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