Smoothness on bubble tree compactified instanton moduli spaces

The bubble tree compactified instanton moduli space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \bar M_\kappa $$\end{document}(X) is introduced. Its singularity set Singκ(X) is described. By the standard gluing theory, one can show that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \bar M_\kappa $$\end{document}(X) — Singκ(X) is a topological orbifold. In this paper, we give an argument to construct smooth structures on it.