Smooth Kuranishi atlases with isotropy

TL;DR

This paper develops a comprehensive theory of Kuranishi atlases that effectively handles nontrivial isotropy, enabling the construction of virtual cycles and fundamental classes for moduli spaces of pseudoholomorphic curves.

Contribution

It extends existing Kuranishi theory by incorporating nontrivial isotropy, providing a transparent framework for virtual cycles and cobordisms.

Findings

Constructed virtual moduli cycles as cobordism classes of weighted branched manifolds.

Defined virtual fundamental classes in Cech homology for moduli spaces.

Resolved challenges of nontrivial isotropy in Kuranishi structures.

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, 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 (VMC - a cobordism class of weighted branched manifolds) and a virtual fundamental class (VFC - a Cech homology class).