Riemann-Roch Theorems for Deligne-Mumford Stacks

TL;DR

This paper extends Riemann-Roch theorems to Deligne-Mumford stacks by developing new cohomology and Chern character tools, leading to generalized formulas for Euler characteristics.

Contribution

It introduces a cohomology theory with coefficients in representations and proves a Grothendieck-Riemann-Roch theorem for stacks, generalizing classical results.

Findings

Proved Riemann-Roch type theorems for Deligne-Mumford stacks.

Derived formulas for Euler characteristics of coherent sheaves.

Established a Hirzebruch-Riemann-Roch formula for stacks.

Abstract

The goal of this paper is to prove Riemann-Roch type theorems for Deligne-Mumford algebraic stacks. To this end, we introduce a "cohomology with coefficients in representations" and a Chern character, and we prove a Grothendieck-Riemann-Roch theorem for the Riemann-Roch transformation it defines. As a corollary we obtain an Hirzebruch-Riemann-Roch formula for the Euler characteristic of a coherent sheaf, and some formulas for the different topological Euler characteristics of complex algebraic stacks.