Manolache — Virtual Pull-Backs (2011)

We propose a generalization of Gysin maps for DM-type morphisms of stacks \(F\to G\) that admit a perfect relative obstruction theory \(E_{ F/G}\textasciicircum{\bullet}\), which we call a “virtual pull-back”. We prove functoriality properties of virtual pull-backs. As applications, we analyze Gromov-Witten invariants of blow-ups and projective bundles.