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Einstein metrics on five-dimensional Seifert bundles. (English) Zbl 1082.53041

Let \(L\) be \(2n+ 1\)-dimensional manifold which admits a differentiable map \(f: L\to X\) to a complex \(n\)-manifold \(X\) such that every fiber is a circle. Such \(L\) is called a higher-dimensional Seifert fibred manifold. For \(n= 1\) there are Seifert fibred 3-manifolds. The author classified all such 5-dimensional manifolds which admit a positive Seifert bundle structure and in a few cases all Seifert bundle structures are classified. These results are then used to construct positive Ricci curvature Einstein metrics on this manifolds.
The methods are not elementary. For example the Leray spectral sequence of the Seifert bundle is used. The construction of Kähler-Einstein metrics on Dell Pezzo orbifolds (it corresponds to \(X\)) uses the algebraic existence criterion of Demailly-Kollar. Then the lifting of the Kähler-Einstein metric on the base of a Seifert bundle to an Einstein metric on the total space uses the Kobayashi-Boyer-Galicki method. There is a special part related to Seifert bundles with a rational homology of the sphere and trivial first homology group over integers. Finally, we have to mention that the paper is very technical.

MSC:

53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
57R22 Topology of vector bundles and fiber bundles
55R20 Spectral sequences and homology of fiber spaces in algebraic topology
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
55R55 Fiberings with singularities in algebraic topology
57R20 Characteristic classes and numbers in differential topology
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