Smooth Kuranishi atlases with isotropy

McDuff, Dusa¹ ; Wehrheim, Katrin²

¹ Mathematics Department, Barnard College, Columbia University, MC4410, 2990 Broadway, New York, NY 10027-6840, United States
² Department of Mathematics, University of California, Berkeley, 907 Evans Hall #3840, Berkeley, CA 94705-3840, United States

Geometry & topology, Tome 21 (2017) no. 5, pp. 2725-2809.

Résumé

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).

Détail

DOI : 10.2140/gt.2017.21.2725

Classification : 53D35, 53D45, 54B15, 57R17, 57R95
Keywords: virtual fundamental cycle, virtual fundamental class, pseudoholomorphic curve, Kuranishi structure, Gromov–Witten invariant, transversality, weighted branched manifold

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