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 C∗ action on X that makes the obstruction theory C∗-equivariant. The C∗-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 C∗-action.