The fundamental class of smooth Kuranishi atlases with trivial isotropy

TL;DR

This paper develops a clear and rigorous theory of Kuranishi atlases with trivial isotropy, enabling the construction of virtual cycles and classes for moduli spaces of pseudoholomorphic curves, addressing key mathematical challenges.

Contribution

It introduces a transparent framework for Kuranishi atlases and cobordisms, providing tools to construct virtual moduli cycles and fundamental classes, applicable to Gromov-Witten moduli spaces.

Findings

Constructed virtual moduli cycles as cobordism classes of smooth manifolds.

Defined virtual fundamental classes in Cech homology.

Proved existence of Kuranishi atlases on simple Gromov-Witten moduli spaces.

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.