Lie groupoid, deformation of unstable curve, and construction of equivariant Kuranishi charts

TL;DR

This paper constructs G-equivariant Kuranishi charts for moduli spaces of pseudo-holomorphic curves in symplectic manifolds with group actions, utilizing Lie groupoid deformation theory and Riemannian techniques.

Contribution

It introduces a detailed, transparent construction of G-equivariant Kuranishi charts using Lie groupoids for unstable curves, correcting previous errors.

Findings

Constructed G-equivariant Kuranishi charts for moduli spaces

Applied Lie groupoid deformation theory to unstable curves

Enhanced clarity and corrected errors in previous methods

Abstract

In this paper we give detailed construction of $G$-equivariant Kuranishi chart of moduli spaces of pseudo-holomorphic curves to a symplectic manifold with $G$-action, for an arbitrary compact Lie group $G$. The proof is based on the deformation theory of {\it unstable} marked curves using the language of Lie groupoid (which is {\it not} necessary etale) and the Riemannnian center of mass technique. This proof is actually similar to [FOn,Sections 13 and 15] except the usage of the language of Lie groupoid makes the argument more transparent. This version correct some errors of the previous version especially those pointed out by the referee.