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

TL;DR

This paper classifies cohomogeneity one Kähler and Kähler-Einstein manifolds with a single singular orbit, describing their structure via painted Dynkin diagrams and moment maps, and identifies those admitting Einstein metrics.

Contribution

It provides a complete classification of invariant Kähler structures on cohomogeneity one manifolds with one singular orbit, using Dynkin diagrams and moment map analysis.

Findings

Classification of all such manifolds using painted Dynkin diagrams.

Identification of conditions for existence of invariant Kähler-Einstein metrics.

Explicit description of the moment map and CR structures on regular orbits.

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 $\mu: S = Gx = G/L \rightarrow F = G/K$ 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.