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

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## Abstract

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.

## Keywords

Kähler–Einstein metrics Cohomogeneity one manifolds Homogeneous vector bundles Flag manifolds Dynkin diagrams## Notes

### Acknowledgements

The authors are grateful to the anonymous referee for her/his very careful comments which helped to improve the exposition of the paper.

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