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

A smooth closed manifold $M$ is called almost Ricci-flat if $$\underset{g}{\inf }\mathrm{\mid }\mathrm{\mid }{\text{Ric}}_g\mathrm{\mid }{\mathrm{\mid }}_{\mathrm{\infty }}\cdot {\text{diam}}_g(M{)}^2=0$$ where ${\text{Ric}}_g$ and ${\text{diam}}_g$ denote the Ricci tensor and the diameter of $g$ respectively 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’s minimal volume vanishes, namely it collapses with sectional curvature bounded.