Cohomogeneity one Kahler and Kahler-Einstein manifolds with one singular orbit, I

Abstract

Let $M$ be a cohomogeneity one manifold of a compact semisimple Lie group $G$ with one singular orbit $S_0 = G/H$. Then $M$ is $G$- diffeomorphic to the total space $G \times_H V$ of the homogeneous vector bundle over $S_0$ defined by a sphere transitive representation of $G$ in a vector space $V$. We describe all such manifolds $M$ which admit an invariant Kahler structure of standard type. This means that the restriction $μ: S = Gx = G/L \rightarrow F = G/K$ of the moment map of $M$ to a regular orbit $S = G/L$ is a holomorphic map of $S$ with the induced CR structure onto a flag manifold $F = G/K$, where $K = N_G(L)$, endowed with an invariant complex structure $J^F$ . We describe all such standard Kahler cohomogeneity one manifolds in terms of the painted Dynkin diagram associated with $(F=G/K; J^F)$ and a parametrized interval in some T-Weyl chamber. We determine which of these manifolds admit invariant Kahler-Einstein metrics.