A Note on the Classification of Compact Homogeneous Locally Conformal K&amp;auml;hler Manifolds

Daniel Guan

doi:10.3844/jmssp.2017.261.267

Abstract

In this study, we apply a result of H. C. Wang and Hano-Kobayashi on the classification of compact complex homogeneous manifolds with a compact reductive Lie group to give some more homogeneous space involved proofs of recent classification of compact complex homogeneous locally conformal Kahler manifolds. In particular, we prove that the semisimple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semisimple Lie group S. We also prove that as an one dimensional complex torus bundle, the metrics on the manifold is completely determined by the metrics (which is the same as the Kahler class) on the base complex manifold and the metrics (same as the Kahler class) on the complex one dimensional torus.

Highlights

Let M be a complex manifold, h be an Hermitian metrics
If h is locally conformal to Kähler metrics, i.e., for any point m ∈ M there is an open neighborhood U such that on U g = efh is a Kähler metrics with a function f, we say that (M, h) is a locally conformal Kähler
A compact complex homogeneous space with an invariant Hermitian structure was classified by Wang (1954), see (Hano and Kobayashi, 1960)

Summary

Introduction

Let M be a complex manifold, h be an Hermitian metrics. If h is locally conformal to Kähler metrics, i.e., for any point m ∈ M there is an open neighborhood U such that on U g = efh is a Kähler metrics with a function f, we say that (M, h) is a locally conformal Kähler. We have (Hano and Kobayashi, 1960 Theorem B): Any compact Hermitian homogeneous manifold is a complex torus bundle over a rational ( connected) projective homogeneous space. A compact homogeneous locally conformal Kähler manifold M = G/H is a complex 1-dimensional torus bundle over a rational projective homogeneous space. We shall prove our Main Theorem of the classification of the locally conformal Kähler metrics

Results

Conclusion