{
  "abstract": "Let $X$ be a smooth projective surface over $\\mathbb{C}$ with an effective action of a finite group $G$, and let $L$ be a $G$-linearized line bundle. We develop a residue-theoretic and birational approach to computing the equivariant Euler characteristic $χ_G(X,L)$. Every such $L$ admits an equivariant divisor presentation. Divisor peeling applies the divisor exact sequences one layer at a time, treating whole $G$-orbits of components together. It reduces the problem to the structure sheaf and lower-dimensional equivariant contributions. On a resolution of $X/G$, the natural extensions of the isotypic direct-image sheaves preserve Euler characteristics, but their induced connections need not be logarithmic, as an explicit $D_4$ calculation shows. Blowing up fixed points before taking the quotient reduces the local calculation to diagonal actions and cyclic Hirzebruch-Jung singularities. We obtain explicit correction class functions for the ADE singularities and a uniform procedure for general diagonal actions, including quasi-reflections. In the diagonal case, the natural extensions are logarithmic, and their residues are computed from monomial valuations on the Hirzebruch-Jung resolution. Combining these residue calculations with the branch-curve terms and divisor peeling gives a procedure for computing the equivariant Euler characteristic of every $G$-linearized line bundle. We compare the local classes with the holomorphic Lefschetz formula and the Kleinian coefficients of Lim and Rota, and give examples on $\\mathbb{P}^2$, including one with fixed curves.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33782",
  "author_total": 1,
  "authors": [
    {
      "name": "Alexandros Kafkas"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ekgxsrx745bbj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33782",
  "fulltext_url": null,
  "id": "ekgxsrx745bbj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33782",
  "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"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214799865",
  "pdf_url": "https://arxiv.org/pdf/2609.33782",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ekgxsrx745bbj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33782",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33782"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Equivariant Riemann-Roch for Surfaces via Connections and Residues",
  "title_text": "Equivariant Riemann-Roch for Surfaces via Connections and Residues",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/ekgxsrx745bbj",
  "venue": "arXiv",
  "versions": []
}