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Podestà 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 Kähler structure (g, J) (“standard G-manifolds”) and studied invariant Kähler and Kähler–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 $$M_{\varphi }$$ of the homogeneous vector bundle $$M = G\times _H V \rightarrow S_0 =G/H$$ over a flag manifold $$S_0$$ and we gave necessary and sufficient conditions for the existence of an invariant Kähler–Einstein metric g on such manifolds M in terms of the existence of an interval in the T-Weyl chamber of the flag manifold $$F = G \times _H PV$$ which satisfies some linear condition. In this paper, we consider standard cohomogeneity one manifolds of a classical simply connected Lie group $$G = SU_n, Sp_n. Spin_n$$ 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 Kähler–Einstein metric reduces to the calculation of the inverse function to a given function of one variable.

Cohomogeneity one Kähler and Kähler–Einstein manifolds with one singular orbit II