Kähler manifolds with large isometry group

Abstract

A Riemannian manifold is called of cohomogeneity one if it is acted on by a closed Lie group G of isometrics with principal orbits of codimension one. This class of manifolds have at least two good reasons for being considered particularly appealing: their degree of symmetry is so high that classification theorems of purely algebraic nature are still possible in several situations (see e.g. [5], [1], [2], [10]); at the same time, they allow to construct non homogeneous examples of Riemannian manifolds with special geometric properties, like Einstein metrics, exceptional holonomy (see e.g. [4]). We are interested in compact non homogeneous Kahler-Einstein manifolds and several of them have been already constructed by Koiso and Sakane in the cohomogeneity one category (see e.g. [15], [13]). The aims of the present paper consist in giving an explicit classification of compact cohomogeneity one Kahler manifolds with vanishing first Betti number and in using it for obtaining a complete list of the Kahler-Einstein manifolds in that family. It is well known (see e.g. [9]) that the vanishing of &ι(M) implies the existence of a G-equivariant moment mapping μ: M —> g* and this fact has an important consequence on the algebraic structure of G. In fact, we prove that (see Lemma 2.2) if G is semisimple and G/L = G(p) is a regular orbit on M, with Q and [ Lie algebras of G and L respectively, then the centralizer G0(l) is of the form