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

A smooth closed manifold M is called almost Ricci-flat if $$\begin{aligned} \inf _g||\text {Ric}_g||_\infty \cdot \text {diam}_g(M)^2=0 \end{aligned}$$ where $$\text {Ric}_g$$ and {diam}_g$$ , respectively, denote the Ricci tensor diameter of g runs over all Riemannian metrics on M. By using Kummer-type method, we construct a nonspin 5-manifold which simply connected. It minimal volume vanishes; namely, it collapses with sectional curvature bounded.