Definition
Suppose \(f:X\to Y\) is a morphism of algebraic stacks. If
- \(f\) is a morphism of DM-type (i.e. \(X\times_Y U\) is Deligne-Mumford whenever \(U\to Y\) is a scheme mapping into \(Y\)) and
- There exists a vector bundle stack \(E\) over \(X\) and a closed embedding \(j:C_f\hookrightarrow E\) where \(C_f\) is the intrinsic normal cone of \(f:X\to Y\).
Then the virtual pullback is a morphism \(f^*:A_*(Y)\to A_*(X)\) defined as the compsite of three maps:
\begin{align*} f^*:A_*(Y)\xrightarrow{\sigma} A_*(C_f) \xrightarrow{j_*} A_*(E) \xrightarrow{0^!_E} A_*(X) \end{align*}where \(\sigma\) is the specialization map defined on the level of cocycles by \([V] \mapsto [C_{V\times_Y X/V}]\), \(j_*\) is (proper or projective) pushforward and \(0^!_E\) is the Gysin morphism associated to the zero section \(0_E:X\to E\).
Remark
We (as in I, Isaac) refer to \((j,E)\) as embedding data, and specifying this data is exactly equivalent to specifying a perfect obstruction theory for \(f\). It is important to note that different choices of embedding data yield different maps \(f^*\).
The virtual pullback is a vast generalization of the virtual fundamental class of Behrend-Fantechi, and if \(Y\) has a fundamental class, then
\begin{align*} f^*[Y] = [X]^{vir}, \end{align*}where \([X]^{vir}\) is defined relative to the perfect obstruction theory corresponding to \((j,E)\). Hence we can take the above formula as the definition of the virtual fundamental class.