Cohomogeneity one Kahler and Kahler-Einstein manifolds with one singular orbit II

摘要

Podesta and Spiro (Osaka J Math 36(4):805-833, 1999) introduced a class of G-manifolds M with a cohomogeneity one action of a compact semisimple Lie group G which admit an invariant Kahler structure (g, J) ("standard G-manifolds") and studied invariant Kahler and Kahler-Einstein metrics on M. In the first part of this paper, we gave a combinatoric description of the standard non-compact G-manifolds as the total space M-phi of the homogeneous vector bundle M = G x (H) V -> S-0 = G/ H over a flag manifold S-0 and we gave necessary and sufficient conditions for the existence of an invariant Kahler-Einstein metric g on such manifolds M in terms of the existence of an interval in the T-Weyl chamber of the flag manifold F = G x (H) PV which satisfies some linear condition. In this paper, we consider standard cohomogeneity one manifolds of a classical simply connected Lie group G = SUn, Sp(n).Spin(n) and reformulate these necessary and sufficient conditions in terms of easily checked arithmetic properties of the Koszul numbers associated with the flag manifold S-0 = G/H. If this condition is fulfilled, the explicit construction of the Kahler-Einstein metric reduces to the calculation of the inverse function to a given function of one variable.