Smoothness on bubble tree compactified instanton moduli spaces

The bubble tree compactified instanton moduli space $ bar M_kappa $ bar M_kappa (X) is introduced. Its singularity set Sing _κ (X) is described. By the standard gluing theory, one can show that $ bar M_kappa $ bar M_kappa (X) — Sing _κ (X) is a topological orbifold. In this paper, we give an argument to construct smooth structures on it.