For some exposition on Atiyah-Bott localization see this blog-style note.
Suppose you have a Deligne-Mumford stack \(M\) which is virtually smooth over a pure dimensional algebraic stack \(\mathcal Y\), that is, you have a morphism \(\pi:M\to \mathcal Y\) and a relative perfect obstruction theory \(\phi:\mathbb E\to L_{M/\mathcal Y}\) for it. This gives you a virtual fundamental class \([M]^{vir} = f^*[\mathcal Y]\). Suppose further that \(T\) is an algebraic torus acting on \(M\) and \(\mathcal Y\) so that \(\pi\) is \(T\)-equivariant. In this situation one would like to use localization to express \([M]^{vir}\) in terms of classes in the fixed locus \(M^T\). However, virtual localization only applies when you have an absolute perfect obstruction theory.
To understand the problem, first recall the content of a “localization formula”. The main localization theorem means that the pushforward of the inclusion \(j:M^T\to M\) is an isomorphism in localized equivariant Chow groups. In smooth situations, the result of pulling back a class \(\alpha \in A^T_*(M)_{Q_T}\) to \(A^T_*(M^T)_{Q_T}\) and then pushing forward to return to \(M\) looks something like
When you \(\mathcal Y = \Spec k\) for some base field, given a perfect obstruction theory \(\mathbb E\) for \(M\to \Spec k\) , the complex \(\mathbb E|_{M^T}^f\) forms a perfect obstruction theory for \(M^T/\Spec k\). The main localization theorem holds for DM stacks
- an example of relative localization for log stable maps