Equivariant Riemann-Roch for Surfaces via Connections and Residues
Abstract
Let X be a smooth projective surface over 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 χG(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 D4 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 ℙ2, including one with fixed curves.
Related
- K-tropical curves, Riemann-Roch and Serre duality
- Riemann-Roch for 0-cycles on a singular variety
- High dimensional Riemann--Roch spaces in linear spaces with small squares
- a-numbers of covers in characteristic two: semigroups and Riemann-Roch spaces
- Riemann-Roch Polynomials, MBM Classes and Poor IHS Manifolds
BibTeX
@misc{kafkas2026equivariant,
title = {{Equivariant Riemann-Roch for Surfaces via Connections and Residues}},
author = {Alexandros Kafkas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33782},
eprint = {2609.33782},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33782},
}
RIS
TY - UNPB
TI - Equivariant Riemann-Roch for Surfaces via Connections and Residues
AU - Alexandros Kafkas
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33782
UR - https://arxiv.org/abs/2609.33782
AB - Let X be a smooth projective surface over 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 ℙ^2, including one with fixed curves.
ER -