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The Chevalley--Weil formula for finite group actions on higher dimensional compact complex manifolds

Wenfei Liu, Renjie Lyu

TL;DR

The paper generalizes the Chevalley–Weil formula to finite-group actions on higher-dimensional compact complex manifolds by introducing ramification modules associated to fixed loci and deriving a fixed-point–type expression for the $G$-Euler characteristic $\chi_G(X,\mathcal{E})$. Central to the approach is the holomorphic Atiyah–Singer fixed-point theorem, which yields correction terms $\Gamma(\mathcal{E})_Z$ attached to strata $Z$, so that $\chi_G(X,\mathcal{E}) = \frac{1}{|G|}\chi(X,\mathcal{E})[\mathbb{C}[G]] + \sum_Z {\Gamma(\mathcal{E})_Z}$. The ramification modules are constructed via cyclic-subgroup data and localization in representation rings, and can be computed in several special cases, with explicit results for $G\cong (\mathbb{Z}/2\mathbb{Z})^n$ acting on complex surfaces. Overall, the work extends curve-based CW formulas to higher dimensions and provides concrete tools for calculating $G$-module structures of cohomology in complex geometry.

Abstract

Building on the Atiyah--Singer holomorphic Lefschetz fixed-point theorem, we define ramification modules associated to the fixed loci of a finite group acting on a compact complex manifold. This allows us to generalize the Chevalley--Weil formula for compact Riemann surfaces to higher dimensions. More precisely, let $G$ be a finite group acting on a compact complex manifold $X$, and let $\mathcal{E}$ be a $G$-equivariant locally free sheaf on $X$. Then, in the representation ring $R(G)_\mathbb{Q}$, we have \[ χ_G(X, \mathcal{E}):=\sum_{i=0}^{\dim X}(-1)^i[H^i(X, \mathcal{E})]=\frac{1}{|G|}χ(X,\mathcal{E})[\mathbb{C}[G]] + \sum_ZΓ(\mathcal{E})_Z \] where $Z$ runs over all connected components of the fixed-point sets $X^g$ for $g\in G$, and each $Γ(\mathcal{E})_Z\in R(X)_\mathbb{Q}$, called the \emph{ramification module} at $Z$, depends only on the restriction $\mathcal{E}|_Z$ and the normal bundle $N_{Z/X}$ as $G_Z$-equivariant bundles. We illustrate the computation of $Γ(\mathcal{E})_Z$ in several special cases and provide a detailed example for faithful actions of $G\cong(\mathbb{Z}/2\mathbb{Z})^n$ on a compact complex surface.

The Chevalley--Weil formula for finite group actions on higher dimensional compact complex manifolds

TL;DR

The paper generalizes the Chevalley–Weil formula to finite-group actions on higher-dimensional compact complex manifolds by introducing ramification modules associated to fixed loci and deriving a fixed-point–type expression for the -Euler characteristic . Central to the approach is the holomorphic Atiyah–Singer fixed-point theorem, which yields correction terms attached to strata , so that . The ramification modules are constructed via cyclic-subgroup data and localization in representation rings, and can be computed in several special cases, with explicit results for acting on complex surfaces. Overall, the work extends curve-based CW formulas to higher dimensions and provides concrete tools for calculating -module structures of cohomology in complex geometry.

Abstract

Building on the Atiyah--Singer holomorphic Lefschetz fixed-point theorem, we define ramification modules associated to the fixed loci of a finite group acting on a compact complex manifold. This allows us to generalize the Chevalley--Weil formula for compact Riemann surfaces to higher dimensions. More precisely, let be a finite group acting on a compact complex manifold , and let be a -equivariant locally free sheaf on . Then, in the representation ring , we have \[ χ_G(X, \mathcal{E}):=\sum_{i=0}^{\dim X}(-1)^i[H^i(X, \mathcal{E})]=\frac{1}{|G|}χ(X,\mathcal{E})[\mathbb{C}[G]] + \sum_ZΓ(\mathcal{E})_Z \] where runs over all connected components of the fixed-point sets for , and each , called the \emph{ramification module} at , depends only on the restriction and the normal bundle as -equivariant bundles. We illustrate the computation of in several special cases and provide a detailed example for faithful actions of on a compact complex surface.

Paper Structure

This paper contains 8 sections, 14 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Constructions around the representation rings
  4. Recovering a representation from its restrictions to cyclic subgroups
  5. G-equivariant K-theory and the holomorphic Lefschetz fixed-point theorem
  6. The Chevalley--Weil formula for compact complex manifolds
  7. Computation of the ramification module in special cases
  8. The action of on a compact complex surface

Key Result

Theorem 1.1

Let $X$ be a compact complex manifold, possibly disconnected and non-equidimensional, and let $G$ be a finite group acting on $X$. Let $\mathcal{E}$ be a $G$-equivariant locally free sheaf on $X$. Let $\chi_G(X, \mathcal{E}):=\sum_j (-1)^j [H^j(X, \mathcal{E})]$ be the $G$-Euler characteristic of $\ where $Z$ runs through the components of the fixed loci $X^g$ with $g\in G$, and $\Gamma(\mathcal{E

Theorems & Definitions (38)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5: Holomorphic Lefschetz fixed-point formula, AS68III
  • Definition 3.2
  • ...and 28 more