Notes on Kuranishi Atlases

TL;DR

This paper clarifies the construction of Kuranishi atlases for Gromov--Witten moduli spaces, providing explicit tools and modifications to handle isotropy, and demonstrates their applicability to finite dimensional orbifolds.

Contribution

It reformulates existing ideas on Kuranishi atlases to clarify formal structures and explicitly address choices, extending the construction to cases with isotropy and finite dimensional orbifolds.

Findings

Constructed a Kuranishi atlas for genus zero Gromov--Witten moduli spaces.

Provided explicit methods to handle isotropy in Kuranishi structures.

Showed that every finite dimensional orbifold admits a Kuranishi atlas.

Abstract

These notes aim to explain a joint project with Katrin Wehrheim that uses finite dimensional reductions to construct a virtual fundamental class for the Gromov--Witten moduli space of closed genus zero curves. Our method is based on work by Fukaya and Ono as well as more recent work by Fukaya, Oh, Ohta, and Ono. We reformulated their ideas in order to clarify the formal structures underlying the construction and make explicit all important choices (of tamings, shrinkings and reductions), thus creating tools with which to give an explicit proof that the virtual fundamental class is independent of these choices. After summarizing the main ideas and proofs in the arXiv preprint 1208.1340, these notes explain the modifications needed to deal with isotropy. Further sections outline the construction of a Kuranishi atlas in the genus zero case, and give some examples of their use. We also show that every finite dimensional orbifold has a Kuranishi atlas.