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Smooth Kuranishi atlases with isotropy

Dusa McDuff and Katrin Wehrheim

Geometry & Topology 21 (2017) 2725–2809

Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic curves that cannot be regularized by geometric methods. Their core idea was to build such a cycle by patching local finite-dimensional reductions, given by smooth sections that are equivariant under a finite isotropy group.

Building on our notions of topological Kuranishi atlases and perturbation constructions in the case of trivial isotropy, we develop a theory of Kuranishi atlases and cobordisms that transparently resolves the challenges posed by nontrivial isotropy. We assign to a cobordism class of weak Kuranishi atlases both a virtual moduli cycle (a cobordism class of weighted branched manifolds) and a virtual fundamental class (a Čech homology class).

virtual fundamental cycle, virtual fundamental class, pseudoholomorphic curve, Kuranishi structure, Gromov–Witten invariant, transversality, weighted branched manifold
Mathematical Subject Classification 2010
Primary: 53D35, 53D45, 54B15, 57R17, 57R95
Received: 28 August 2015
Revised: 8 September 2016
Accepted: 8 October 2016
Published: 15 August 2017
Proposed: Leonid Polterovich
Seconded: Yasha Eliashberg, Peter Teichner
Dusa McDuff
Mathematics Department
Barnard College, Columbia University
2990 Broadway
New York, NY 10027-6840
United States
Katrin Wehrheim
Department of Mathematics
University of California, Berkeley
907 Evans Hall #3840
Berkeley, CA 94705-3840
United States