Symmetric Semi-perfect Obstruction Theory Revisited.

In this paper we survey some results on the symmetric semi-perfect obstruction theory on a Deligne—Mumford stack X constructed by Chang—Li, and Behrend's theorem equating the weighted Euler characteristic of X and the virtual count of X by symmetric semi-perfect obstruction theories. As an application, we prove that Joyce's d-critical scheme admits a symmetric semi-perfect obstruction theory, which can be applied to the virtual Euler characteristics by Jiang—Thomas. [ABSTRACT FROM AUTHOR]