Erratum to “Â-genus on non-spin manifolds with $S^1$ actions and the classification of positive quaternion-Kähler 12-manifolds”

Abstract

We prove that the Â-genus vanishes on certain non-spin manifolds. Namely, Â(M) vanishes on any oriented, compact, connected, smooth manifold M with finite second homotopy group and endowed with non-trivial (isometric) smooth S1 actions. This result extends that of Atiyah and Hirzebruch on spin manifolds endowed with smooth S1 [1] to manifolds which are not necessarily spin. We prove such vanishing by means of the elliptic genus defined by Ochanine [23, 24], showing that it also has the special property of being rigid under S1 actions on these (not necessarily spin) manifolds. We conclude with a non-trivial application of this new vanishing theorem by classifying the positive quaternion-Kahler 12-manifolds. Namely, we prove that every quaternion-Kahler 12-manifold with a complete metric of positive scalar curvature must be a symmetric space.