Kummer-type constructions of almost Ricci-flat 5-manifolds

A smooth closed manifold M is called almost Ricci-flat if infg||Ricg||∞·diamg(M)2=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \inf _g||\text {Ric}_g||_\infty \cdot \text {diam}_g(M)^2=0 \end{aligned}$$\end{document}where Ricg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {Ric}_g$$\end{document} and diamg\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text {diam}_g$$\end{document}, respectively, denote the Ricci tensor and the diameter of g and g runs over all Riemannian metrics on M. By using Kummer-type method, we construct a smooth closed almost Ricci-flat nonspin 5-manifold M which is simply connected. It is minimal volume vanishes; namely, it collapses with sectional curvature bounded.