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

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 \omega_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 \omega_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.