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PreprintarXiv licencearXiv

Counting Invariants for Torus Quotients under FPF Group Actions

  • Xue Zhang
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Abstract

Let G be a finite group admitting a rational fixed-point-free (FPF) representation V, with a G-invariant full-rank lattice L⊂ V. We study the induced G-action on the complex torus Vℂ/L and the quotient orbifold Vℂ/(Gltimes L). We prove that every FPF group admits a unique irreducible rational FPF representation. We derive closed-form formulas for three core geometric invariants: the number of non-free points of the torus action, the number of singular points of the quotient orbifold, and its orbifold Euler characteristic.

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Cite

BibTeXRIS
BibTeX
@misc{zhang2026counting,
  title = {{Counting Invariants for Torus Quotients under FPF Group Actions}},
  author = {Xue Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.27736},
  eprint = {2609.27736},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.27736},
}
RIS
TY  - UNPB
TI  - Counting Invariants for Torus Quotients under FPF Group Actions
AU  - Xue Zhang
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.27736
UR  - https://arxiv.org/abs/2609.27736
AB  - Let G be a finite group admitting a rational fixed-point-free (FPF) representation V, with a G-invariant full-rank lattice L⊂ V. We study the induced G-action on the complex torus V_ℂ/L and the quotient orbifold V_ℂ/(Gltimes L). We prove that every FPF group admits a unique irreducible rational FPF representation. We derive closed-form formulas for three core geometric invariants: the number of non-free points of the torus action, the number of singular points of the quotient orbifold, and its orbifold Euler characteristic.
ER  -