Positive scalar curvature with point singularities

We show that in every dimension n 8, there exists a smooth closed manifold Mⁿ which does not admit a smooth positive scalar curvature ("psc") metric, but M admits an L^-metric which is smooth and has psc outside a singular set of codimension 8. This provides counterexamples to a conjecture of Schoen. In fact, there are such examples of arbitrarily high dimension with only single point singularities. In addition, we provide examples of L^-metrics on Rⁿ for certain n 8 which are smooth and have psc outside the origin, but cannot be smoothly approximated away from the origin by everywhere smooth Riemannian metrics of non-negative scalar curvature. This stands in precise contrast to established smoothing results via Ricci--DeTurck flow for singular metrics with stronger regularity assumptions. Finally, as a positive result, we describe a KO-theoretic condition which obstructs the existence of L^-metrics that are smooth and of psc outside a finite subset. This shows that closed enlargeable spin manifolds do not carry such metrics.