## The Michigan Mathematical Journal

- Michigan Math. J.
- Volume 68, Issue 2 (2019), 227-250.

### Note on MacPherson’s Local Euler Obstruction

#### Abstract

This is a note on MacPherson’s local Euler obstruction, which plays an important role recently in the Donaldson–Thomas theory by the work of Behrend.

We introduce MacPherson’s original definition and prove that it is equivalent to the algebraic definition used by Behrend, following the method of González-Sprinberg. We also give a formula of the local Euler obstruction in terms of Lagrangian intersections. As an application, we consider a scheme or DM stack $X$ admitting a symmetric obstruction theory. Furthermore, we assume that there is a ${\mathbb{C}}^{\ast}$ action on $X$ that makes the obstruction theory ${\mathbb{C}}^{\ast}$-equivariant. The ${\mathbb{C}}^{\ast}$-action on the obstruction theory naturally gives rise to a cosection map in the Kiem–Li sense. We prove that Behrend’s weighted Euler characteristic of $X$ is the same as the Kiem–Li localized invariant of $X$ by the ${\mathbb{C}}^{\ast}$-action.

#### Article information

**Source**

Michigan Math. J., Volume 68, Issue 2 (2019), 227-250.

**Dates**

Received: 11 April 2017

Revised: 3 July 2017

First available in Project Euclid: 30 January 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.mmj/1548817530

**Digital Object Identifier**

doi:10.1307/mmj/1548817530

**Mathematical Reviews number (MathSciNet)**

MR3961215

**Zentralblatt MATH identifier**

07084761

**Subjects**

Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]

Secondary: 14A20: Generalizations (algebraic spaces, stacks)

#### Citation

Jiang, Yunfeng. Note on MacPherson’s Local Euler Obstruction. Michigan Math. J. 68 (2019), no. 2, 227--250. doi:10.1307/mmj/1548817530. https://projecteuclid.org/euclid.mmj/1548817530