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Kuranishi structures were introduced in the 1990s by Fukaya and Ono for thepurpose of assigning a virtual cycle to moduli spaces of pseudoholomorphiccurves that cannot be regularized by geometric methods. Their core idea was tobuild such a cycle by patching local finite dimensional reductions, given bysmooth sections that are equivariant under a finite isotropy group.  Building on our notions of topological Kuranishi atlases and perturbationconstructions in the case of trivial isotropy, we develop a theory of Kuranishiatlases and cobordisms that transparently resolves the challenges posed bynontrivial isotropy. We assign to a cobordism class of weak Kuranishi atlasesboth a virtual moduli cycle (VMC - a cobordism class of weighted branchedmanifolds) and a virtual fundamental class (VFC - a Cech homology class).

Smooth Kuranishi atlases with isotropy