Research Article Open Access

A Note on the Classification of Compact Homogeneous Locally Conformal Kähler Manifolds

Daniel Guan1
  • 1 University of California at Riverside, United States

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 Kähler 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 Kähler class) on the base complex manifold and the metrics (same as the Kähler class) on the complex one dimensional torus.

Journal of Mathematics and Statistics
Volume 13 No. 3, 2017, 261-267

DOI: https://doi.org/10.3844/jmssp.2017.261.267

Submitted On: 28 April 2017 Published On: 2 August 2017

How to Cite: Guan, D. (2017). A Note on the Classification of Compact Homogeneous Locally Conformal Kähler Manifolds. Journal of Mathematics and Statistics, 13(3), 261-267. https://doi.org/10.3844/jmssp.2017.261.267

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Keywords

  • Cohomology
  • Invariant Structure
  • Homogeneous Space
  • Complex Torus Bundles
  • Hermitian Manifolds
  • Reductive Lie Group
  • Compact Manifolds
  • Ricci Form
  • Locally Conformal Kähler Manifolds
  • 1991 Mathematics Subject Classification. 53C15, 57S25, 53C30, 22E99, 15A75