A Classification of Compact Cohomogeneity One Locally Conformal Kähler Manifolds

Daniel Guan ()
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Daniel Guan: Department of Mathematics, School of Mathematics and Statistics, Henan University, Kaifeng 475004, China

Mathematics, 2024, vol. 12, issue 11, 1-12

Abstract: In this paper, we apply a result of the classification of a compact cohomogeneity one Riemannian manifold with a compact Lie group G to obtain a classification of compact cohomogeneity one locally conformal Kähler manifolds. In particular, we prove that the compact complex manifold is a complex one-dimensional torus bundle over a projective rational homogeneous, or cohomogeneity one manifold except of a class of manifolds with a generalized Hopf surface bundle over a projective rational homogeneous space. Additionally, it is a homogeneous compact complex manifold under the complexification G C of the given compact Lie group G under an extra condition that the related closed one form is cohomologous to zero on the generic G orbit. Moreover, the semi-simple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semi-simple Lie group S in that special case.

Keywords: cohomology; invariant structure; homogeneous space; cohomogeneity one; complex torus bundles; Hermitian manifolds; reductive Lie group; compact manifolds; locally conformal Kähler manifolds (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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