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Smooth Kuranishi atlases
with isotropy
Dusa McDuff and Katrin Wehrheim |
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Geometry & Topology 21 (2017) 2725–2809
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Abstract |
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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). |
Keywords
virtual fundamental cycle, virtual fundamental class,
pseudoholomorphic curve, Kuranishi structure, Gromov–Witten
invariant, transversality, weighted branched manifold
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Mathematical Subject Classification 2010
Primary: 53D35, 53D45, 54B15, 57R17, 57R95
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References |
Publication
Received: 28 August 2015
Revised: 8 September 2016
Accepted: 8 October 2016
Published: 15 August 2017
Proposed: Leonid Polterovich
Seconded: Yasha Eliashberg, Peter Teichner
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Authors |
| Dusa McDuff | |
| Mathematics Department Barnard College, Columbia University MC4410 2990 Broadway New York, NY 10027-6840 United States |
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| Katrin Wehrheim | |
| Department of Mathematics University of California, Berkeley 907 Evans Hall #3840 Berkeley, CA 94705-3840 United States |
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