Virtual push-forwards

Abstract

Let $p:F\to G$ be a morphism of DM stacks of positive virtual relative dimension $k$ and let $\gamma \in A^k\left(F\right)$. We give sufficient conditions for $p_{\ast }\left(\gamma \cdot {\left[F\right]}^{virt}\right)$ to be a multiple of ${\left[G\right]}^{virt}$. We show an analogue of the conservation of number for virtually smooth families. We show implications to Gromov–Witten invariants and give a new proof of a theorem of Marian, Oprea and Pandharipande which compares the virtual classes of moduli spaces of stable maps and moduli spaces of stable quotients.