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Compact 3-Sasakian 7-manifolds with arbitrary second Betti number. (English) Zbl 0901.53033

The authors give an affirmative answer, by explicit construction, to the following questions. (1) Do there exist 3-Sasakian manifolds with second Betti number greater than one? (2) Are there singular Fano varieties with arbitrary large Picard number? (3) Are there compact Einstein manifolds of positive scalar curvature with arbitrarily large total Betti number? (4) Are there compact Einstein manifolds of positive scalar curvature which do not admit metrics of nonnegative sectional curvature?

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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