2014/12/11 by Jiang, Yunfeng
#Algebraic Geometry (math.AG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1412.3720
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.