Note on MacPherson's local Euler obstruction

TL;DR

This paper clarifies the definition of MacPherson's local Euler obstruction, proves its equivalence to Behrend's algebraic version, and relates it to Kiem-Li's localized invariants in the context of schemes or stacks with symmetric obstruction theories and $ ext{C}^*$ actions.

Contribution

It introduces MacPherson's original local Euler obstruction, proves its equivalence to Behrend's algebraic definition, and connects it to Kiem-Li's localized invariants under $ ext{C}^*$-actions.

Findings

Proved the equivalence of MacPherson's and Behrend's definitions.

Derived a formula for the local Euler obstruction via Lagrangian intersections.

Showed that Behrend's weighted Euler characteristic equals Kiem-Li's localized invariant under certain conditions.

Abstract

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 $\CC^*$ action on $X$, which makes the obstruction theory $\CC^*$-equivariant. The $\CC^*$-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 $\CC^*$-action.