Smoothness of Kuranishi atlases on Gromov-Witten moduli spaces

TL;DR

This paper demonstrates that Gromov-Witten moduli spaces admit sufficiently smooth Kuranishi atlases, enabling the construction of a virtual fundamental class across all virtual dimensions, based on a new gluing theorem.

Contribution

It proves the existence of smooth enough Kuranishi atlases on Gromov-Witten moduli spaces, facilitating the definition of their virtual fundamental classes.

Findings

Gromov-Witten moduli spaces admit smooth Kuranishi atlases

A stronger gluing theorem is established

Virtual fundamental classes can be constructed in any virtual dimension

Abstract

Kuranishi atlases were introduced by McDuff and Wehrheim to build a virtual fundamental class on moduli spaces of J-holomorphic curves and resolve some of the challenges in this field. This paper considers Gromov-Witten moduli spaces and shows they admit a smooth enough Kuranishi atlas to be able to define a Gromov-Witten virtual fundamental class in any virtual dimension. The key step for this result is the proof of a stronger gluing theorem.