Riemann-Roch coefficients for Kleinian orbisurfaces

TL;DR

This paper computes correction terms in To"en's Grothendieck-Riemann-Roch theorem specifically for Kleinian orbisurfaces with ADE stabilizers, extending the understanding of Riemann-Roch for orbifold surfaces.

Contribution

It explicitly calculates the correction terms for ADE-type Kleinian orbifold surfaces, providing concrete formulas for To"en's Riemann-Roch theorem in this setting.

Findings

Explicit correction terms for ADE orbifold surfaces

Extension of Riemann-Roch theorem to Kleinian orbisurfaces

Clarification of inertia stack components in the formula

Abstract

Suppose $\mathcal{S}$ is a smooth, proper, and tame Deligne-Mumford stack. To\"en's Grothendieck-Riemann-Roch theorem requires correction terms, involving components of the inertia stack, to the standard formula for schemes. We give a brief overview of To\"en's Grothendieck-Riemann-Roch theorem, and explicitly compute the correction terms in the case of an orbifold surface with stabilizers of types ADE.