On a formula for the equivariant Euler characteristic of a $G$-sheaf

TL;DR

This paper extends the computation of the equivariant Euler characteristic of G-sheaves from curves to higher-dimensional varieties using a Riemann-Roch theorem for quotient stacks.

Contribution

It generalizes previous results on G-sheaves from curves to higher-dimensional varieties by employing a Riemann-Roch theorem for quotient stacks.

Findings

Extended equivariant Euler characteristic computations to higher dimensions.

Applied Riemann-Roch theorem for quotient stacks in the new context.

Provided a framework for future research on G-sheaves in higher-dimensional geometry.

Abstract

H. Fischbacher-Weitz and B. K\"ock computed the equivariant Euler characteristic of a $G-$sheaf on a $G$-curve $X$ over a field. Using a form of the Riemann-Roch theorem for quotient stacks proved by the second author we extend their computations to the cases where $dim(X) >1$.