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

2025/10/12 by Liu, Wenfei, Lyu, Renjie · 1 citation
#14J50 #20C15 #Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2510.10430

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 E be a G-equivariant locally free sheaf on X. Then, in the representation ring R(G)_ℚ, we have χ_G(X, E):=∑_i=0^{dim X}(-1)ⁱ[Hⁱ(X, E)]=(1)/(|G|)χ(X,E)[ℂ[G]] + ∑_ZΓ(E)_Z where Z runs over all connected components of the fixed-point sets X^g for g∈ G, and each Γ(E)_Z∈ R(X)_ℚ, called the ramification module at Z, depends only on the restriction E|_Z and the normal bundle N_Z/X as G_Z-equivariant bundles. We illustrate the computation of Γ(E)_Z in several special cases and provide a detailed example for faithful actions of G≅(ℤ/2ℤ)ⁿ on a compact complex surface.

Citations

• On the cohomological representations of finite automorphism groups of singular curves and compact complex spaces
• On regular surfaces of general type with numerically trivial automorphism group of order 4
• The Chevalley-Weil formula on nodal curves
• Residues of connections and the Chevalley-Weil formula for curves
• Computing the equivariant Euler characteristic of Zariski and etale sheaves on curves

Cited by

• Pluricanonical Geometry of Varieties Isogenous to a Product and Abelian Covers

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