In this paper we give the detailed construction of a -equivariant Kuranishi chart of moduli spaces of pseudo-holomorphic curves to a symplectic manifold with -action, for an arbitrary compact Lie group . The proof is based on the deformation theory of unstable marked curves using the language of Lie groupoids (which is not necessarily étale) and the Riemannian center of mass technique. This proof is actually similar to Fukaya and Ono (Arnold conjecture and Gromov–Witten invariant, Topology 38 (1999), 933–1048, Sects. 13 and 15), except that the usage of the language of Lie groupoids makes the argument more transparent.
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Kenji Fukaya, Lie Groupoids, Deformation of Unstable Curves, and Construction of Equivariant Kuranishi Charts. Publ. Res. Inst. Math. Sci. 57 (2021), no. 3/4, pp. 1109–1225DOI 10.4171/PRIMS/57-3-13