We give examples of two inequivalent smooth structures on the complex projective 9-space such that one admits a metric of nonnegative scalar curvature and the other does not. Following this example and the work of Thomas Farrell and Lowell Jones, we also construct examples of closed negatively curved Riemannian 18-manifolds, which are homeomorphic but not diffeomorphic to complex hyperbolic manifolds. (Joint work with Samik Basu.)

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Last updated: 28 May 2024