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

## TL;DR

This paper characterizes when certain cohomogeneity one Kaehler-Einstein manifolds with one singular orbit admit invariant metrics, providing explicit conditions and constructions based on arithmetic properties of associated Koszul numbers.

## Contribution

It reformulates existence conditions for invariant Kaehler-Einstein metrics on standard cohomogeneity one manifolds in terms of arithmetic properties of Koszul numbers, extending previous combinatorial descriptions.

## Key findings

- Necessary and sufficient conditions for invariant Kaehler-Einstein metrics are given in terms of Koszul numbers.
- Explicit construction of metrics reduces to inverting a specific function of one variable.
- Conditions are reformulated for classical Lie groups like SU_n, Sp_n, Spin_n.

## Abstract

F. Podest\`a 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 $M_{\varphi}$ 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 $G = SU_n, Sp_n. Spin_n$ 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.