The fundamental class of smooth Kuranishi atlases with trivial isotropy

Abstract

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. The first sections of this paper discuss topological, algebraic and analytic challenges that arise in this program.We then develop a theory of Kuranishi atlases and cobordisms that transparently resolves these challenges, for simplicity concentrating on the case of trivial isotropy. In this case, we assign to a cobordism class of additive weak Kuranishi atlases both a virtual moduli cycle (VMC — a cobordism class of smooth manifolds) and a virtual fundamental class (VFC — a Cech homology class). We, moreover, show that such Kuranishi atlases exist on simple Gromov–Witten moduli spaces and develop the technical results in a manner that easily transfers to more general settings.