
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 be a morphism of stacks of positive \emph{virtual} relative dimension and let . We give sufficient conditions for to be a multiple of . 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.
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