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\(\hat A\)-genus on non-spin manifolds with \(S^ 1\) actions and the classification of positive quaternion-Kähler 12-manifolds. (English) Zbl 1071.53027

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

MSC:

53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
58J26 Elliptic genera
53C35 Differential geometry of symmetric spaces
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