Motives of Deligne-Mumford Stacks

TL;DR

This paper constructs motives for smooth, separated Deligne-Mumford stacks within Voevodsky's framework, compares them to Chow motives in proper cases, and shows they relate to motives of quasi-projective varieties in general.

Contribution

It introduces a motive $M(F)$ for Deligne-Mumford stacks and compares it with existing Chow motives, extending motive theory to a broader class of stacks.

Findings

$M(F)$ is a direct summand of a motive of a smooth quasi-projective variety.

When $F$ is proper over a characteristic 0 field, $M(F)$ matches the Chow motive.

The construction applies to all smooth, separated Deligne-Mumford stacks.

Abstract

For every smooth and separated Deligne-Mumford stack $F$, we associate a motive $M(F)$ in Voevodsky's category of mixed motives with rational coefficients $\mathbf{DM}^{\eff}(k,\mathbb{Q})$. When $F$ is proper over a field of characteristic 0, we compare $M(F)$ with the Chow motive associated to $F$ by Toen (\cite{t}). Without the properness condition we show that $M(F)$ is a direct summand of the motive of a smooth quasi-projective variety.