Note on MacPherson's local Euler obstruction

This is a note on MacPherson’s local Euler obstruction, which plays an important role recently in 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 Gonzalez-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 ${\text{CC}}^{\ast }$ action on $X$, which makes the obstruction theory ${\text{CC}}^{\ast }$-equivariant. The ${\text{CC}}^{\ast }$-action on the obstruction theory naturally gives rise to a cosection map in the sense of Kiem-Li. We prove that Behrend’s weighted Euler characteristic of $X$ is the same as Kiem-Li localized invariant of $X$ by the ${\text{CC}}^{\ast }$-action.