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On the cohomological representations of finite automorphism groups of singular curves and compact complex spaces

Qing Liu, Wenfei Liu

TL;DR

The paper develops a comprehensive framework to understand how finite groups G act on cohomology of G-equivariant sheaves on singular curves and related complex spaces. It extends the Chevalley–Weil paradigm to proper reduced curves with nodal singularities, via ramification modules Γ_G(𝓔)_Z and an equivariant Riemann–Roch theorem, yielding explicit expressions for χ_G(𝓔) and H^0(ω_C(T)^{⊗m})^G in terms of quotient data and inertia. It also connects cohomological invariants to combinatorial topology through the dual graph, and applies these results to the equivariant deformation theory of stable G-curves and to the singular cohomology of G-CW complexes. The results reveal new phenomena specific to singular settings, provide numerical criteria for irreducible representations in global sections, and have implications for moduli spaces and product-quotient constructions. Overall, the work offers a robust toolkit for computing G-actions on cohomology in singular and complex settings, with broad algebraic and geometric implications.

Abstract

Let G be a finite group acting tamely on a proper reduced curve C over an algebraically closed field. We study the G-module structure on the cohomology groups of a G-equivariant locally free sheaf F on C, and give formulas of Chevalley--Weil type, with values in the Grothendieck ring R_k(G)_Q of finitely generated G-modules. We also give a similar formula for the singular cohomology of compact complex spaces. The focus is on the case where C is nodal. Using the Chevalley--Weil formula, we compute the G-invariant part of the global sections of the pluricanonical bundle ω_C^{\otimes m}. In turn, we use the formula for m=2 to compute the equivariant deformation space of a stable G-curve C. We also obtain numerical criteria for the presence of any given irreducible representation in space of the global sections of ω_C\otimes F, where F is an ample locally free G-sheaf on C. Some new phenomena, pathological compared to the smooth curve case, are discussed.

On the cohomological representations of finite automorphism groups of singular curves and compact complex spaces

TL;DR

The paper develops a comprehensive framework to understand how finite groups G act on cohomology of G-equivariant sheaves on singular curves and related complex spaces. It extends the Chevalley–Weil paradigm to proper reduced curves with nodal singularities, via ramification modules Γ_G(𝓔)_Z and an equivariant Riemann–Roch theorem, yielding explicit expressions for χ_G(𝓔) and H^0(ω_C(T)^{⊗m})^G in terms of quotient data and inertia. It also connects cohomological invariants to combinatorial topology through the dual graph, and applies these results to the equivariant deformation theory of stable G-curves and to the singular cohomology of G-CW complexes. The results reveal new phenomena specific to singular settings, provide numerical criteria for irreducible representations in global sections, and have implications for moduli spaces and product-quotient constructions. Overall, the work offers a robust toolkit for computing G-actions on cohomology in singular and complex settings, with broad algebraic and geometric implications.

Abstract

Let G be a finite group acting tamely on a proper reduced curve C over an algebraically closed field. We study the G-module structure on the cohomology groups of a G-equivariant locally free sheaf F on C, and give formulas of Chevalley--Weil type, with values in the Grothendieck ring R_k(G)_Q of finitely generated G-modules. We also give a similar formula for the singular cohomology of compact complex spaces. The focus is on the case where C is nodal. Using the Chevalley--Weil formula, we compute the G-invariant part of the global sections of the pluricanonical bundle ω_C^{\otimes m}. In turn, we use the formula for m=2 to compute the equivariant deformation space of a stable G-curve C. We also obtain numerical criteria for the presence of any given irreducible representation in space of the global sections of ω_C\otimes F, where F is an ample locally free G-sheaf on C. Some new phenomena, pathological compared to the smooth curve case, are discussed.

Paper Structure

This paper contains 20 sections, 36 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representation theory of finite groups
  4. The Grothendieck ring of finitely generated -modules
  5. Brauer character
  6. The -equivariant Euler characteristic and reduction to normal connected varieties
  7. Connected components.
  8. Normalization
  9. The Chevalley--Weil formula for proper reduced curves
  10. The Chevalley--Weil formula for smooth curves
  11. The Chevalley--Weil formula for singular curves
  12. An equivariant Riemann--Roch theorem
  13. The case of nodal curves
  14. Computation of at nodes with smoothable actions
  15. Lower bound on for ample sheaves
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $C$ be a proper reduced curve over an algebraically closed field $k$, and $G$ a finite group acting tamely on $C$ (see Definition defn:tame_action). Then for any locally free $G$-sheaf $\mathcal{E}$ of rank $r$ on $C$, we have the following equality in $R_k(G)_\mathbb{Q}$: where $Z$ ranges over all irreducible subvarieties of $C$ with nontrivial inertia group $G_Z$ as well as the singular poi

Theorems & Definitions (86)

  • Theorem 1.1: = Theorem \ref{['thm: CW singular']}
  • Theorem 1.2: = Corollary \ref{['cor: Equiv-RR']}
  • Theorem 1.3: = Corollary \ref{['cor: pluricanonical G-inv']}
  • Theorem 1.4: = Theorem \ref{['thm: CW for CW']}
  • Theorem 1.5: = Corollary \ref{['cor: chi Q irreducible']}
  • Remark 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 76 more