Kahler manifolds with large isometry group

TL;DR

This paper classifies certain compact Kahler manifolds with large symmetry groups, providing a correspondence with abstract models and characterizing Kahler-Einstein manifolds with specific symmetry properties.

Contribution

It establishes a one-to-one correspondence between these manifolds and abstract models, and fully characterizes Kahler-Einstein manifolds with cohomogeneity one actions.

Findings

Derived the Ricci tensor for these manifolds.

Characterized Kahler-Einstein manifolds with symmetry.

Provided a classification framework for such manifolds.

Abstract

We investigate compact Kahler manifolds, which are acted on by a semisimple compact Lie group G of isometries with one hypersurface orbit. In case of ordinary action and projectable complex structure, we set up a one to one correspondence between such manifolds and abstract models. The Ricci tensor is then computed and we fully characterize the Kahler-Einstein manifolds with an ordinary cohomogeneity one action and projectable complex structure.