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 $$M_{\varphi }$$ 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.

中文翻译：

---

同质性一 Kähler 和 Kähler-Einstein 流形，具有一个奇异轨道 II

Podestà 和 Spiro (Osaka J Math 36(4):805–833, 1999) 介绍了一类 G 流形 M 具有同质性的一个紧致半单李群 G 的作用，它承认不变的 Kähler 结构 ( g , J ) ( “标准 G 流形”）并研究了 M 上的不变 Kähler 和 Kähler-Einstein 度量。在本文的第一部分，我们给出了标准非紧 G 流形的组合描述为齐次向量丛的总空间 $$M_{\varphi }$$ M φ $$M = G\times _HV \rightarrow S_0 =G/H $$ M = G × HV → S 0 = G / H 在标志流形 $$S_0$$ S 0 上，我们给出了在这种流形 M 上存在不变 Kähler-Einstein 度量 g 的充要条件标记流形的 T-Weyl 室中存在一个区间 $$F = G \times _H PV$$ F = G × HPV，满足一些线性条件。在本文中，我们考虑经典单连通李群$$G = SU_n, Sp_n 的标准同质流形。Spin_n$$ G = SU n ，S pn 。根据与标志流形 $$S_0 = G/H$$ S 0 = G / H 相关联的 Koszul 数的易于检查的算术性质，S pinn 并重新表述这些必要和充分条件。如果满足此条件，则 Kähler-Einstein 度量的显式构造简化为计算一个变量的给定函数的反函数。