Einstein metrics on five-dimensional Seifert bundles

The aim of this article is to study Seifert bundle structures on simply connected 5-manifolds. We classify all such 5-manifolds which admit a positive Seifert bundle structure, and in a few cases all Seifert bundle structures are also classified. These results are then used to construct positive Ricci curvature Einstein metrics on these manifolds. The proof has 4 main steps. First, the study of the Leray spectral sequence of the Seifert bundle, based on work of Orlik-Wagreich. Second, the study of log Del Pezzo surfaces. Third, the construction of Kähler-Einstein metrics on Del Pezzo orbifolds using the algebraic existence criterion of Demailly-Kollár. Fourth, the lifting of the Kähler-Einstein metric on the base of a Seifert bundle to an Einstein metric on the total space using the Kobayashi-Boyer-Galicki method.