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Affine compact almost-homogeneous manifolds of cohomogeneity one

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Guan D.

Open Mathematics

2009

tom 7

nr 1

84-123

This paper is one in a series generalizing our results in [12, 14, 15, 20] on the existence of extremal metrics to the general almost-homogeneous manifolds of cohomogeneity one. In this paper, we consider the affine cases with hypersurface ends. In particular, we study the existence of Kähler-Einstein metrics on these manifolds and obtain new Kähler-Einstein manifolds as well as Fano manifolds without Kähler-Einstein metrics. As a consequence of our study, we also give a solution to the problem posted by Ahiezer on the nonhomogeneity of compact almost-homogeneous manifolds of cohomogeneity one; this clarifies the classification of these manifolds as complex manifolds. We also consider Fano properties of the affine compact manifolds.

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1 Open Mathematics

1 Guan D.

1 2009

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Affine compact almost-homogeneous manifolds of cohomogeneity one