Locally conformally Kaehler structures on homogeneous spaces

A homogeneous Hermitian manifold $M$ with its homogeneous Hermitian structure $h$, defining a locally conformally Kaehler structure $w$ is called a homogeneous locally conformally Kaehler or shortly a homogeneous l.c.K. manifold. If a simply connected homogeneous l.c.K. manifold $M=G/H$, where $G$ is a connected Lie group and $H$ a closed subgroup of $G$, admits a free action of a discrete subgroup $D$ of $G$ from the left, then a double coset space $D\setminus G/H$ is called a locally homogeneous l.c.K. manifold. We discuss explicitly homogeneous and locally homogeneous l.c.K. structures on Hopf surfaces and Inoue surfaces, and their deformations. We also classify all complex surfaces admitting locally homogeneous l.c.K. structures.
We show as a main result a structure theorem of compact homogeneous l.c.K. manifolds, asserting that it has a structure of a holomorphic principal fiber bundle over a flag manifold with fiber a 1-dimensional complex torus. As an application of the theorem, we see that only compact homogeneous l.c.K. manifolds of complex dimension 2 are Hopf surfaces of homogeneous type. We also see that there exist no compact complex homogeneous l.c.K. manifolds; in particular neither complex Lie groups nor complex paralellizable manifolds admit their compatible l.c.K. structures.
We show as a main result a structure theorem of compact homogeneous l.c.K. manifolds, asserting that it has a structure of a holomorphic principal fiber bundle over a flag manifold with fiber a 1-dimensional complex torus. As an application of the theorem, we see that only compact homogeneous l.c.K. manifolds of complex dimension 2 are Hopf surfaces of homogeneous type. We also see that there exist no compact complex homogeneous l.c.K. manifolds; in particular neither complex Lie groups nor complex paralellizable manifolds admit their compatible l.c.K. structures.
This talk is based on a joint work with Y. Kamishima “Locally conformally Kaehler structures on homogeneous spaces” (arXiv:1101.3693).