Pro-Chern-Schwartz-MacPherson class for DM stacks

TL;DR

This paper extends the Pro-Chern-Schwartz-MacPherson class concept from schemes to Deligne-Mumford stacks, linking it to Behrend functions and weighted Euler characteristics, and confirming a conjecture of Behrend.

Contribution

It generalizes the Pro-CSM class to DM stacks and establishes its equivalence to the CSM class for Behrend functions, proving Behrend's conjecture.

Findings

Pro-CSM class defined for DM stacks.

Equivalence of Pro-CSM and CSM classes for Behrend functions.

Integration yields weighted Euler characteristic.

Abstract

We generalize the definition of Pro-Chern-Schwartz-MacPherson (Pro-CSM) class of Aluffi for schemes to not necessarily proper DM stacks. The Pro-CSM class of constructible functions on a DM stack $\cX$ can be similarly defined. In the case that $\cX$ is proper, the Pro-CSM class of the Behrend function of $\cX$ is the same as the Chern-Schwartz-MacPherson (CSM) class for the Behrend function. The integration of this class over $\cX$ gives rise to the weighted Euler characteristic corresponding to the Behrend function, thus proving a conjecture of Behrend.