Cohomogeneity one Kaehler and Kaehler-Einstein manifolds with one singular orbit, II

Abstract

F. Podestà and A. Spiro introduced a class of $G$-manifolds $M$ with a cohomogeneity one action of a compact semisimple Lie group $G$ which admit an invariant Kaehler structure $(g,J)$ (``standard $G$-manifolds") and studied invariant Kaehler and Kaehler-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 $M=G{\times }_{H}V\to S_0=G/H$ over a flag manifold $S_0$ and we gave necessary and sufficient conditions for the existence of an invariant Kaehler-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 $G=S{U}_n,S{p}_n.Spi{n}_n$ and reformulate these necessary and sufficient conditions in terms of easily checked arithmetic properties of the Koszul numbers associated with the flag manifold $S_0=G/H$. If this conditions is fulfilled, the explicit construction of the Kaehler-Einstein metric reduces to the calculation of the inverse function to a given function of one variable.

27 pages

References in corpus (2)

• Spin structures on compact homogeneous pseudo-Riemannian manifolds
• Calabi-Yau metrics on canonical bundles of flag varieties