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

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 \[ \chi_G(X, \mathcal{E}):=\sum_{i=0}^{\dim X}(-1)^i[H^i(X, \mathcal{E})]=\frac{1}{|G|}\chi(X,\mathcal{E})[\mathbb{C}[G]] + \sum_Z\Gamma(\mathcal{E})_Z \] where $Z$ runs over all connected components of the fixed-point sets $X^g$ for $g\in G$, and each $\Gamma(\mathcal{E})_Z\in R(X)_\mathbb{Q}$, 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 $\Gamma(\mathcal{E})_Z$ 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.