Here is proposition 5 as given in Tom Graber’s notes. Need to write the proofs.
There is an induced \(T\)-action on \(\widetilde{M}\), \red{and \(j\) induces a local isomorphism of \(M^T\) with the \(T\)-fixed locus of \(\widetilde{M}\). This follows from the fact that they have the same deformation theory by the previous proposition.} The map \(j\) is also proper (since \(i\) is proper and \(\eta\) is separated) implying that \(M^T\) is a union of connected components of \((\widetilde{M})^T\).
Proposition
Virtual pullback commutes with taking equivariant residues; explicitly, we have an equality of two maps \(A^T_*(\mathcal Y) \to A^T_*(\widetilde M)\),
\begin{align*}\tilde \pi^!_{\mathbb E|_{\widetilde M}} \circ \Res_{\mathcal Y^T}^{C_{\mathcal Y^T/\mathcal Y}} \circ \sigma_{\mathcal Y} = \Res^{C_{\widetilde M/M}}_{\widetilde M} \circ \sigma_{M} \circ \pi^!_{\mathbb E}. \end{align*}In particular, if \(\mathcal Y\) has a fundamental class,
\begin{align*} \tilde \pi^!_{\mathbb E}([\mathcal Y^T]) = \Res^{C_{\widetilde M/M}}_{\widetilde M}(\sigma_M(\pi^!_{\mathbb E}([\mathcal Y]))) \end{align*}in \(W^{-1}A_*(\widetilde M)\).
The restriction map is defined whenever we have a moving \(T\)-cone \(C\to Z\) over \(Z\subset X^T\). If \(j:C\to E\) embeds \(C\) into \(E\) a vector bundle over \(Z\) and \(s:Z\to E\) is the zero section, we write \(\Res^C_Z:A^T_*(C)\to A^T_*(Z)\) (technically we need localized Chow) defined by \[\Res^C_Z(\alpha) = e(E)^{-1}\cap s^*(j_*(\alpha)).\] Notice that this close to a virtual pullback of Manolache. Denote by \(f:Z\to X\) the embedding of \(Z\) into \(X\) (or the map \(f:X^T\to X\) restricted to \(Z\), which may fail to be an embedding) then if \(\sigma:A^T_*(X)\to A^T_*(C)\) is the specialization of \(X\) to \(C\), then \[\Res^C_Z\circ \sigma (\alpha) = e(E)^{-1}\cap f^!(\alpha).\] This means that the equality we wish to prove can instead be written in terms of virtual pullbacks, i.e. we notice that in both equalities we have a \(\Res\) next to a specialization \(\sigma\) and hence basically have a virtual pullback. However it might be the case that \(\eta\) and \(\overline i\) do not satisfy Condition 3.3 in Manolache, in which case the virtual pullback isn’t defined. We can instead just expand \(\Res\) and move the pullbacks \(\pi^!\) through the composition.
\begin{align*} \tilde \pi^!_{\mathbb E|_{\widetilde M}} \circ \Res_{\mathcal Y^T}^{C_{\mathcal Y^T/\mathcal Y}} \circ ~\sigma_{\mathcal Y} &= \tilde \pi^!_{\mathbb E|_{\widetilde M}} \circ \big(e(\mathbb E|_{\widetilde M})^{-1}\cap (s^*\circ j_*\circ \sigma_{\mathcal Y})\big)\\ &= \Res^{C_{\widetilde M/M}}_{\widetilde M} \circ ~\sigma_{M} \circ \pi^!_{\mathbb E} \end{align*}I think we may as well argue using the virtual pullback idea. We need to confirm that the virtual pullback exists, which means checking that \(\iota:\mathcal Y^T\to \mathcal Y\) is of DM-type (relatively Deligne-Mumford) and that we can find a relative perfect obstruction theory \(E^\bullet_{\mathcal Y^T/\mathcal Y}\).