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

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 $${\chi }_G(X,\mathcal{E}):=\sum _{i=0}^{\dim X}(-1{)}^i[H^i(X,\mathcal{E})]=\frac{1}{\mathrm{\mid }G\mathrm{\mid }}\chi (X,\mathcal{E})[\mathbb{C}[G]]+\sum _Z\mathrm{\Gamma }(\mathcal{E}{)}_Z$$ where $Z$ runs over all connected components of the fixed-point sets $X^g$ for $g\in G$, and each $\mathrm{\Gamma }(\mathcal{E}{)}_Z\in R(X{)}_{\mathbb{Q}}$, called the \emph{ramification module} at $Z$, depends only on the restriction $\mathcal{E}{\mathrm{\mid }}_Z$ and the normal bundle $N_{Z\mathrm{/}X}$ as $G_Z$-equivariant bundles. We illustrate the computation of $\mathrm{\Gamma }(\mathcal{E}{)}_Z$ in several special cases and provide a detailed example for faithful actions of $G\cong (\mathbb{Z}\mathrm{/}2\mathbb{Z}{)}^n$ on a compact complex surface.