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# Pure and Applied Mathematics Quarterly

## Volume 11 (2015)

### Number 1

### The Pro-Chern-Schwartz-Macpherson class for DM stacks

Pages: 87 – 114

DOI: http://dx.doi.org/10.4310/PAMQ.2015.v11.n1.a4

#### Author

#### 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 $\mathcal{X}$ can be similarly defined. In the case that $\mathcal{X}$ is proper, the Pro-CSM class of the Behrend function of $\mathcal{X}$ is the same as the Chern-Schwartz-MacPherson (CSM) class for the Behrend function. The integration of this class over $\mathcal{X}$ gives rise to the weighted Euler characteristic corresponding to the Behrend function, thus proving a conjecture of Behrend.

#### Keywords

Pro-CSM class, Chern-Mather class, Behrend function, weighted Euler characteristic, Lagrangian intersection

#### 2010 Mathematics Subject Classification

Primary 14N35. Secondary 14A20.