Constructing the virtual fundamental class of a Kuranishi atlas

McDuff, Dusa

doi:10.2140/agt.2019.19.151

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Constructing the virtual fundamental class of a Kuranishi atlas

Dusa McDuff

Algebraic & Geometric Topology 19 (2019) 151–238

DOI: 10.2140/agt.2019.19.151

Abstract

Consider a space $X$ , such as a compact space of $J$ –holomorphic stable maps, that is the zero set of a Kuranishi atlas. This note explains how to define the virtual fundamental class of $X$ by representing $X$ via the zero set of a map $S_{M} : M \to E$ , where $E$ is a finite-dimensional vector space and the domain $M$ is an oriented, weighted branched topological manifold. Moreover, $S_{M}$ is equivariant under the action of the global isotropy group $Γ$ on $M$ and $E$ . This tuple $(M, E, Γ, S_{M})$ together with a homeomorphism from $S_{M}^{- 1} (0) ∕ Γ$ to $X$ forms a single finite-dimensional model (or chart) for $X$ . The construction assumes only that the atlas satisfies a topological version of the index condition that can be obtained from a standard, rather than a smooth, gluing theorem. However, if $X$ is presented as the zero set of an $sc$ –Fredholm operator on a strong polyfold bundle, we outline a much more direct construction of the branched manifold $M$ that uses an $sc$ –smooth partition of unity.

Keywords

virtual fundamental cycle, virtual fundamental class, pseudoholomorphic curve, Kuranishi structure, weighted branched manifold, polyfold

Mathematical Subject Classification 2010

Primary: 18B30, 53D35, 53D45, 57R17, 57R95

References

Publication

Received: 16 October 2017

Revised: 5 September 2018

Accepted: 15 September 2018

Published: 6 February 2019

Authors

Dusa McDuff
Mathematics Department Barnard College Columbia University New York, NY United States

Volume 19, issue 1 (2019)