The Michigan Mathematical Journal
- Michigan Math. J.
- Volume 68, Issue 2 (2019), 227-250.
Note on MacPherson’s Local Euler Obstruction
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 admitting a symmetric obstruction theory. Furthermore, we assume that there is a action on that makes the obstruction theory -equivariant. The -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 is the same as the Kiem–Li localized invariant of by the -action.
Michigan Math. J., Volume 68, Issue 2 (2019), 227-250.
Received: 11 April 2017
Revised: 3 July 2017
First available in Project Euclid: 30 January 2019
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 14A20: Generalizations (algebraic spaces, stacks)
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