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Quaternionic Kähler 8-manifolds with positive scalar curvature. (English) Zbl 0733.53035
All known examples of quaternionic Kähler manifolds with positive scalar curvature are exhausted by symmetric Wolf spaces that may be identified with the coset space \(G/N(G_{\alpha})\) of a simple Lie group G over the normalizer \(N(G_{\alpha})_{\alpha}\) of the three- dimensional subgroup \(G_{\alpha}\) associated with highest root \(\alpha\) of G. The authors prove that in dimension 8 there are no other manifolds of this type: Theorem. A complete connected quaternionic Kähler 8- manifold with positive scalar curvature is isometric to the quaternionic projective plane \({\mathbb{H}}{\mathbb{P}}^ 2\), the complex Grassmannian \({\mathbb{G}}r_ 2({\mathbb{C}}^ 4)\) or the exceptional space \(G_ 2/SO_ 4\). The proof is complicated and diverse arguments and results from algebraic and differential geometries, topology, Lie group theory are used. The starting point is the identification of the twistor space of M with a Fano 5-fold of coindex 3 and the establishing that the isometry group I of M has dimension not less than 6.

53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C35 Differential geometry of symmetric spaces
14J45 Fano varieties
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