{
  "abstract": "Let $X$ be a compact connected Riemann surface equipped with two nonconstant meromorphic projections $q,z:X\\to\\mathbb{P}^1$. Their differentials determine a canonical meromorphic relative differential morphism $Z:T_q=q^*T\\mathbb{P}^1\\to T_z=z^*T\\mathbb{P}^1$, whose local coefficient in coordinate frames is $Z=z_q\\,\\partial_z\\otimes dq$. Its divisor is $\\operatorname{Div}(Z)=R_z-R_q$, and its local logarithmic derivative defines a rank-one logarithmic connection on $L=\\operatorname{Hom}(T_q,T_z)$. The general theory of logarithmic connections is classical. The main point of the present paper is that, in the dual conformal-net setting, the Levi--Civita curvature data of the two nets realize this logarithmic connection by the symmetric formula $H^{-1}\\widetildeω\\,dz-Hω\\,dq =2i\\,d\\log z_q =2i\\,Z^{-1}dZ$, where the last expression is understood through local trivializations of $L$. Thus the two dual conformal connection differentials occur with opposite signs, and their difference is the logarithmic Maurer--Cartan form of the relative differential morphism. Its residues recover the relative ramification multiplicities, $\\operatorname{Res}_pΘ=2i(r_z(p)-r_q(p))$, and globally $\\sum_{p\\in X}\\operatorname{Res}_pΘ=4i(\\operatorname{deg}z-\\operatorname{deg}q)$. Thus local conformal connection data determine both relative ramification and the difference of the degrees of the two projections. We also discuss the dual morphism, the square-root issue, and a hyperelliptic example.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26803",
  "author_total": 1,
  "authors": [
    {
      "name": "A. G. Nuramatov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7g2zf0athw3gh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26803",
  "fulltext_url": null,
  "id": "7g2zf0athw3gh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26803",
  "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"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214153381",
  "pdf_url": "https://arxiv.org/pdf/2609.26803",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7g2zf0athw3gh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.26803",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26803"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The Relative Differential Morphism and the Logarithmic Connection of Dual Conformal Nets",
  "title_text": "The Relative Differential Morphism and the Logarithmic Connection of Dual Conformal Nets",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/7g2zf0athw3gh",
  "venue": "arXiv",
  "versions": []
}