  "authors": [
    {
      "name": "César Camacho"
    },
    {
      "name": "Rudy Rosas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/78veph8y8te6v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22279",
  "fulltext_url": null,
  "id": "78veph8y8te6v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22279",
  "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"
    ],
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      "arxiv",
      "math.DG"
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      "arxiv",
      "math.DS"
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      "openalex",
      "2608|Geometry and complex manifolds"
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  "openalex": "W7214042421",
  "pdf_url": "https://arxiv.org/pdf/2609.22279",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/78veph8y8te6v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22279",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22279"
    }
  ],
  "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.DS",
      "subfield": "dynamical-systems"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Geometric index theorems for holomorphic foliations and curves",
  "title_text": "Geometric index theorems for holomorphic foliations and curves",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/78veph8y8te6v",
  "venue": "arXiv",
  "versions": []
}