Notes on Kuranishi atlases

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.