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A finiteness theorem for quaternionic-Kähler manifolds with positive scalar curvature. (English) Zbl 0806.53065
Eastwood, Michael (ed.) et al., The Penrose transform and analytic cohomology in representation theory. AMS-IMS-SIAM summer research conference, June 27 - July 3, 1992, South Hadley, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 154, 89-101 (1993).
The class of all complete quaternionic-Kähler (QK) \(4n\)-manifolds of positive scalar curvature is studied in this paper. In the case of dimension 8 it was proved by Poon and Salamon that each of these manifolds is symmetric, see Y. S. Poon and S. M. Salamon [J. Differ. Geom. 33, No. 2, 363-378 (1991; Zbl 0733.53035)]. It was conjectured by the above mentioned authors that this situation persists in higher dimensions. In the paper under review two important theorems in this direction are proved. The first one states that for any \(n\) there are only finitely many compact \(QK\) \(4n\)-manifolds of positive scalar curvature. In the second (called strong rigidity theorem) the groups \(\pi_ 1(M)\) and \(\pi_ 2(M)\) are calculated for every \(M \in (QK): \pi_ 1(M) = 0\) for each \(M \in (QK)\); \(\pi_ 2(\mathbb{H} \mathbb{P}_ n) = 0\), \(\pi_ 2(\text{Gr}_ 2 \mathbb{C}^{n+2}) = \mathbb{Z}\), and for other \(M \in (QK)\) of \(\dim 4n\) \(\pi_ 2(M)\) is finite containing the group \(\mathbb{Z}_ 2\). The question of symmetricity remains open for \(4n > 8\).
For the entire collection see [Zbl 0780.00026].

MSC:
53C55 Global differential geometry of Hermitian and Kählerian manifolds
14J45 Fano varieties
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