Annals of Global Analysis and Geometry | 2019
Cohomogeneity one Kähler and Kähler–Einstein manifolds with one singular orbit II
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 ,\xa0 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 $$M_{\\varphi }$$ M φ of the homogeneous vector bundle $$M = G\\times _H V \\rightarrow S_0 =G/H$$ M = G × H V → S 0 = G / H over a flag manifold $$S_0$$ 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 $$F = G \\times _H PV$$ F = G × H P V which satisfies some linear condition. In this paper, we consider standard cohomogeneity one manifolds of a classical simply connected Lie group $$G = SU_n, Sp_n. Spin_n$$ G = S U n , S p n . S p i n n and reformulate these necessary and sufficient conditions in terms of easily checked arithmetic properties of the Koszul numbers associated with the flag manifold $$S_0 = G/H$$ 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.