Volume 16, issue 4 (2012)

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Virtual push-forwards

Cristina Manolache

Geometry & Topology 16 (2012) 2003–2036
Abstract

Let $p:\phantom{\rule{0.3em}{0ex}}F\to G$ 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.

Keywords
virtual classes, Gromov–Witten invariants
Primary: 14C17
Secondary: 14N35
Publication
Accepted: 6 May 2012
Published: 31 August 2012
Proposed: Richard Thomas
Seconded: Jim Bryan, Simon Donaldson
Authors
 Cristina Manolache Institut für Mathematik Humboldt-Universität zu Berlin Unter den Linden 6 10099 Berlin Germany