Virtual push-forwards

TL;DR

This paper establishes conditions under which push-forwards of virtual classes behave predictably, with applications to Gromov-Witten invariants and a new proof relating different moduli spaces.

Contribution

It provides a new criterion for virtual push-forwards and applies it to conservation of number, Gromov-Witten invariants, and moduli space comparisons.

Findings

Derived sufficient conditions for virtual push-forwards to be multiples of virtual classes.

Proved an analogue of conservation of number for virtually smooth families.

Provided a new proof relating virtual classes of stable maps and stable quotients.

Abstract

Let $p:F\to G$ be a morphism of stacks of positive \emph{virtual} relative dimension $k$ and let $\gamma\in H^k(F)$. We give sufficient conditions for $p_*\gamma\cdot[F]^{virt}$ to be a multiple of $[G]^{virt}$. We apply this result to 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.